seismic analysis of masonry infilled rc frames

Transcript

FACOLTÀ DI INGEGNERIA
CORSO DI LAUREA MAGISTRALE IN
INGEGNERIA CIVILE PER LA PROTEZIONE DAI RISCHI NATURALI
Tesi di Laurea
SEISMIC ANALYSIS OF MASONRY INFILLED R.C. FRAMES
Relatori:
Prof. Gianmarco de Felice
Prof. Paulo B. Lourenço
Correlatore:
Ing. Alberto Mauro
Anno Accademico 2008 – 2009
Candidato:
Francesco Ferraguti
Facoltà di Ingegneria
L.M. in Ingegneria Civile per la Protezione dai Rischi Naturali
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Ai miei genitori,
esempio e presenza costante nella mia vita.
Tutto quello che di positivo è in me
proviene essenzialmente da voi:
sappiate che, qualunque sia il destino a me riservato,
vi porterò sempre nel cuore.
A.A. 2008-09
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RINGRAZIAMENTI
Con questa pagina, l’ultima scritta in ordine cronologico, si conclude non solo una tesi
frutto di mesi e mesi di lavoro, ma anche un percorso universitario iniziato ormai da più
di sei anni e più in generale un pluriennale ciclo di formazione che vede in questo lavoro
l’ultimo gradino prima del confronto con il mondo del lavoro. È quindi doverosa una
pagina per ringraziare chiunque, in diversi modi ed in diversi tempi, ha accompagnato il
mio cammino fino a questo istante.
Vorrei allora iniziare col ringraziare il mio relatore, il prof. de Felice, per avermi dato
modo di sviluppare una tesi stimolante fuori dai canoni classici delle tesi magistrali,
grazie soprattutto all’opportunità di svolgerne una cospicua parte in un altro ateneo, ed il
mio correlatore, l’Ing. Alberto Mauro, amico prima ancora che correlatore, la cui
pazienza e il cui pc sono stati messi a dura prova specie in questi ultimi tempi.
Non posso poi fare a meno di ringraziare il prof. Paulo Lourenço, che mi ha accolto a
braccia aperte all’Universidade do Minho di Guimarães supervisionando costantemente il
mio lavoro, e tutti i suoi ragazzi del Departamento de Estruturas che mi hanno in qualche
modo aiutato: certamente Pedro Medeiros con i suoi consigli sempre puntuali e Nuno
Mendez con i suoi spunti, ma anche João Leite e Tommaso Scappaticci.
C’è poi chi ha condiviso con me tutto il percorso accademico, o comunque buona parte di
esso: grazie a voi, grazie al nostro vivere quotidianamente l’ambiente universitario
insieme, sicuramente è risultato tutto più di facile; con molti di voi il rapporto è andato
oltre il semplice essere colleghi, e l’amicizia che ne è nata spero possa durare anche in là
con il tempo. Nominare tutti è impossibile, mi limito allora a ringraziare a nome di tutti
quanti Marco, compagno inossidabile di esami e progetti, Stefano, autentica bandiera
della nostra università, Alessandro, senza la cui allegria sarebbe stato tutto molto più
triste, e poi Jacopo lo zingaro, Daniele, Giorgio, Tamburrino, Gabriele, Alessio, Oreste,
Leo, e ancora Vito, Simone, Gigi, Francesco e tutti gli altri che non ho dimenticato ma
per cui servirebbe un’altra mezza pagina…
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Ci sono poi persone che hanno condiviso con me due periodi particolari ed
indimenticabili della mia vita all’interno di questi ultimi anni: sto parlando delle mie due
esperienze di studio all’estero, prima a Santander e poi a Guimarães. In misura diversa,
sono state esperienze indelebili entrambe, più incosciente e spensierata la prima, più
matura e formativa la seconda. Non posso allora fare a meno di citare almeno le persone
più importanti, senza comunque dimenticare le altre: un ringraziamento particolare va
allora a Mary, Giuseppe, Orazio, Simone, Ludo, Giulia, Simona, Riccardo, Elia, Pedro,
Doris, Gerry, Nino e Alex, compagni del semestre erasmus a Santander, il più incredibile
che abbia mai vissuto, ed un altro ringraziamento va anche a Jaime, Lucas, Michela,
Maria Agnese, Francesca, Tommaso, Vincenzo, Fon, Julia, Zé e Guillerme, per la
compagnia delle serate passate insieme.
Gli amici di sempre, grazie di esserci ancora nonostante gli anni che passano e le strade
diverse che ognuno di noi sta percorrendo: so che su di voi posso contare ancora adesso, e
non chiedo di più. Siete pochi, ma buoni, ed allora la citazione personale è d’obbligo:
Gabriele, fratello né maggiore né minore, semplicemente un fratello, magari di poche
parole come me, ma sempre pronto a capirmi anche solo con uno sguardo; Marco, ora che
questo cammino sta finendo, sono convinto che l’amicizia che ci lega continuerà a vivere
anche al di là di queste aule; Antonio, per te le parole sono superflue e non renderanno
mai l’idea di quello che sei…; Damiano, Bernardo, Creatura, spero che le nostre strade si
possano intrecciare di nuovo, perché ogni volta che è capitato non è mai successo nulla di
noioso e banale; Valentina, grazie del sostegno che mi hai fornito nei momenti tristi e
dell’allegria che hai portato in quelli più felici.
La mia famiglia, i miei genitori, la mia sorellina, i nonni, gli zii e tutti i parenti: vi
ringrazio per aver avuto sempre piena fiducia in me, lasciandomi sbagliare quando era
necessario e sgridandomi quando invece era il momento; se sono qui è anche grazie a voi.
Infine vorrei concludere ringraziando chiunque abbia portato, anche per un solo istante, il
sorriso sulle mie labbra: non importa se poi a quel sorriso sono seguite lacrime e tristezza,
grazie comunque perché ogni istante di felicità vale sempre la pena di essere vissuto.
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SOMMARIO
Uno dei principali campi di ricerca degli ultimi decenni nell’ingegneria civile è quello
inerente l’influenza delle tamponature nella risposta sismica delle strutture in c.a.
Analizzando l’ampia letteratura scientifica in materia (frutto di più pi 40 anni di ricerche),
possiamo affermare che l’effetto dei pannelli murari è lontano dall’essere trascurabile, sia
per quanto riguarda gli effetti positivi sulla struttura (incremento di rigidezza, incremento
di resistenza ai carichi laterali, migliore comportamento anelastico) sia per quanto
riguarda quelli negativi (incremento della domanda sismica, possibili meccanismi di
collasso per taglio dei pilastri); d’altra parte, a causa dell’elevato numero dei parametri
che entrano in gioco in questa tematica, a causa della loro alta variabilità ed anche a causa
dell’influenza della manodopera umana, non è stata raggiunta una convergenza da parte
degli esperti su un unico modello da adottare per svolgere le analisi.
Usualmente le strutture in c.a. sono progettate facendo riferimento semplicemente al
telaio non tamponato, e utilizzando prevalentemente metodi lineari elastici: ma ora che
l’uso degli Eurocodici nella progettazione strutturale sta prendendo sempre più piede, è
importante sviluppare modelli che riescano a tener conto anche dell’incidenza delle
tamponature (come richiesto dall’Eurocodice 8) senza aumentare di troppo la complessità
del problema; allo stesso tempo è richiesta la prevenzione di collassi fragili e
dell’instaurarsi di pericolosi meccanismi fuori dal piano. Così, una corretta progettazione
che tenga conto degli elementi non strutturali potrebbe portare anche a una riduzione dei
costi di ricostruzione post-sisma (Lourenço P. et al. [2009]).
Esattamente in questa direzione saranno condotti dei test su tavola vibrante al L.N.E.C.
(National Laboratory of Civil Engineering) di Lisbona: saranno testati tre edifici in c.a.
tamponati in muratura, uno con muratura non rinforzata e due con diversi tipi di rinforzi.
Il presente lavoro si focalizza sull’implementazione (grazie al supporto del software
DIANA) di modelli a bielle della struttura con muratura non rinforzata e sui risultati
ottenuti da analisi non lineari statiche (pushover) e dinamiche (time-history). L’obiettivo
della tesi è infatti la descrizione dei modelli adottati per realizzare le analisi con il codice
agli elementi finiti DIANA ed il commento dei risultati ottenuti dalle suddette analisi.
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Come normativa di riferimento sono stati considerati gli Eurocodici attualmente in vigore
(EN 1990, EN 1991, EN 1992, EN 1996, EN 1998), supportati dove necessario dalla
normative portoghesi per le strutture in c.a. (R.E.B.A.P., 1983), per l’acciaio (R.E.A.E.,
1983) e per le azioni (R.S.A., 1983), e da altri codici internazionali (Model Code 1990).
A causa della regolarità in pianta dell’edificio, è possibile usare due modelli piani (uno
per ogni direzione principale) invece di un modello spaziale [EN 1998-1:2003 – 4.3.3.1];
travi e pilastri sono stati modellati con elementi beam, mentre le tamponature sono state
sostituite da bielle diagonali. All’inizio, le analisi sono state svolte su modelli in cui ogni
tamponatura è sostituita da una biella diagonale (“modelli a singola biella”, in accordo
con quanto proposto da Fardis M.N. [1996] e da Safina S. [2002]); successivamente sono
stati adottati modelli più accurati, nei quali ogni tamponatura è sostituita da uno schema
di tre bielle diagonali (“modelli a tre bielle”, seguendo gli studi svolti da Bergami A.V.
[2008]), così da poter tener conto degli effetti negativi derivanti dell’interazione tra i
pannelli e i pilastri, come ad esempio meccanismi di collasso fragili causati dal contributo
di taglio trasferito dalla tamponatura al pilastro.
Il primo tipo di analisi condotta (sia per modelli a singola che a tripla biella) è l’analisi
modale, il cui scopo è la comprensione del comportamento modale dell’edificio nelle due
direzioni principali.
Quindi sono state realizzate, sempre su entrambe i tipi di modelli, analisi statiche non
lineari (pushover), corredate da analisi di sensibilità che hanno l’obiettivo di calibrare
alcuni parametri del modello così da poter scegliere per le successive analisi il migliore in
termini di accuratezza e onere computazionale.
Infine, il comportamento dinamico della struttura è stato studiato grazie ad analisi
dinamiche non lineari (time-history) sui soli modelli a tre bielle: come input sismico sono
stati impiegati accelerogrammi sia artificiali che naturali (non scalati e scalati). Le curve
ottenute sono poi confrontate con quelle delle analisi di pushover, così come sono stati
confrontati il massimo spostamento con il target displacement, ed i relativi drift
interpiano. Per analizzare l’influenza nei risultati di altri parametri di primaria
importanza, sono state condotte in conclusione altre analisi di sensibilità.
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Come accennato, lo scopo principale di questo lavoro è la realizzazione di modelli che
possano essere rappresentativi della struttura che verrà testata, così da poter prevedere i
risultati dei test su tavola vibrante; allo stesso tempo, poiché l’edificio da testare è
rappresentativo dell’edilizia tipica portoghese (ma in una certa misura anche italiana), i
modelli realizzati potrebbero anche essere visti come un agevole strumento per la
progettazione strutturale nel rispetto degli Eurocodici.
Per raggiungere questo obiettivo, sono stati creati diversi tipi di modelli e sono state
realizzate diverse tipologie di analisi, così da poter individuare un modello che sia
sufficientemente accurato senza però aumentare eccessivamente la complessità delle
analisi: in questa direzione, il poter realizzare due modelli piani per rappresentare
l’edificio può considerarsi un buon inizio. Poi, è fondamentale valutare quanti e quali
siano i benefici dati dall’adozione di modelli a tre bielle (rispetto a quelli a singola biella):
nel dettaglio per esempio, bisogna valutare se il contributo del taglio portato dalle
tamponature risulta determinante in una eventuale rottura fragile dei pilastri, e più in
generale, se la differenza nei risultati è tale da giustificare l’uso di questi modelli.
Condurre analisi non lineari sia statiche che dinamiche è utile per comprendere alcuni
aspetti importanti e per studiare le eventuali differenze nei risultati conseguiti tramiti i
due tipi di analisi: riguardo alla determinazione del target displacement sulla curva di
capacità ottenuta dalle analisi di pushover, è sicuramente interessante confrontarlo con il
massimo spostamento ottenuto dalle analisi time-history; è inoltre possibile evidenziare
divergenze o convergenze nei risultati in termini di drift interpiano; infine, per analizzare
l’affidabilità delle analisi dinamiche non lineari, può essere rilevante confrontare i
risultati ottenuti a partire da accelerogrammi artificiali, con quelli ottenuti a partire da
accelerogrammi naturali non scalati e da accelerogrammi naturali scalati. Le analisi di
sensibilità sono state realizzate con lo scopo di tarare caratteristiche meccaniche dei
materiali non esplicitamente indicate dalle normative, e per accertare l’influenza di altri
parametri che regolano l’accuratezza dei risultati così da trovare un buon compromesso
tra precisione e affidabilità del modello da una parte, e semplicità e oneri computazionali
dall’altra.
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L’ultimo obiettivo della tesi è la verifica di alcuni dei requisiti di sicurezza richiesti
dall’Eurocodice, in modo tale da poter capire se l’uso di tamponature rinforzate (come
quelle degli altri due edifici da testare) si rende indispensabile per una adeguata
progettazione strutturale.
La tesi è divisa in sette capitoli e otto allegati.
Il capitolo 1 è un’introduzione generale ai temi che verranno trattati: sono presentati gli
aspetti principali, lo scopo del lavoro e un breve sommario.
Il capitolo 2 presenta gli edifici da testare su tavola vibrante, la loro progettazione
strutturale e l’attrezzatura sperimentale.
Il capitolo 3 tratta le caratteristiche dei modelli usati per le analisi: dopo una breve
descrizione del codice di calcolo DIANA, sono spiegati i modelli a biella usati nelle
analisi, le caratteristiche degli elementi adottati in fase di modellazione, le proprietà dei
materiali impiegati, i carichi verticali ed orizzontali applicati alla struttura, le condizioni
di vincolo imposte al modello ed il processo di meshing di cui ci si è serviti.
Il capitolo 4 illustra le analisi modali, studiando il comportamento modale dell’edificio
attraverso l’analisi dei modi di vibrare, dei periodi propri, delle masse partecipanti, ecc.
Il capitolo 5 espone i risultati delle analisi di pushover, espressi in termini di curve di
capacità, target displacement, drift interpiano e sollecitazioni sugli elementi strutturali,
effettuando confronti tra il modello a singola biella e quello a tre bielle; sono presentati
anche i risultati delle analisi di sensibilità a cui si accennava in precedenza.
Il capitolo 6 mostra i risultati delle analisi dinamiche non lineari, espressi ora in termini di
curve time-history, spostamento massimo e drift interpiano: particolare attenzione è
dedicata al confronto tra i risultati ottenuti a partire dagli accelerogrammi artificiali e
quelli ottenuti partire dagli accelerogrammi naturali, eseguendo poi un paragone più
ampio tra i risultati delle analisi di pushover e quelli delle analisi time-history.
Il capitolo 7 infine presenta le conclusioni della tesi e le considerazioni finali a cui si è
giunti, fornendo eventuali spunti per successive ricerche.
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ABSTRACT
One of the main field of investigation in civil engineering in last decades is the one
concerning the infills’ influence on the seismic response of r.c. (reinforced concrete)
structures. Analyzing the vast scientific literature on this topic (proceedings of more than
40 years of researches), we can asses that the effect of infill panels is far to be negligible,
both for their positive effects (higher stiffness, higher resistance to horizontal loads and a
better inelastic behaviour of the structure) and for their negative ones (increasing of the
seismic demand, possible shear failure mechanisms in the columns); on the other hand,
because of the elevate number of the parameters presents in that theme, because of their
high variability and also because of workmanship’s influence, no convergence on a
unique model to adopt in the analysis has been reached by the experts.
Usually r.c. structures are designed referring just to the bare frame, and mostly using
linear elastic methods: but now that the use of Eurocodes in structural design is
increasing, is important to develop models that manage to take into account the infills’
effects (like Eurocode 8 requires) without arise too much the problem’s complexity; as
well safety assessments of the walls and the prevention of a brittle failure or their out-ofplane collapse are required. So, a correct design taking into account non-structural
elements could lead also to lower post-earthquake reconstruction and repair costs
(Lourenço P. et al. [2009]).
Exactly in this way, experimental tests on shaking table will be carry out at L.N.E.C.
(National Laboratory of Civil Engineering) in Lisbon: three scaled r.c. buildings, one with
unreinforced masonry infills and two with different types of reinforcements will be tested.
The present work focus on the implementation (with the support of the software DIANA)
of strut models of the structure with unreinforced masonry infills and on the results of
nonlinear static (pushover) and dynamic (time-history) analysis.
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TABLE OF CONTENTS
RINGRAZIAMENTI
5
SOMMARIO
7
ABSTRACT
11
LIST OF FIGURES
16
LIST OF TABLES
19
1
21
INTRODUCTION
1.1 AIM OF THE WORK
22
1.2 THESIS LAYOUT
23
2
PRESENTATION OF THE EXPERIMENTAL TEST
24
2.1 GEOMETRY OF THE STRUCTURE
24
2.2 STRUCTURAL DESIGN
27
2.3 TEST EQUIPMENT
28
3
FEATURES OF THE MODELS
29
3.1 DIANA – FINITE ELEMENT CODE
29
3.2 STRUT MODELS
32
3.3 GEOMETRY OF THE MODELS
37
3.3.1 STRUCTURAL ELEMENTS
39
3.3.2 CROSS SECTIONS
44
3.3.3 INTEGRATION SCHEME
45
3.4 MATERIALS’ PROPERTIES
48
3.4.1 CONCRETE
48
3.4.2 MASONRY
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3.4.3 STEEL
52
3.5 LOADS
53
3.5.1 VERTICAL LOADS
54
3.5.2 HORIZONTAL LOADS: STATIC LOADS
56
3.5.3 HORIZONTAL LOADS: DYNAMIC LOADS
57
3.6 BOUNDARY CONDITIONS
67
3.7 MESHING
68
4
EIGENVALUE ANALYSIS
69
4.1 ANALYSIS PROCEDURE
69
4.2 ANALYSIS RESULTS
72
5
PUSHOVER ANALYSIS
78
78
82
5.2.1 CAPACITY CURVE
82
5.2.2 TARGET DISPLACEMENT
87
5.2.3 INTERSTOREY DRIFT
100
5.2.4 SOLICITATIONS
101
5.3 SENSITIVE ANALYSIS
102
5.4 SAFETY ASSESSMENTS
107
5.4.1 LIMITATION OF INTERSTOREY DRIFT
107
5.4.2 SHEAR RESISTANCE
108
6
TIME-HISTORY ANALYSIS
116
117
6.2 DAMPING EFFECTS
119
122
6.3.1 TIME-HISTORY CURVE
122
6.3.2 MAXIMUM DISPLACEMENT
123
6.3.3 INTERSTOREY DRIFT
125
6.4 SAFETY ASSESSMENTS
126
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6.4.1 LIMITATION OF INTERSTOREY DRIFT
126
6.5 SENSITIVE ANALYSIS
127
7
SUMMARY AND CONCLUSIONS
132
ANNEXES
135
REFERENCES
187
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LIST OF FIGURES
Figure 2-1. Geometry of the structure’s prototype (in metres) ..................................................... 24
Figure 2-2. Geometry of the structure’s scaled model (in metres) ............................................... 26
Figure 2-3. Geometry of the infill’s panels (in centimetres) ......................................................... 26
Figure 2-4. Detail of the infill in the column’s zone (in centimetres) ........................................... 26
Figure 2-5. Scheme of the shaking table ....................................................................................... 28
Figure 3-1. DIANA program architecture .................................................................................... 31
Figure 3-2. Global models (a) and local models (b) for infilled structures.................................. 33
Figure 3-3. Possible failure mechanism due to infill’s collapse................................................... 34
Figure 3-4. Load-displacement curves in relation at the different infill failure mechanism ........ 34
Figure 3-5. Idealized parallel systems for strut models................................................................ 35
Figure 3-6. Scheme of the single strut model................................................................................ 36
Figure 3-7. Equivalence between the single strut model and the triple strut model..................... 37
Figure 3-8. Geometry of single strut models: model 1 (left) and model 2 (right) ........................ 38
Figure 3-9. Geometry of triple strut models: model 1 (left) and model 2 (right) ......................... 38
Figure 3-10. Geometry of reinforcement bars: model 1 (left) and model 2 (right) ...................... 39
Figure 3-11. Displacements in 2D beams: class-I and class-II (left); class-III (right) ................ 40
Figure 3-12. Strains in two-dimensional beams ........................................................................... 40
Figure 3-13. Moments, forces and stresses in two-dimensional beams ........................................ 40
Figure 3-14. Displacements, strain, stresses and forces in truss elements................................... 42
Figure 3-15. Geometry of an embedded reinforcement in beam elements ................................... 44
Figure 3-16. Integration schemes along the elements axis ........................................................... 46
Figure 3-17. Integration schemes on the cross-section of the elements ....................................... 47
Figure 3-18. Concrete behaviour in tension and compression ..................................................... 50
Figure 3-19. Constant shear retention for total strain fixed models ............................................ 50
Figure 3-20. Masonry behaviour in compression for the two models .......................................... 52
Figure 3-21. Steel behaviour ........................................................................................................ 53
Figure 3-22. Distribution of vertical load in plan ........................................................................ 54
Figure 3-23. Horizontal load patterns for pushover analysis....................................................... 56
Figure 3-24. Seismic zones in continental Portugal ..................................................................... 58
Figure 3-25. Horizontal elastic response spectra ......................................................................... 60
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Figure 3-26. One of the artificial accelerograms generated: spectral acceleration..................... 61
Figure 3-27. ESD records database .............................................................................................. 63
Figure 3-28. Interface of the software REXEL .............................................................................. 65
Figure 3-29. Spectral waves of the recorded unscaled accelerograms: 1ST combination ............. 65
Figure 3-30. Spectral waves of the recorded scaled accelerograms: 1ST combination ................. 66
Figure 3-31. Mesh of the triple strut model 2................................................................................ 68
Figure 4-1. Mode shapes of the single strut model 1: mode 1 (left) and mode 2 (right) ............... 73
Figure 4-2. Mode shapes of the single strut model 2: mode 1 (left) and mode 2 (right) ............... 74
Figure 4-3. Mode shapes of the triple strut model 1: mode 1 (left) and mode 3 (right)................ 75
Figure 4-4. Mode shapes of the triple strut model 2: mode 1 (left) and mode 2 (right)................ 76
Figure 5-1. Arc-length control ...................................................................................................... 80
Figure 5-2. Regular Newton-Raphson iteration ............................................................................ 81
Figure 5-3. Convergence criterion ................................................................................................ 81
Figure 5-4. Capacity curve: single strut model 1 .......................................................................... 82
Figure 5-5. Capacity curve: single strut model 2 .......................................................................... 82
Figure 5-6. Capacity curve and its components: single strut model 1 – uniform load pattern ..... 83
Figure 5-7. Capacity curve and its components: single strut model 1 – modal load pattern ....... 84
Figure 5-8. Capacity curve and its components: single strut model 2 – uniform load pattern ..... 84
Figure 5-9. Capacity curve and its components: single strut model 2 – modal load pattern ....... 84
Figure 5-10. Capacity curve: single and triple strut model 1 – uniform load pattern .................. 85
Figure 5-11. Capacity curve: single and triple strut model 1 – modal load pattern..................... 85
Figure 5-12. Capacity curve: single and triple strut model 2 – uniform load pattern .................. 86
Figure 5-13. Capacity curve: single and triple strut model 2 – modal load pattern..................... 86
Figure 5-14. N2 method: bi-linearization of the S.D.O.F. capacity curve .................................... 88
Figure 5-15. Hysteretic behaviour of the equivalent S.D.O.F. system .......................................... 88
Figure 5-16. Idealization of the capacity curve in the N2 extended method ................................. 89
Figure 5-17. Elastic and inelastic spectra versus capacity diagram ............................................ 91
Figure 5-18. Idealization of the capacity curve: model 1 (left) and model 2 (right)..................... 92
Figure 5-19. Individuation of the target displacement in the N2 extended method ...................... 93
Figure 5-20. Choose of the displacement of first attempt ............................................................. 94
Figure 5-21. Bilinear representation of the capacity spectrum corresponding to dC,i .................. 95
Figure 5-22. Equivalent viscous damping associated to energy hysteretic dissipation ................ 95
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Figure 5-23. κ factor diagram ...................................................................................................... 96
Figure 5-24. Determination of the target displacement................................................................ 97
Figure 5-25. Individuation of the target displacement in the CSM: model 1, type 1 spectrum ..... 99
Figure 5-26. T.D. def. shape: type 1 spectrum - triple strut model 1 (left) and 2 (right). CSM . 100
Figure 5-27. T.D. def. shape: type 2 spectrum - triple strut model 1 (left) and 2 (right). CSM . 101
Figure 5-28. T.D.: bending moment MZ (left); NX and NY force resultant (right). CSM............ 101
Figure 5-29. T.D.: NX force (left); NY force (right). CSM ......................................................... 102
Figure 5-30. Sensitive analysis results: capacity curves ............................................................ 104
Figure 5-31. Influence of the infill’s compressive strength: capacity curves ............................. 106
Figure 5-32. Mesh nodes of the models ...................................................................................... 108
Figure 5-33. Truss model proposed by the EC 2 to represent the shear resistant mechanism... 109
Figure 5-34. Degradation of concrete shear strength with ductility .......................................... 110
Figure 5-35. Column shear strength due to axial force: reverse (a) and single bending (b) ..... 111
Figure 5-36. Shear verification: model 1 – uniform load patter ................................................ 112
Figure 5-37. Shear verification: model 1 – modal load patter ................................................... 112
Figure 5-38. Shear verification: model 2 – uniform load patter ................................................ 113
Figure 5-39. Shear verification: model 2 – modal load patter ................................................... 114
Figure 5-40. Components of shear design values ....................................................................... 115
Figure 5-41. Components of Priestley shear strength ................................................................ 115
Figure 6-1. Mass-proportional damping (left); stiffness-proportional damping (right) ............ 120
Figure 6-2. Variation of modal damping ratios with natural frequencies .................................. 121
Figure 6-3. Example of a time-history curve .............................................................................. 122
Figure 6-4. Comparison between the time-history and the capacity curve ................................ 123
Figure 6-5. Influence on results of concrete tensile strength...................................................... 128
Figure 6-6. Influence on results of viscous damping .................................................................. 128
Figure 6-7. Influence on results of seismic intensity level .......................................................... 130
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LIST OF TABLES
Table 2-1. Scale factors of the parameters in according to the Cauchy’s similarity law ............. 25
Table 3-1. Overview of beam elements.......................................................................................... 41
Table 3-2. Overview of truss elements........................................................................................... 43
Table 3-3. Permanent and imposed vertical loads: summary ....................................................... 54
Table 3-4. Permanent and imposed vertical loads applied to the models ..................................... 55
Table 3-5. Reference peak ground acceleration ............................................................................ 59
Table 3-6. Importance factor ......................................................................................................... 59
Table 3-7. Parameters of horizontal elastic response spectra: zone 1.3 (left) and 2.3 (right)...... 59
Table 3-8. Informations about the recorded unscaled accelerograms: 1ST combination .............. 66
Table 3-9. Informations about the recorded scaled accelerograms: 1ST combination .................. 66
Table 4-1. Eigenvalue analysis: single strut model 1 results ........................................................ 72
Table 4-2. Eigenvalue analysis: single strut model 2 results ........................................................ 73
Table 4-3. Eigenvalue analysis: triple strut model 1 results ......................................................... 74
Table 4-4. Eigenvalue analysis: triple strut model 2 results ......................................................... 75
Table 5-1. Capacity curves: maximum force values...................................................................... 86
Table 5-2. Parameters defining the idealized pushover curve in the N2 extended method ........... 91
Table 5-3. Parameters of R-µ-T relation in the N2 extended method ........................................... 92
Table 5-4. Model 1 type 1 spectrum: determination of the T.D. with CSM................................... 97
Table 5-8. T.D.: N2 extended method vs. CSM ........................................................................... 100
Table 5-9. T.D.: interstorey drift values: triple strut model, uniform load pattern. CSM ........... 100
Table 5-10. Sensitive analysis results: capacity curves peak values ........................................... 105
Table 5-11. Influence of the infill’s compressive strength: capacity curves peak values ............ 106
Table 5-12. Interstorey drift safety assessment ........................................................................... 107
Table 6-1. Maximum displacements: artificial accelerograms ................................................... 123
Table 6-2. Maximum displacements: recorded unscaled accelerograms ................................... 124
Table 6-3. Maximum displacements: recorded scaled accelerograms ....................................... 124
Table 6-4. Target displacement (pushover) vs. Maximum displacement (time-history) ............. 125
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Table 6-5. Maximum interstorey drifts of time-history analysis ................................................. 126
Table 6-6. Maximum interstorey drifts of pushover analysis...................................................... 126
Table 6-7. Interstorey drift safety assessment ............................................................................. 127
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1 INTRODUCTION
In this thesis the models adopted for the analysis carried out with the finite element code
DIANA are described and the results are also commented.
The references regulations adopted are essentially the European Standards (EN 1990, EN
1991, EN 1992, EN 1996, EN 1998), supported by portuguese codes for concrete
structures (R.E.B.A.P., 1983), for steel (R.E.A.E., 1983) and for the actions (R.S.A.,
1983), and by other international codes (Model Code 1990).
Because of the regularity in plan of the building it was possible to use two planar models,
one for each main direction [EN 1998-1:2003 – 4.3.3.1]; beam and column have been
modelled with beam element, while the infills have been substituted by diagonal struts. At
the beginning the analysis have been carried out on models with the infills replaced by
one diagonal strut (“single strut model”, in according with the models proposed by Fardis
M.N. [1996] and by Safina S. [2002]); then has been used a more advanced model, where
the infills were replaced by three diagonal struts (“triple strut model”, in according with
the model proposed by Bergami A.V. [2008]) and is so possible to take into account also
the negative effects of the interaction between the panels and the frames, like brittle
failure of the columns due to shear contribute that the walls pass to them.
Firstly, both of single and triple strut models, eigenvalue analysis have been realized with
the purpose to value the building modal behaviour in the two main directions.
Then, static nonlinear (pushover) analysis on both types of models have been carried out
and some sensitive analysis have been carried out too, with the goal to calibrate some
parameters of the model and to chose be best model to be used in further analysis.
Finally, dynamic behaviour of the structure has been investigate with dynamic nonlinear
(time-history) analysis, just on triple strut models: artificial and recorded accelerograms
(unscaled and scaled) have been used for seismic input. The time-history resulting curves
have been compared with the capacity curves, the maximum displacement compared with
the target displacement, and the interstorey drifts have been compared too. Some sensitive
analysis have been also done to check other parameters of primary importance.
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1.1
AIM OF THE WORK
The main purpose of this thesis is to create a model that could be representative of the
structure that will be tested so to forecast the results of the shaking table tests; at the same
time, because of that building is representative of the typical portuguese (but also italian)
constructions, the model created could also be considered a suitable tool for the structural
design with respect to the European Standards.
To reach this objective different types of model have been set and different types of
analysis have been run, how previously stated, to find a model that could be as reliable as
possible without being too much complex: in this direction, the possibility to use two
planar model because of the plan regularity of the structure is a first good step. Then it’s
of main importance to check how much are the benefits of the triple strut model: namely
if the shear contribution that is taken into account is determinant for a shear failure of the
columns and if the global results are so different to justify the use this model.
To carry out both static and dynamic nonlinear analysis have been useful to understand
some remarkable aspects and to analyze the differences: with regarding to the
determination of the target displacement on the capacity curve (proceeding of pushover
analysis), could be interesting compare it with maximum displacement got by timehistory curve; it’s also possible to evidence divergences or convergences on results of
interstorey drift calculated in the two analysis; moreover, to study the reliability of the
time-history analysis, different kind of accelerograms’ set have been used: artificial,
recorded unscaled, recorded scaled. Both in static and dynamic nonlinear analysis some
sensitive analysis have been carried out with the aim to calibrate parameters that are no
explicitly mentioned in the codes and to check the importance of other parameters that
control the accuracy of the results, so to get a good compromise between models’
precision and reliability on one side, and models’ simplicity and computational time on
the other side.
The last goal of this work is to verify the safety assessments required by the Eurocodes,
and so understand if the use of reinforced panels, like the ones of the other two buildings
to be tested, are actually essential to a good structural design of the structures.
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1.2
THESIS LAYOUT
This thesis is divided into seven chapters and eight annexes.
Chapter 1 is a general introduction to the themes that will be dealt: the main topic is
described, the purposes of the work are set and a brief summary of the present job is
presented.
Chapter 2 presents the scaled building to be tested on the shaking table, its structural
design and the experimental equipment.
Chapter 3 discusses the features of the models used for pushover and time-history
analysis: after a short description of the finite element code DIANA, the strut models
used in the analysis, the elements properties adopted in the modelling phase, the materials
properties assumed, the vertical and horizontal loads applied to the structures, the
boundary condition imposed in the models and the meshing process are explained.
Chapter 4 deals with eigenvalue analysis: the modal properties are studied to analyze the
shape modes of the structure.
Chapters 5 argues about the results of the pushover analysis expressed in terms of
capacity curves, target displacement, interstorey drift and solicitations on the structural
elements, with comparisons between single and triple strut models; sensitive analysis
carried out to calibrate some parameters of the models are presented so to choose the
features of the models to be used in the furthers analysis.
Chapter 6 talks about the dynamic nonlinear analysis: the results are expressed now in
terms of time-history curves, maximum displacement and interstorey drifts: particular
attention is paid on comparisons between the results got from single and triple strut
models, from artificial and recorded accelerograms, and a general comparison is done
between pushover and time-history analysis.
Chapter 7 presents the conclusions of the thesis and the final considerations achieved,
giving some suggestion for further works on this topic.
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2 PRESENTATION OF THE EXPERIMENTAL TEST
How previously stated, three r.c. structures with masonry infills will be tested on the
L.N.E.C.’s shaking table: one with unreinforced masonry infills (designed in according to
the portuguese codes with the purpose to be representative of the ordinary national design
practice in the period subsequent the came into effect of the codes) and the others with
two different kind of reinforced infills (designed in according to the Eurocodes with the
purpose to study new solutions in reducing the damages dued to seismic events): in this
work just the first structure is analyzed.
The buildings to be tested have been scaled 1:1.5 because of the shaking table’s limits
(dimension and capacity) and because of the laboratory gates’ height. More detailed
references on the design phase of the buildings and on the relative pushover analysis’
results can be found in Leite J. [2009]; this chapter aims just to resume the main
informations on the structure studied in the present thesis.
2.1
GEOMETRY OF THE STRUCTURE
The structure is a two-storey building, with an interstorey height of 3.00 m; the shorter
frame
(5.70 m) has just one span, while the longer (6.45 m) has two spans.
Figure 2-1. Geometry of the structure’s prototype (in metres)
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The criterion used to scale the building is the Cauchy’s similarity law, a simple rule that
allows to scale all the physical parameters just by using a single number λ, the scale
factor of the model, in this case 1.5. In the following table relationship between the
prototype’s variables and the scaled model’s variables are presented.
Table 2-1. Scale factors of the parameters in according to the Cauchy’s similarity law
Parameter
Scale Factor
Length (L)
LP / LM = λ
Area (A)
AP / AM = λ2
Volume (V)
VP / VM = λ3
Displacement (d)
dP / dM = λ
Velocity (v)
vP / vM = 1
Acceleration (a)
aP / aM = λ-1
Mass (m)
m P / m M = λ3
Weight (w)
wP / wM = λ3
Density (ρ)
ρP / ρM = 1
Force (F)
FP / FM = λ2
Moment (M)
MP / MM = λ 3
Tension (τ)
τP / τM = 1
Deformation (ε)
εP / εM = 1
Module of elasticity (E)
EP / EM = 1
Time (t)
Frequency (f)
tP / tM = λ
fP / fM = λ-1
Once that the similarity relationship have been decide set, the prototype could be scaled
and the dimension of beams, columns, foundations and infills have been fixed: so in the
model the beams have a section of 15 x 30 cm2, the columns of 15 x 15 cm2, the
foundation is an reversed T beam with an height of 30 cm, the slab has an height of 12
cm, and the infills are composed by a double leaf of hollow clay bricks with horizontal
perforations, the inner 7 cm depth and the outer 9 cm depth, with air space between the
two leaves. In the next figures the geometry of the scaled model and of its structural and
non-structural elements is presented.
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Figure 2-2. Geometry of the structure’s scaled model (in metres)
Figure 2-3. Geometry of the infill’s panels (in centimetres)
Figure 2-4. Detail of the infill in the column’s zone (in centimetres)
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2.2
STRUCTURAL DESIGN
To get the solicitations’ values to be used in the design phase, a response spectrum
analysis with the software SAP2000 has been carried out a 3D model of the building:
frame elements have been used to represent beams and columns, diagonal frame element
have been also choose to replace the infills, and rigid diaphragms have been set in
correspondence of the slabs to simulate their real behaviour regarding to the
displacements.
The vertical loads have been applied to the beams. Regarding to the spectrum employed
(just the horizontal components have been used [EN 1998-1:2003 – 4.3.3.5.2]) the
portuguese code, like also EC8 does, defines two kind of spectra to use: a closer and a
farer one; in both cases, the parameters required to characterize the spectra are: the return
period that characterizes the considered limit state (475 years because the L.S. of
Significant Damage is supposed to be the most suitable for the investigated kind of
building [EN 1998-3:2003 – 2.1]), the zone in which the building is located (Lisbon), the
ground type of that zone (very consistent soil); then spectra’s acceleration have been
divided for a behaviour factor that take into account the linearity of the analysis and the
energy dissipation capacity (also EC8 requires that, but sets this parameter is a different
way). Obviously, all the loads have been scaled in according to the Cauchy’s similarity
law.
The values of bending moment and shear force obtained from that analysis, and used to
design the reinforcement of the structural elements, are perfectly scaled as shown from a
comparison with the analysis’ results in SAP2000 of the building’s prototype. The
materials used in the design phase are those, within the portuguese code’s ones, that are
supposed to be the most representative of the constructions that the building object of this
work aims to represent. In the ANNEX 1 it’s possible to look at the detailed structural
design of the building.
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2.3
TEST EQUIPMENT
The division of L.N.E.C. that deals with experimental researches in the seismic field is
the N.E.S.D.E. (Earthquake Engineering and Structural Dynamic Division). Within that
section a triaxial shaking tables is present. It’s one of the most capable existing in a civil
engineering laboratory and it’s dated 1995: it’s a 4.6 x 5.6 m2 steel shaking table, with a
392 kN maximum load capacity; the actuators’ system is hydraulic, while the control’s
type is mixed analogue-digital.
Besides the platform’s dimensions, the other feature that make this shaking table so
performing is the earthquake motions’ severity that is capable to apply to the specimens
(maximum nominal acceleration values: aTRASV = 15 m/s2, aLONG = 25 m/s2, aVERT = 7.5 m/s2;
maximum nominal velocity values: vTRASV = 70.1 cm/s, vLONG = 41.9 cm/s, vVERT = 42.4 cm/s).
To access the laboratory there are two gates, whose height is 4.5 m.
The building objective of the present thesis was built outside and later moved by bridge
cranes (maximum load 392 kN, useful height 8.0 m) on the shaking table where it was
fixed by metallic tubes inserted in holes presents on the platform, and hence realized also
on the structure’s foundation.
All the references on the datas exposed in this paragraph and on more other datas can be
found in http://www.lnec.pt.
Figure 2-5. Scheme of the shaking table
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3 FEATURES OF THE MODELS
A presentation of the software employed in the analysis it was supposed to be useful to
the reader: so, the first paragraph of the present chapter is dedicated to provide a general
approach of what the finite element code DIANA is, to present its main features, the
fields of application in which is most applied, its basic principles and its scheme of work.
Next, the two types of models used to represent infilled frames (local and global models)
are discussed, arguing on their features and on their suitability in characterizing the main
problems present in the kind of construction objective of this thesis: particular attention is
paid in presenting the two global models adopted (single and triple strut model).
Hence, the geometry of the models is described in detail: the element types chosen, the
cross-section assigned and the integration schemes adopted.
In the following paragraph, the mechanical characteristics of concrete, masonry and steel
are examined, paid particular attention on their nonlinear behaviour.
Then, the vertical and horizontal loads assigned to the models are discussed, explaining
the difference between horizontal loads for pushover and for time-history analysis.
The last two paragraphs briefly summarize the boundary conditions applied to the models
and the meshing procedure.
3.1
DIANA – FINITE ELEMENT CODE
DIANA is a multi-purpose finite element code, based on the displacement method. It has
been under development at TNO since 1972. In the beginning of 2003 a new organisation
around DIANA was founded: TNO DIANA bv.
DIANA is a well proven and tested software package with a reputation for handling
difficult technical problems relating to various design and assessment activities: civil,
mechanical, biomechanical, and other engineering problems can be solved with the
DIANA program. Standard application work includes: concrete cracking, excavations,
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tunnelling, composites, plasticity, creep, cooling of concrete, engineering plastics, various
rubbers, groundwater flow, fluid-structure interactions, temperature-dependent material
behaviour, heat conduction, stability analysis, buckling, phased analysis, etc. The
program’s robust functionality includes extensive element, material and procedure
libraries based on advanced database techniques, linear and non-linear capabilities, full
2D and 3D modelling features and tools for CAD interoperability.
Concerning the element types, DIANA offers a great variety of this, such as beams
(straight and curved), solids, membranes, axisymmetric and plane strain elements, plates,
shells, springs, and interface elements (gap). All these elements may be combined in a
particular finite element model.
Relating to the material models, here the most important are presented: elasticity (linear
isotropic and orthotropic elasticity, nonlinear elasticity, hyper-elasticity, visco-elasticity,
regular plasticity, orthotropic plasticity, visco-plasticity); cracking (smeared crack, total
strain fixed and rotating crack); soil mechanics (initial stress ratio, undrained behaviour,
liquefaction); interface nonlinearities (discrete cracking, crack dilatancy, bond-slip,
friction, nonlinear elasticity, and a general user-supplied interface model); user-supplied
(to let the user specifies a general nonlinear material behaviour).
The wide range of analysis modules includes: linear static analysis, nonlinear analysis,
dynamic analysis, Euler stability analysis, potential flow analysis, coupled flow-stress
analysis, phased analysis, parameter estimation and lattice analysis.
Nevertheless, one of the most notable benefits is its power in the field of concrete and soil
where excellent material models are available, developed by researchers in the
Netherlands since the early 1970's: most notably are the models for smeared and discrete
cracking, and for reduction of prestress due to special effects. For the design and
assessment of concrete and reinforced concrete structures, DIANA offers a wide range of
material models for the analysis of the non-linear behaviour of concrete, which comprises
cracking, crushing and shearing effects in cracks and joints, special techniques for
modelling reinforcement and prestressed cables, determination and integration of creep
and shrinkage and advanced solutions for the analysis of young hardening concrete.
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Moreover, special elements may be used to model embedded reinforcement in concrete
structures: bars, grids and prestressed cables. To model these reinforcements DIANA has
a built-in pre-processor
processor in which reinforcement can be defined globally.
The architecture of the DIANA system, as seen from the user's point of view consists of a
number of modules,, indicated with M1 to Mn in Figure 3.1.
Figure 3-1. DIANA program architecture
Each module fulfils a clearly defined task in the Finite Element Analysis. For instance,
module INPUT (M1)) reads the description of the finite element model. All modules have
data communication with a central database, the FILOS file. After the analysis DIANA
can produce output of the analysis results.
To have access to this software architecture, there are three
three basic user-interfaces:
user
a batch
interface, an interactive graphical user interface (GUI), and an interface with usersupplied subroutines. In the batch interface the user defines the finite element model via
an input data file; furthermore, analysis commands
commands must be supplied to indicate how the
analysis should be performed; DIANA will then load the appropriate modules to perform
the analysis; output can be obtained in tabular form for printing or viewing.
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The interactive graphics interface, called iDIANA, is a fully integrated pre- and postprocessing environment to DIANA: the user has to specify the basic model geometry,
loading, materials and other data interactively; this data is stored in a database for preprocessing from which iDIANA can automatically generate the finite element model in
the form of the input data file: moreover, the necessary analysis commands may be
generated via user-friendly interactive forms; analysis results are written to a database for
interactive post-processing and may then be presented in various styles like coloured
contour plots, diagrams, tables etc. Finally, DIANA offers a user-supplied subroutine
option to the advanced user, with skill in programming; via this option the code of various
subroutines with pre-defined arguments may be supplied to define special material
models, interface behaviour, etc.
All the references on the informations exposed in this paragraph about DIANA features
can be found in http://www.tnodiana.com and in Manie J., Wolthers A. [2008].
3.2
STRUT MODELS
Many years of researches and experimental tests in the field of infilled frames consent to
asses that the influence of the infills on response of r.c. structures subjected to lateral
loads isn’t negligible, on the contrary of what in the common structural design is assumed
up to now.
However, there are some problems to understand the interaction between the infills and
the boundary frame, and this is one of the main reason that led the researchers to propose
several models to try to fit the experimental results; is possible to divide these models into
two big classes: the local models (where the infills are modelled adopting discrete or
continuum models for masonry) and global models (where the infills are replaced by
single or multiple compression strut).
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Figure 3-2. Global models (a) and local models (b) for infilled structures
How previously stated, the main aspect that affects the characterization of the infilled
frames under seismic loads is the interaction between the infills and the boundary frames;
experimental evidences have shown that the phenomena is influenced essentially by the
strength of the two materials, concrete and masonry, and by the level of horizontal load
applied to the structure: so it’s possible to analyze the pre-peak phase by dividing it into
three stages. At the beginning, when low forces (and thus low deformations) are applied,
there is no separation between the boundary frames and the wall (if there are no gaps
between the two component), and its contribute in terms of stiffness is very high: this
stage lasts just for very low values of load, and so it’s supposed to be no such essential.
Successively, when forces start to increase to consistent values, a separation occurs
between the wall and the frames (both columns and beams), and so the resistant
mechanism of the infills becomes very similar to a compression strut, with compressive
stresses concentrated at the compressed corner and rapidly decaying in the central zone:
in this stage there is a quite small energy dissipation because cracking is still not reached.
Finally, once the crack strength has been reached, two cases are possible: shear collapse
in the concrete element if the infill is very resistant and the frame is very poor detailed, or
diffusion of the cracks in the infill panel with consequent growth of energy dissipation in
hysteretic cycles; three types of crack pattern have been in the most of the experimental
tests: horizontal slip crack (when the mortar is very weak), diagonal cracks (stair-step
configuration when the bricks are very strong or diagonal configuration when also the
mortar is of good quality), corner crushing (when both masonry and frame are strong, and
the strut failure mechanism is so fully developed).
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Figure 3-3. Possible failure mechanism due to infill’s collapse
Figure 3-4. Load-displacement curves in relation at the different infill failure mechanism
The main difficulties that came up in applying local models was about the correct
representation of the behaviour of infill-frame interaction, of the behaviour of masonry
subjected to plane state of stress (smeared crack model), and of the hysteretic behaviour
of infilled frames (discrete crack models). On the other side, global models always
represented an attractive solution to the researchers because of their simplicity, their
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computational efficiency and their good appropriateness in performing both static and
dynamic nonlinear analysis; their principal limit lays just in their simplicity: it means that
if they are calibrated to well represent some features of the infills (initial stiffness,
strength, collapse mechanism, hysteretic behaviour, etc.), maybe they won’t represent so
well other features. However, they could be not adapt in representations of complex
buildings or also simple buildings that anyway present some particular feature (i.e.
openings, shear connections), where maybe local models are more preferably.
A recent state of art dealing with the main aspects of these topics can be found in Mauro
A. [2008]. In the present work two different global models have been used, a single and a
triple strut model.
Usually in the strut models the tension strength of the infill is considered equal to zero.
Another feature that joins the strut models is to place the strut (or the struts) in both
direction, in order to simulate the infill’s behaviour under cyclical loads: this aspect is
fundamental in dynamic nonlinear analysis, whereas could be not considered in static
nonlinear analysis. Moreover, according to Stafford Smith, B. [1966], the global initial
stiffness of the infilled structure it’s supposed to be obtained considering two parallel
systems: a bare cracked frame and a diagonal strut-taut column; in this way, the global
initial stiffness is achieved by adding the flexural stiffness of the first system and the
stiffness to horizontal translation of the second system’s free node: in this stage the
stiffness of the infilled frame could be much higher than the one of the bare frame, up to
20 times and more.
Figure 3-5. Idealized parallel systems for strut models
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The principal parameters that rule the strut mechanism are the width of the strut and the
length of the contact area between the panel and the beam or the column: there are a lot of
variable that influence these two parameters, experiments led to scattered results and the
experts formulated different proposals.
The single strut model adopted in the present work, is the one suggested by Fardis [1996]
and later recommend also by Safina [2002]. It’s a quite simple model, where the width of
the equivalent diagonal compression strut is considered to be equal to 20% of his length.
Figure 3-6. Scheme of the single strut model
Anyway, all the single strut models don’t take into account a phenomena that sometimes
could be of primary relevance: the interaction between the idealized compression strut
that takes place in the elastic phase and the boundary concrete frames; in fact, in the
contact area, a secondary moment and shear (not considered in single strut models) is
produced that could led the system to a brittle failure because of shear in the columns. To
reproduce this aspect triple strut models have been developed by researchers.
In this work the one proposed by Bergami A. [2008] has been used. The two main
variables to determinate are: the width of the struts (one on the main diagonal and the
other parallel above and below) and the position of the lateral struts. The model is based
on the criterion of equivalent horizontal displacement between the infill panel and the set
of struts: in this way, the equivalence between the global initial stiffness of the single strut
model and the one of the triple strut model is guaranteed. From this correspondence a
relation between the width of the single strut and the width of the central and lateral struts
of the triple strut model is provided.
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Figure 3-7. Equivalence between the single strut model and the triple strut model
=
cos = cos = cos + 2
1 − cos ℎ
= + 2 1 − cos ℎ
The position of the lateral struts, and so also the width of the struts, are found by
assuming a distribution of the stresses along the contact area: in this case, a linear
distribution is assumed.
=
2
=
41 − ⁄ℎ
where:
1 + − 1 + − 8
=
ℎ
2
3.3
=
6ℎ cos GEOMETRY OF THE MODELS
First of all, define the units’ system wherewith the finite element code is going to work is
essential; the following set has been chosen: mm (length), kg (mass), N (force), s (time),
K (temperature). Then in the analysis selection a Structural 2D model type has been
decided to use because planar analysis have to be ran.
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Now, draw the geometry of the model is finally possible: using mono-dimensional
elements the frame was so drawn. Four models have been created: two frames (one for
each main direction) to represent the building using a single strut model, and two frames
using a triple strut model.
Figure 3-8. Geometry of single strut models: model 1 (left) and model 2 (right)
Figure 3-9. Geometry of triple strut models: model 1 (left) and model 2 (right)
In this phase also the reinforcements have been modelled using appropriate commands
that let the user define the steel bars embedded in the concrete parts, thanks to the theory
of fibre model that is one of the main concept adopted by the finite element code.
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Figure 3-10. Geometry of reinforcement bars: model 1 (left) and model 2 (right)
3.3.1 Structural Elements
DIANA offers different kind of elements to model the geometry in a structural analysis:
beam e., truss e., plane stress e., plane strain e., axisymmetric e., plate bending e., flat
shell e., curved shell e., solid e., interface e., embedded reinforcements and other special
elements. In this work, beam elements have been used to model the concrete parts (beams
and columns), truss elements for masonry parts (struts) and embedded reinforcements to
represent the steel bars.
Beam elements are bars which must fulfil the condition that the dimensions d
perpendicular to the bar axis are small in relation to the bar's length l. Beam elements may
have axial deformation ∆l, shear deformation γ, curvature κ and torsion: therefore they
can describe axial force, shear force and moment. DIANA offers three classes of beam
elements:
•
CLASS-I: classical beam elements with directly integrated cross-sections. These
elements may be used in linear and in geometric nonlinear analysis;
•
CLASS-II: fully numerically integrated classical beam elements. These elements
may be used in linear and in geometric and physic nonlinear analysis;
•
CLASS-III: fully numerically integrated Mindlin beam elements. These elements
may be used in linear and in geometric and physic nonlinear analysis.
A.A. 2008-09
39
The basic variables of beam elements are the displacements in the nodes: translations u
and rotations Φ. The orientation of the displacements depends on the beam class and on
the dimensionality. For two-dimensional beams, displacements are oriented in the local
xyz directions:
Figure 3-11. Displacements in 2D beams: class-I and class-II (left); class-III (right)
For beam elements, DIANA can calculate strains and Cauchy stresses in so-called “stress
point”: for fully numerically integrated beam elements the stress points are equivalent
with the integration points (see §3.3.3), whereas for directly integrated beam elements the
stress points must be specified explicitly. Forces and moments are evaluated in nodes and
cross-sections. The set of forces, moments and stresses depends on the dimensionality of
the element: for two-dimensional beams Nx, Qy, Mz, σxx, σxy=σyx are computed.
Figure 3-12. Strains in two-dimensional beams
Figure 3-13. Moments, forces and stresses in two-dimensional beams
The sign convention for bending is that a positive moment yields a positive stress
(tension) in the positive area (+Mz works in the -z direction). The sign convention for
forces is that a positive force yields a positive stress. The differences between the class
types are shown in the next table:
40
A.A. 2008-09
Table 3-1. Overview of beam elements
Class
Theory
Class-I
Class-II
Class-III
Bernoulli
Bernoulli
Mindlin-Reissner
L6
L12
L7 L13 CL9 CL12 CL15 CL18 CL24 CL30
BEN BEN BEN BEN BEN BEN BEN BEN BEN BEN
Type
2D
3D
2D
3D
2D
2D
2D
3D
3D
3D
Straight or curved
str.
str.
str.
str.
cur.
cur.
cur.
cur.
cur.
cur.
Nr. of nodes
2
2
2
2
3
4
5
3
4
5
Nr. of d.o.f.
6
12
7
13
9
12
15
18
24
30
ux
x
x
x
x
x
x
x
x
x
x
uy
x
x
x
x
x
x
x
x
x
x
Variables
Dimension
uz
x
x
x
x
x
Φx
x
x
x
x
x
Φy
x
x
x
x
x
Φz
x
x
∆u x
Primary strains
x
x
x
x
x
x
x
x
x
x
∆u
∆u
ε
ε
ε
ε
ε
ε
ε
ε
κ
∆Φ
γ
γ
γ
γ
γ
γ
γ
γ
σ
σ
σ
σ
σ
σ
σ
σ
κ
Primary stresses
N
N
M
M
Q
Q
Shear deformation
opt.
opt.
no
no
yes
yes
yes
yes
yes
yes
Num int crosssection
no
no
yes
yes
yes
yes
yes
yes
yes
yes
Num int along
beam axis
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
In this phase, class-II L7BEN beam elements have been chosen to model beams and
columns: this decision have been taken because class-I elements cannot be used in physic
nonlinear analysis, and class-III elements (that of course are more accurate with respect to
class-II ones) maybe don’t lead significant changes in global results. However, sensitive
analysis on element types have been carried out to analyze their influence (see chapter 5).
A.A. 2008-09
41
Also truss elements are bars which must fulfil the condition that the dimensions d
perpendicular to the bar axis are small in relation to the bar's length l. Nevertheless, their
deformation can only be the axial elongation ∆l: there is neither bending nor shear
deformation.
Analogously to beam elements, DIANA proposes three families:
•
regular truss element: with directly integrated cross-sections, are suitable for
linear static and physical nonlinear analysis;
•
enhanced elements: with directly integrated cross-sections, have additional d.o.f.
perpendicular to the bar axis: can be used in geometrically nonlinear and dynamic
analysis;
•
cable elements: fully numerically integrated elements that can be used in
geometrically nonlinear analysis of cables, and nonlinear analysis of prestressed
cables (tendons) in r.c. They don’t have initial stiffness in transverse direction.
The fundamental variables of truss elements are the translations ux of the nodes in x
direction; from these DIANA derives the deformations εxx and calculates the normal
Cauchy stresses σxx and the generalized axial forces Nx:
Figure 3-14. Displacements, strain, stresses and forces in truss elements
To model the struts representing the masonry panels, enhanced truss elements have been
chosen: in fact, the regular ones are not suitable for dynamic analysis, whereas no
assurances are given by cables because of their lack of initial stiffness in transverse
direction.
42
A.A. 2008-09
Table 3-2. Overview of truss elements
Class
Regular Enhanced
L2
TRU
Type
Cable
L6 CL6 CL8 CL10 CL9 CL12 CL15
L4
TRU TRU TRU TRU TRU TRU TRU TRU
2D
2D
3D
2D
2D
2D
3D
3D
3D
Straight or curved
str.
str.
str.
cur.
cur.
cur.
cur.
cur.
cur.
Nr. of nodes
2
2
2
3
4
5
3
4
5
Nr. of d.o.f.
2
4
6
6
8
10
9
12
15
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Variable
s
Dimension
ux
uy
uz
x
Primary strains
εxx
εxx
εxx
εxx
εxx
εxx
εxx
εxx
εxx
Primary stresses
σxx
σxx
σxx
σxx
σxx
σxx
σxx
σxx
σxx
Nx
Nx
Nx
Nx
Nx
Nx
Nx
Nx
Nx
Num int cross-section
no
no
No
yes
yes
yes
yes
yes
yes
Num int along beam axis
yes
yes
yes
yes
yes
yes
yes
yes
yes
Discussing now about the modelling of reinforcements, DIANA offers appropriate
elements created also to this purpose: embedded reinforcements to model steel bars and
grid reinforcements to model shear reinforcements: they have the characteristic to add
stiffness to the finite element model. Reinforcements are element embedded in structural
elements, the so-called mother elements: the code ignores the space occupied by an
embedded reinforcement and the mother element neither diminishes in stiffness, nor in
weight; consequently reinforcement doesn’t contribute to the weight (mass) of the
element, and don’t have degrees of freedom of their own. Nevertheless grids cannot be
embedded mono-dimensional elements like beams, so just embedded reinforcements have
been used. Reinforcement bars may be embedded in various families of elements: beams,
plane stress, curved shell and solid: the total length of the bar is considered to be divided
in several particles and the so-called location points define the position of the particles in
the finite element model and the curvature of the bar; some location points are the
intersections of the bar with the element boundaries, other location points are in-between
these intersections
A.A. 2008-09
43
Figure 3-15. Geometry of an embedded reinforcement in beam elements
By default, reinforcement strains are computed from the displacement field of the mother
elements. This implies perfect bond between the reinforcement and the surrounding
material. However, is possible specify that the reinforcement is not bonded to the
embedding elements. The variables for a bar reinforcement are the strains εxx and the
stresses σxx. The strains and stresses are coupled to the degrees of freedom of the
surrounding element. Bar reinforcement can be embedded only in beam elements of
class-II and class-III.
Finally, to represent the correct distribution of the masses (fundamental in dynamic
analysis), point elements have been used to take into account the masses of the orthogonal
frames; this kind of elements proposed by DIANA are just concentrated masses that don’t
influence static behaviour of the model: they don't have stiffness, strain or stress. In static
analysis, the concentrated mass acts as concentrated loading for dead weight. Two kind of
point masses are available: translational (PT3T, whose d.o.f. are ux, uy, uz) and rotational
(PT3RO, whose d.o.f. are Φx, Φy, Φz): in these models the first ones have been selected
because it is supposed to represent in a better way the influence of the orthogonal frames.
It is important to remark that the direction without stiffness (Z) has to be supported.
3.3.2 Cross Sections
The next step consists in defining the cross-sections of the elements.
Regarding to beam elements, three ways are possible: choosing in a profile library
(available just for class-I elements), using one of the predefined shapes, or create an
arbitrary shape. Because rectangular sections needs to be assigned to concrete elements
(beams and columns), predefined rectangular shapes have been used: it’s so necessary
specify just height and width of the elements:
44
A.A. 2008-09
MODEL 1
MODEL 2
# = 300 &&
# = 150 &&
# = 300 &&
# = 150 &&
' = 150 &&
' = 225 &&
' = 150 &&
' = 150 &&
beams
columns
beams
columns
The models want to represents half structure: so the width of the columns of model 1 has
been increased of 50% to take into account also half column of the orthogonal frames
(both in terms of mass and stiffness).
For truss elements simply the area is required:
MODEL 1
SINGLE STRUT
MODEL 2
MODEL 1
TRIPLE STRUT
MODEL 2
) = 70200 && ) = 45900 && ) = 35100 && ) = 22950 && ) = 20959 && ) = 13139 && How mentioned in previous paragraphs, infills are composed by a double leaf of hollow
clay bricks with horizontal perforations, the inner 7 cm depth and the outer 9 cm depth;
but in the computation of strut areas, just the outer deepness has been considered because
of the it’s supposed that the two leaves don’t work like a unique resistant element: in fact,
they are separated by 2.0 cm of air space and no connections are put between them.
To complete this section, it’s important to say that also the areas of embedded bar
reinforcements have been inserted into the physical properties panel of the models.
3.3.3 Integration Scheme
Numerical integration is based upon the evaluation of the function to be integrated in a
number of specific points, the so called integration points. These function values in the
integration points are then weighted and summed to obtain the value of the integral. The
weight function depends on the method of integration. For finite element integration
A.A. 2008-09
45
usually the Gauss integration scheme is applied, as this method requires the least number
of integration points. Now the integration of a function f (x) can be written numerically as:
:
67
, -./. = 0 123 -45 ;
589
where <23 describes the weight function of the applied method for the specific integration
interval, =2 the number of integration points and ξi the coordinate of the integration point.
A minimum number of integration points is required by the numerical integration method
and depends on the order of the interpolation polynomial. In order to integrate all of the
terms in the integrand a full integration scheme is necessary.
DIANA applies a default scheme appropriate for most types of meshes and analyses, but
sometimes it may be useful to choose another scheme by means of input data. For
integration along the axis of line elements, i.e. in the isoparametric ξ direction, DIANA
offers integration rules according to Gauss, Simpson, Newton-Cotes, and Lobatto.
Figure 3-16. Integration schemes along the elements axis
Referring to beam elements, class-II and class-III types are not only integrated along the
bar axis ξ, but also in the area of the cross-section. The integration in the area of the
46
A.A. 2008-09
cross-section depends on its shape (for rectangular zones, like the ones of the models in
exam, DIANA defines two isoparametric axes of the zones) and also on the
dimensionality of the beam element. For a two-dimensional beam element the zones are
integrated in η direction only, for a three-dimensional beam element in η and ζ direction.
Available rules on the cross-section are Gauss and Simpson.
Figure 3-17. Integration schemes on the cross-section of the elements
Hence, to have accurate results, 7-point Gauss integration scheme along the element axis
5-point Gauss integration scheme on the cross-section: otherwise DIANA default
integration schemes are 2-point Gauss and 3-point Simpson. In chapter 5 the results of
sensitive analysis on the number of integration points adopted will be presented to show
its influence.
How exposed in Table 3.2, numerical integration on the cross-section is not possible for
truss enhanced elements because they aren’t fully integrated elements: so they are just
integrated on the element axis with and 7-point Gauss integration scheme. Once again
DIANA default integration scheme is 2-point Gauss.
For embedded reinforcements (that can be integrated only along the element axis)
DIANA default integration scheme, 2-point Gauss, it was supposed to be appropriate.
A.A. 2008-09
47
3.4
MATERIALS’ PROPERTIES
The materials’ data required by the program are subdivided into 4 parts: linear elasticity,
mass, damping (differently by other programs this feature is required in the materials’
properties), static nonlinearity (also used in time-history analysis), transient nonlinearity
(creep and visco-elasticity), expansion and Wohler diagrams. The last three
characteristics are related to phenomena not present in this study, while the damping data
will be explained later, when dynamic analysis will be faced (chapters 6). Thus, in the
next paragraphs, the linear elasticity, mass and static nonlinearity properties assumed for
concrete, masonry and steel will be explained.
3.4.1 Concrete
According to the portuguese code [R.E.B.A.P. – art.13], a concrete class B25 has been
chosen in the design phase: referring to the European Standards, it correspond to a class
C20/25 with the following strength characteristics [EN 1992-1-1:2004 – Table 3.1]:
->? = 20 @AB
->?,>D:E = 25 @AB
->G?,I = 1.5 @AB
->F = 28 @AB
->G?,.JI = 2.9 @AB
->GF = 2.2 @AB
K>F = 30000 @AB
required data are the Young’s modulus ECM and the Poisson’s ratio L = 0.20 [EN 1992-1-
In the linear elasticity section, an isotropic behaviour has been chosen, and thus the
1:2004 – 3.1.3].
In the mass section, a density of M = 2.5N OP QR⁄&&S has been set [CEB-FIP, MC
1990 – 2.1.2].
In the static nonlinearity section, a “total strain fixed crack” behaviour has been selected
in the wide range of DIANA’s library proposal: in fact, it is appropriate for a quasi-brittle
material like concrete is, and moreover it makes the structure model well suited for state
limit analysis. In a total strain concept the stresses are described as a function of the
strains: in the fixed crack approach, the stress-strain relationship is evaluated in a fixed
coordinate system which is fixed upon cracking, whereas in a rotating crack approach is
evaluated in the principal direction of the strain vector: however, the basic idea of the
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A.A. 2008-09
total strain approaches is that the stress are evaluated in the directions which are given by
the crack directions. The tensile behaviour is modelled using an exponential softening
curve, based on fracture energy and also related to the crack bandwidth, as in usual
smeared crack model (Elfgren L. [1989]). The concrete compressive behaviour could be
influenced by the lateral confinement (strength and ductility increase with increasing
isotropic stress) and by the lateral cracking (peak stress and peak strain are reduced if the
material is cracked in the lateral direction): the model could be so improved, but in this
work these aspects haven’t been taken into account; the compressive behaviour is just
modelled with a parabolic function based on, once again, fracture energy and crack
bandwidth. The parameters that rule the concrete behaviour in tension and compression
(->F = 28 @AB, ->GF = 2.2 @AB), the tensile fracture energy, the compressive fracture
are, apart from the tensile and the compressive strength already previously defined
energy and the crack bandwidth; the European Standards don’t give references to
determinate these parameters. So, the Model Code [CEB-FIP, Model Code 1990 –
2.1.3.3.2] suggestion have been used to determinate the tensile fracture energy, as a
function of the concrete class and of the maximum aggregate size (dMAX = 8mm in this
case):
TU V = TU V WX YZ \
X
YZ[
.]
= 0.0514 @AB ⋅ &&
where TU V = 0.025 @AB ∙ && (for dMAX = 8mm) and ->F = 10 @AB. Regarding to
the compressive fracture energy, regulations don’t give indications: then, a formula
proposed by Lourenço P.B. [2008], depending on the compressive strength has been used:
T` = 15 + 0.43 ->F − 0.0036 ->F = 24.22 @AB ⋅ &&
(12 @AB < ->F < 80 @AB)
For the crack bandwidth, a value corresponding to the length of the element in the mesh
ℎ = 200 &&.
discretization (this is also the default value of DIANA) it was supposed to be suitable: so
A.A. 2008-09
49
30
[MPa]
20
stress
25
10
15
5
0
-0,0010
-5
0,0010
0,0030
0,0050
strain
0,0070
0,0090
[%]
Figure 3-18.. Concrete behaviour in tension and compression
To complete the static nonlinearity section, in the fixed crack concept a shear retention
variable value is possible to define; in the present work = 0.15 has been adopted.
factor must be set to consider the shear stiffness reduction after the cracking: a constant or
Figure 3-19. Constant shear retention for total strain fixed models
3.4.2 Masonry
In the design phase, a double leaf of hollow clay bricks with horizontal perforations
(group 4 [EN 1996-1-1:2005
1:2005 – Table 3.1]) has been used for the infills; but, how
previously stated, just the external leaf is taken into
in account in the structural analysis: the
dimension of these bricks are 30 x 20 x 9 cm3; for the compressive strength of the units, a
value of -: = 2.9 @AB is given by the fabricant. A M10 class mortar was used to fix the
bricks: its compressive strength is -F = 10 @AB [EN 1996-1-1:2005 – 3.2.2].
3.2.2]
50
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Starting from the values of compressive strength of brick and mortar, the characteristic
compressive strength of masonry can be defined [EN 1996-1-1:2005 – 3.6.1.2]:
-? = -:.] -F.S = 1.47 @AB
supposing = 0.35 [EN 1996-1-1:2005 – Table 3.3]. Because of the slenderness of the
infill, this value should be reduced by multiplying it by a factor Φ that take into account
this aspect [EN 1996-3:2005 – 4.2.2.3]:
Φ = 0.85 − 0.0011
ℎEX
= 0.63
cEX
considering ℎEX = M6 ∙ ℎ, with M6 = M = 0.75 [EN 1996-3:2005 – 4.2.2.4], ℎ = 1.70 &
is the height of the infill and cEX = 0.09 & is its thickness.
The characteristic compressive strength become in this way -? = 0.93 @AB. The tensile
strength in the direction perpendicular to bed joints is zero [EN 1996-1-1:2005 – 6.1.1].
It’s possible now to define the properties in the linear elasticity section: an isotropic
characteristic compressive strength K>F = 925.9 @AB [EN 1996-1-1:2005 – 3.7.2], while
behaviour has again been chosen: the Young’s modulus has been taken 1000 times the
for the Poisson’s ratio a reasonable value is L = 0.15. In fact, neither the European
Standards and the Model Code give references for this parameter: nevertheless this
variable is useless in the DIANA model, because the truss elements are just sensitive to
axial deformations.
In the mass section, despite the density value of M = 1.35N OP QR⁄&&S proposed by
Brazão Farinha J.S., Correia dos Reis, A. [1996] in their studies on portuguese materials,
a null value has been set because the struts must have just a stiffening function in the
model; their self-weight anyway is considered in the loads (see §3.5.1).
For the static nonlinearity aspects, the same behaviour adopted for the concrete has been
behaviour, fictitious values of ->GF = 1.0N O9 @AB and TU V = 0.01 @AB ⋅ && should
defined: however, because of the impossibility to put a zero strength in the tensile
have to be put; for the compressive fracture energy, a different formula from the one
A.A. 2008-09
51
adopted for concrete has to be used, because the masonry strength doesn’t lays in its
representative range; thus, for materials with compressive strength lower than 12 MPa the
following formula is suggested:
T` = /D,> -? = 1.48 @AB ∙ &&
where /D,> = 1.6 && is the recommended value of the average ductility index in
compression. For the crack bandwidth, the length of the struts,
strut different for the two
concrete has been used: = 0.15.
0
stress
[MPa]
models, has been set. Finally, for the shear retention factor the same value used for the
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
model 1
model 2
-0,0005 0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030
strain
[%]
Figure 3-20. Masonry behaviour in compression for the two models
3.4.3 Steel
According to the portuguese code [R.E.A.E. – art.22.1] a steel class A400NR
A400
has been
decided to use for the bars; it is characterized by the following features, analogous to the
ones prescribed by the European Standards [EN 1992-1-1:2004 – Annex C]:
-d? = 400 @AB
-G? = 460 @AB
eD? = 14%
In the liner elasticity section,, a bonded reinforcement concept has been selected;
selected it means
that the bars are embedded in the beam elements (the so called “mother elements”):
elements”) the
stiffness of the reinforcements doesn’t contribute to the stiffness of the mother element,
nor do its strains and stresses
sses change with deformation of the mother element. For this
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A.A. 2008-09
reason, just the Young’s modulus is required: K>F = 200000 @AB [R.E.A.E. – art.22.1
and EN 1992-1-1:2004 – 3.2.7].
Thus, a density of M = 7.85N OP QR⁄&&S has been put [EN 1992-1-1:2004 – 3.2.7].
At last, for the static nonlinear properties, a Von Mises plasticity is available for
embedded reinforcements: it is possible to define an hardening diagram or an ideal
plasticity; the last one was chosen in this work to describe the steel’s inelastic behaviour.
stress
[MPa]
In this case, just the yielding strength, earlier defined, has to be supply to the code.
450
400
350
300
250
200
150
100
50
0
0,0000 0,0200 0,0400 0,0600 0,0800 0,1000
strain
[%]
Figure 3-21. Steel behaviour
3.5
LOADS
One of the features still needed to be inserted to describe the buildings’ models are the
loads (both vertical and horizontal). Whilst in eigenvalue analysis loads can be omitted
because just masses are taken into account to determinate the modal response of the
structure, it’s important to remark the profound difference in horizontal loads between
those applied in the models used for pushover analysis and those applied in the models
used for time-history analysis: a combination of storey forces’ schemes has to be
considered in the first case, whereas a combination of accelerograms compatible with the
reference spectrum has to be considered.
A.A. 2008-09
53
3.5.1 Vertical Loads
The permanent loads considered are: self-weight of concrete structural elements,
permanent loads dued to partitions, linings, slabs and infills. The imposed loads regard
the floor loads, and are those considered in the European Standard: category A (areas for
domestic and residential activities) has been taken into account for the internal floor [EN
1991-1-1:2001 – Table 6.1, Table 6.2], and category H (roofs not accessible except for
normal maintenance and repair) has been taken into account for the roof [EN 1991-11:2001 – Table 6.9, Table 6.10].
Table 3-3. Permanent and imposed vertical loads: summary
Permanent Loads
Partitions
Internal floor lining
Roof lining
Slabs
Infills
Beams
Columns
Imposed Loads
Internal floor
Roof
kN/m2
1.11
1.07
1.07
3.00
kN/m2
1.33
0.67
kN/m0
3.74
1.10
0.55
kN/m0
-
To define the loads to be applied to the models, distributed loads borne by the beams and
concentrated loads applied to the external beam-column joints, a load distribution
(represented in the next figure) has to be considered.
Figure 3-22. Distribution of vertical load in plan
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A.A. 2008-09
Therefore, considering the presence of the frames in the orthogonal direction (loads borne
by orthogonal beams), and the combination of action for seismic design situation [EN
1990:2002 – 6.4.3.4], the loads to apply to the DIANA models have been evaluated:
Table 3-4. Permanent and imposed vertical loads applied to the models
MODEL 1
Storey 1
(internal floor level)
Partitions
Infill
Slab
Imposed load
Orthogonal beam
Partitions (orth. beam)
(orth. beam)
Infill (orth. beam)
Slab (orth. beam)
Imposed load
(orth. beam)
TOT
Storey 2
(roof level)
Roof lining
Slab
Imposed load
Orthogonal beam
Roof lining (orth. beam)
Slab (orth. beam)
Imposed load
(orth. beam)
TOT
MODEL 2
kN/m
kN
1.05
1.02
3.74
2.85
0.25
-
2.37
2.53
-
2.44
-
8.04
6.84
-
0.61
8.91 22.83
kN/m
kN
1.02
2.85
0.00
-
2.37
2.44
6.84
-
0.00
3.87 11.65
Storey 1
(internal floor level)
Partitions
Infill
Slab
Imposed load
Orthogonal beam
Partitions (orth. beam)
(orth. beam)
Infill (orth. beam)
Slab (orth. beam)
Imposed load
(orth. beam)
TOT
Storey 2
(roof level)
Roof lining
Slab
Imposed load
Orthogonal beam
Roof lining (orth. beam)
Slab (orth. beam)
Imposed load
(orth. beam)
TOT
kN/m
kN
1.18
1.13
3.74
3.18
0.28
-
2.10
2.00
-
1.93
-
7.11
5.42
-
0.48
9.52 19.03
kN/m
kN
1.13
3.18
0.00
-
2.10
1.93
5.42
-
0.00
4.32
9.44
Because of the importance in dynamic analysis to consider not just the loads but also the
masses, in applying the loads the following procedure has been adopted: the distributed
loads have been divided for the gravity acceleration, and then applied to the beams using
A.A. 2008-09
55
an equivalent density (considering it in addition to the density of the beams, §3.4.1); the
concentrated loads have been divided for the gravity loads too, and then applied like
concentrated masses in the external beam-column joints (see §3.3.1). In this way, all the
masses of the structure are correctly contemplated; to transform the masses in loads, a
gravity load case has been selected, imposing a vertical acceleration of -9.81 m/s2.
3.5.2 Horizontal Loads: static loads
To perform static nonlinear analysis, monotonically increasing lateral load patterns have
to be applied to the model of the building. The European Standards prescribe to take into
account at least two vertical distributions of lateral loads: a uniform pattern, based on
lateral forces that are proportional to mass regardless of elevation; a modal pattern,
proportional to lateral forces consistent with the lateral force distribution in the direction
under consideration determined in elastic analysis [EN 1998-1:2003 – 4.3.3.4.2.2].
last paragraph; they are: &9 = 8.53 c , & = 4.30 c . It’s now possible determinate the
So the first step consist in calculate the storey masses, in according to what detailed in the
values of the concentrated forces corresponding to the lateral loads pattern to be applied
to DIANA models.
Figure 3-23. Horizontal load patterns for pushover analysis
A gravity load case has been selected, imposing an horizontal acceleration of 1.00 m/s2.
56
A.A. 2008-09
3.5.3 Horizontal Loads: dynamic loads
In nonlinear dynamic analysis, seismic motion is represented by ground acceleration
time-histories: both artificial and recorded accelerograms can be employed [EN 19981:2003 – 3.2.3.1.1]. Anyway, some rules in finding out the accelerograms must be
satisfied [EN 1998-1:2003 – 3.2.3.1.2]:
a) a minimum of 3 accelerograms should be used;
b) the mean of the zero period spectral response acceleration values (calculated from
the individual time-histories) should not be smaller than the value of agS for the
site in question;
c) in the range of periods between 0.2 T1 and 2 T1 (T1 fundamental period of the
structure) no value of the mean 5% damping elastic spectrum, calculated from all
time-histories, should be less than 90% of the corresponding value of the 5%
damping elastic response spectrum.
In dynamic nonlinear analysis carried out in the present work, sets of 7 accelerograms
have been used: in this way, the average of the response quantities from all the analysis
can be selected; otherwise, the most unfavourable of the response quantity among the
analysis should be selected [EN 1998-1:2003 – 4.3.3.4.3].
In any case, the first step consist in defining the horizontal elastic response spectrum
wherewith the accelerograms have to be compatible (how previously stated the vertical
component is not necessary to be taken into account, §2.2). The reference return period of
the Limit State of Significant Damage: Thij = 475 years [EN 1998-3:2003 – 2.1].
the seismic action to be assumed for the no-collapse requirement is the one characterizing
One o more alternative shapes of the spectrum may be adopted depending on the seismic
sources and on the earthquake magnitude generated from them [EN 1998-1:2003 –
3.2.2.1], in according to the National Annexes: the portuguese one considers two shapes,
a closer (type 1) and a farer one (type 2) [NP EN 1998-1:2006 – NA.2.3 e]. For the
horizontal component of the seismic action, the elastic response spectrum Se(T) is defined
by the following expressions [EN 1998-1:2003 – 3.2.2.1]:
A.A. 2008-09
57
0 ≤ p ≤ pq :
T
∙ η ∙ 2.5 − 1
Tv
pq ≤ p ≤ p` :
St T = au ∙ S ∙ η ∙ 2.5
p` ≤ p ≤ 4y:
St T = au ∙ S ∙ η ∙ 2.5 ∙ p` ≤ p ≤ px :
where:
ag:
St T = au ∙ S ∙ 1 +
Tj
St T = au ∙ S ∙ η ∙ 2.5 ∙ T
Tj∙ Tz
T
design ground acceleration on type A ground (ag = agR · γI)
TB: lower limit of the constant spectral acceleration branch
TC: upper limit of the constant spectral acceleration branch
TD: value that define the beginning of the constant displacement response range of the
spectrum;
S:
soil factor;
η:
damping correction factor with a reference value of η=1 for 5% viscous damping.
{ = 10⁄5 + 4 , with ξ viscous damping ratio of the structure (in percentage).
The National Annex gives the values of the other parameters needed to define the spectra:
the seismic zone where the building is placed (Lisbon lays in zone 1.3 and 2.3) and
consequently the reference peak ground acceleration [NP EN 1998-1:2006 – NA.2.3 c].
Figure 3-24. Seismic zones in continental Portugal
58
A.A. 2008-09
Table 3-5. Reference peak ground acceleration
Type 1 seismic action Type 2 seismic action
Seismic
Zone
agR
(m/s2)
Seismic
Zone
agR
(m/s2)
1.1
1.2
1.3
1.4
1.5
2.5
2.0
1.5
1.0
0.5
2.1
2.2
2.3
2.4
2.5
2.5
2.0
1.7
1.1
0.8
To get the design the peak ground acceleration ag, the reference value must be multiplied
by the importance factor γI [EN 1998-1:2003 – 3.2.2.2]: it is provided by the National
Annex and depends on the building importance class [NP EN 1998-1:2006 – NA.2.3 h];
in this case, a class II (ordinary buildings) it was supposed to be suitable [EN 19981:2003 – 4.2.5].
Table 3-6. Importance factor
Type 1
Importance
seismic
class
action
I
II
III
IV
0.6
1.0
1.6
2.1
Type 2 seismic action
Continental
Portugal
Azores
0.8
1.0
1.3
1.6
0.8
1.0
1.2
1.4
As a result, considering a type A ground, is possible to get the value of the soil factor S
and of the characteristic periods TB, TC, TD [NP EN 1998-1:2006 – NA.2.3 f].
Table 3-7. Parameters of horizontal elastic response spectra: zone 1.3 (left) and 2.3 (right)
Ground
type
S
A
B
C
D
E
1.00
1.20
1.50
1.80
1.70
A.A. 2008-09
TB (s) TC (s) TD (s)
0.10
0.10
0.10
0.10
0.10
0.60
0.60
0.60
0.80
0.60
2.00
2.00
2.00
2.00
2.00
Ground
type
A
B
C
D
E
S
1.00
1.35
1.50
1.80
1.60
TB (s) TC (s) TD (s)
0.10
0.10
0.10
0.10
0.10
0.25
0.25
0.25
0.30
0.25
2.00
2.00
2.00
2.00
2.00
59
The horizontal elastic response spectra are thus defined; to get the respective ones to be
applied to the scaled model, they must be scaled in according to Cauchy similarity law:
Se [m/s2]
5,0
type1 (prototype)
4,0
type2 (prototype)
3,0
2,0
1,0
0,0
0,0
1,0
2,0
3,0
4,0
T [s]
Figure 3-25. Horizontal elastic response spectra
Now, the procedures operated to get the accelerograms, artificial and recorded, will be
explained.
To generate artificial accelerograms has been employed an apposite software
(http://nisee.berkeley.edu/elibrary/getpkg?id=SIMQKE1): SIMQKE-1, SIMulation of
earthQuaKE ground motions, developed at the University of California, Berkeley
(U.S.A.); this software has been modified in a new version, SIMQKE-GR, by prof. Gelfi
at
the
University
of
Brescia
(Italy)
(freely
available
on
http://dicata.ing.unibs.it/gelfi/software/programmi_studenti.html). With this program is
possible to create artificial earthquakes compatible with response spectra; imported
spectra can be used (like done in the present work), or spectra in according to italian
codes (both N.T.C.2008 and O.P.C.M.3274) setting the relative input parameters. The
following datas are required in input (the adopted values are reported in brags):
TS: smallest period of desired response spectrum (0.02 s – cannot be zero);
TL: Largest period of desired response spectrum (4.00 s);
TRISE: Start of the stationary part of the accelerogram (5.00 s);
TLVL: Duration of the stationary part (10 s, in according to [EN 1998-1:2003 –
3.2.3.1.2]);
60
DUR: Total duration (20 s);
A.A. 2008-09
NCYCLE: Number of cycles to smoothen the response spectrum (50);
AGMX: Maximum ground acceleration (it’s automatically calculated by the
program);
NPA: Number of artificial earthquakes (7);
IIX: Arbitrary odd integer;
AMOR: Damping coefficient (0.05, in according to [EN 1998-1:2003 –
3.2.3.1.2]).
Figure 3-26. One of the artificial accelerograms generated: spectral acceleration
A.A. 2008-09
61
Once the accelerograms have been created, it should be checked that in the range of
period between 0.2 T1 and 2 T1 (T1 fundamental period of the structure) no value of the
mean 5% damping elastic spectrum should be less than 90% of the corresponding value
of the 5% damping elastic response spectrum [EN 1998-1:2003 – 3.2.3.1.2]. The
eigenvalue analysis, whence fundamental period of the structure is determinate, will be
exposed in chapter 4: the smallest period of the two infilled frames has been assumed for
the lower bound and the biggest period of the two bare frames for the upper bound has
been taken into account, so that the range is 0.025s – 0.700s.
All the references on the informations exposed in this paragraph about SIMQKE features
can be found in Gasparini, D.A., Vanmarcke E.H. [1976] and in Gasparini, D.A.,
Vanmarcke E.H., Nau R.F. et al. [1976].
To get sets of recorded accelerograms, the free software REXEL 2.61 beta, available on
the ReLUIS internet website (www.reluis.it), has been used. The programs allows to find
out sets of 7 records (from free-field conditions) compatible, in the average, with an input
design spectrum. The datasets included in REXEL, are the European Strong-motion
Database (ESD), and the Italian Accelerometric Archive (ITACA). Initially, the input
design spectrum has to be defined; it could be: a spectrum in according to the European
Standards [EN, 1998-1:2003], to the Italian Standards (N.T.C.2008) or a completely userdefined spectrum (like done in the present work); in the first two cases the necessary
parameters defining the spectrum have to be set so that the program can calculate it. The
following input stages required to the user are below described:
indicate the site class (A-E): like stated in this paragraph the ground type is A;
specify if just the two horizontal components (characterized by the same elastic
response spectrum) are wanted or if also the vertical component has to be find out
by the program: the vertical component has been here ignored (see §2.2);
choose the magnitude range (moment magnitude for A-D site class records of
ESD and local magnitude for E site class records; moment magnitude for all
ITACA records): 4 < M < 8 has been chosen;
62
set the epicentral distance range: 0 km < R < 600 km has been set;
decide the records of database (ESD or ITACA): ESD was decided to use;
A.A. 2008-09
select if the accelerograms to consider have to be from the same local geology of
the site under consideration or from any local geology: accelerograms the first
option has been selected for the unscaled set and the second for the scaled, so to
get in this case a combination with low values of scale factor.
As a result of this preliminary research, the software returns the number of records
(and the corresponding number of events) available in these ranges and to be
considered in the compatibility analysis: 164 events and 2 x 393 records has been
found by the program in the case of unscaled accelerograms and 387 events and 2 x
1383 records. All the ESD database has been investigated in terms of magnitude and
epicentral distance, and also in terms of local geology for scaled accelerograms.
Figure 3-27. ESD records database
Now, the research has to be continued by setting other datas:
the tolerance (lower and upper limit) with which the average spectrum of the
combination must match the target spectrum in an arbitrary interval of periods
[T1, T2] have to be specified: regarding to the lower bound, 10% was imposed in
according to the limitation prescribed in the European Standard explained
previously in this paragraph, whereas regarding to the upper bound, 10% is
supposed to be a fine value; concerning the intervals of period, the chosen interval
is 0.10 s – 0.70 s because values lower than 0.10 s aren’t accepted;
A.A. 2008-09
63
if obtain combinations of accelerograms compatible with the code spectrum which
doesn’t need to be scaled, or combination of accelerograms compatible with the
reference spectrum if scaled linearly (the records have to be normalized by
dividing the spectral ordinates to their PGA; combinations of these spectra have to
be compared to the non-dimensional code spectrum): if this second option is
chosen, the user have to select the maximum mean scale factor to use. Both sets of
accelerograms, unscaled and scaled (maximum mean scale factor = 2), have been
find out in this work in order to check the differences in the analysis’ results;
if stop the analysis after the first compatible combination was found selecting the
option I’m feeling lucky; otherwise a maximum number of compatible
combinations to find has to be set: the option has been used because, how will be
explained later, is sufficient the first set;
if the compatible combination have to be formed by 7 accelerograms (for planar
analysis), by 7 pairs of accelerograms (two horizontal components for a spatial
analysis) or by 3 pairs of accelerograms (for a complete spatial analysis):
obviously the first case has been selected.
Finally, the program can find out the list of records, sorted in ascending order of a
parameter which measures how much the average spectrum deviates from the reference
spectrum, so that it’s possible to take the first combinations (i.e., the one found with the
I’m feeling lucky option) being sure it is the one with the smallest individual scattering
regarding to the reference spectrum:
;|E};~E
;,;|E};~E p5 − ;,G;}~EG p5 1
= 0

;,G;}~EG p5
589
For all the combinations found, it is also possible to calculate the deviation of each
accelerogram:
;, p5 − ;,G;}~EG p5 1
5 = 0

;,G;}~EG p5
589
64
A.A. 2008-09
Figure 3-28. Interface of the software REXEL
Figure 3-29. Spectral waves of the recorded unscaled accelerograms: 1ST combination
A.A. 2008-09
65
Table 3-8. Informations about the recorded unscaled accelerograms: 1ST combination
Earthquake
Name
Izmit
Izmit
Campano-Lucano
Kalamata
Valnerina
South Iceland
Friuli
Date
17/18/1999
17/08/1999
23/11/1980
13/10/1997
19/09/1979
21/06/2000
06/05/1976
AVERAGE VALUES
Mom.
Magn.
[MW]
7.6
7.6
6.9
6.4
5.8
6.4
6.5
6.7
Epic. Wave Earth
Station
Dist. form quake
ID
[km]
ID
ID
47
1228
472
ST561
47
1228
472
ST561
23
287
146
ST93
48
5819 1885 ST1321
5
242
115
ST225
20
6342 2142 ST2556
23
55
34
ST20
30.4
EC8
Site
class
A
A
A
A
A
A
A
Figure 3-30. Spectral waves of the recorded scaled accelerograms: 1ST combination
Table 3-9. Informations about the recorded scaled accelerograms: 1ST combination
Earthquake
Name
Kalamata
Umbria Marche
Montenegro
Spitak
Friuli
Umbria Marche
Valnerina
AVERAGE VALUES
66
Date
13/09/1986
14/10/1997
24/05/1979
07/12/1988
16/09/1977
26/09/1997
19/09/1979
Mom.
Magn.
[MW]
5.9
5.6
6.2
6.7
5.4
8.0
5.8
5.9
Epic. Wave Earth
Station
Dist. form quake
ID
[km]
ID
ID
10
413
192
ST164
23
642
292
ST225
20
232
108
ST77
36
439
213
ST173
11
981
72
ST1043
23
596
286
ST83
5
242
115
ST225
18.3
EC8
Site
class
B
A
B
C
A
B
A
A.A. 2008-09
All the references on what exposed in this paragraph about REXEL features can be found
in Iervolino I., Galasso C., Cosenza E. [2010], Iervolino I., Galasso C. [2010] and
Ambraseys, N., Smit, P., Sigbjornsson, R., Suhadolc, P. and Margaris, B. [2002].
Both for artificial and recorded accelerograms, two steps are essential. At first apply to
the time-histories a baseline correction to remove from the input motion spurious baseline
trends, usually well noticeable in the displacement time-histories obtained from double
time-integration of uncorrected acceleration records: this procedure can be carried out
thanks
to
the
aid
of
the
software
SeismoSignal
v3.3.0
(available
on
www.seismosoft.com); the method as implemented there consists in: (i) determining,
through regression analysis (least-squares-fit method), the polynomial curve that best fits
the time-acceleration pairs of values and then (ii) subtracting from the actual acceleration
values their corresponding counterparts as obtained with the regression-derived equation.
All the time-histories of the accelerograms are detailed in ANNEX 2.
The accelerogram so corrected must be scaled in according to Cauchy similarity law;
moreover, because of the system unit adopted, times need to be multiplied by √1000.
In DIANA models accelerograms have been set loading the time-histories files and
selecting a gravity load case with an horizontal acceleration of 1.00 m/s2.
3.6
BOUNDARY CONDITIONS
The structure has been supported by applying restrains at the bottom joints of the column:
translations in vertical and horizontal directions (TR1, TR2) and rotation in the plane
(RO3) have been restrained. In triple struts models, also the end joints of the lateral struts
that lays at Y = 0 level have been restrained regarding to the translational degrees of
freedom (TR1, TR2).
Then, the joints where concentrated masses have been applied have been restrained
regarding to the translation in the direction orthogonal to the plane (TR3), for the reason
explained in §3.3.1.
A.A. 2008-09
67
Finally, to represent the rigid plane constrain that should take into account the stiffness of
the slabs, linear constrains (called TYING in the code) that equals the horizontal
displacements has been applied to nodes belonging to the beams (EQUAL TR1).
3.7
MESHING
The meshing procedure consist in subdivide the lines that define the geometry of the
structure in smaller elements, so that the code can refine the results. The number of
elements that DIANA create has been set differently for concrete and masonry parts: in
the first case, the fineness of the mesh has been set by imposing a specific length of 200
mm, in according to the value of the concrete crack bandwidth (see §3.4.1); in the second
case, the process was a little different, because has been imposed to the code that the lines
representing the struts to be divided in just one element, in according to the idea that the
axial force on the struts has to be constant.
Figure 3-31. Mesh of the triple strut model 2
The data files of the models are detailed in ANNEX 3.
68
A.A. 2008-09
4 EIGENVALUE ANALYSIS
With the objective to study the modal features of the structure and his dynamic behaviour,
eigenvalue analysis have been carried out on the models previously subjected to pushover
analysis.
At the beginning of this chapter the attention is paid on the analysis procedure, describing
the main characteristics of an eigenvalue analysis as implemented in DIANA.
Then the results, expressed in terms of natural frequencies, periods, participation factors,
mass participation and shape modes are analyzed and commented.
4.1
ANALYSIS PROCEDURE
After the calling of the data files and the verify of its correctness, the type of analysis has
be chosen: in that case structural eigenvalue. The settings are divided in four blocks: the
evaluation of the finite element model, the specification of the type of eigenvalue
problem, the detailing of the analysis execution and the selection of the results wanted in
output.
The first panel has been left unaltered, because all the primary flags (selected by default)
are necessary: evaluation of geometric and material properties for elements and
reinforcements..
In the following module is necessary to specify the problem type, depending on the kind
of matrices that are used to determinate the eigenvalues: free vibration (mass matrices),
standard (identity matrix) and linearized buckling (element geometric stress-stiffness
matrices) are available. Starting from the governing equation of motion for a linear
dynamic finite element system:
A.A. 2008-09
+ + = c
69
where , , (N is the number of equations in the problem) are
c9 is the external forces vector and 9 , 9 , 9 are respectively the
respectively the mass, damping and stiffness matrices of the finite element model,
damping and external forces, and supposing to look for a solution in the form c =
acceleration, velocity and displacement unknown vectors. Considering the absence of
ΦN 5G , the previous equation becomes an eigenvalue problem that assumes, in the three
cases, the following form:
= 1
=
=
free vibration eigenproblem
standard eigenproblem
linearized buckling eigenproblem
where 9 is the eigenvector, ω is the circular natural frequency (ω2 = λ), λ is the
eigenvalue, is the identity matrix and , is the geometric stress-stiffness matrix
of the finite element model. The free vibration problem has been investigated in the
present thesis; considering P circular natural frequencies squared, the problem can be
rewritten as:
£ = £¤
where ¥h¦§ is the matrix with eigenvectors and ¨ §¦§ is the diagonal matrix with the
corresponding eigenvalues. Once the problem type has been decided, in this panel is
necessary to set stiffness matrix type: linear elastic or tangential (from a previous
executed nonlinear analysis); linear elastic is the default option and it has been let
unaltered. It’s also required to specify the kind of mass matrices: consistent or lumped;
the first one, that is the default option, has been considered.
In the execution block the number of eigenpairs to be calculated and the maximum
number of iterations are required: 10 eigenpairs and a maximum of 30 iterations has been
set; DIANA calculates the eigenvalues in ascending order; at most N eigenpairs can be
calculated. DIANA takes reasonable defaults for the maximum number of iterations and
the convergence criterion.
70
A.A. 2008-09
The output datas asked to DIANA for the present free vibration problem are the default
ones: eigenvalues (with relative errors), generalized masses, participation factors,
direction dependent participation vectors, effective masses and eigenmodes. For each
calculated frequency fi , DIANA determines the corresponding generalized mass mii by:
©ªª = «ª ª
with the eigenvectors ¬ normalized such that ©ªª = 1. The participation factors are
given by:
«ª ® ¬
ª =
©ªª
where i is the unity vector, i.e., a vector with a unity displacement for each degree of
freedom. The direction dependent participation factors for the translational and rotational
degrees of freedom in global X , Y , and Z direction are given by:
¯°±ª =
²°±ª
©ªª
=
¯°³ª =
²µ´ª
©ªª
²°³ª
©ªª
¯°´ª =
²°´ª
©ªª
¯µ±ª =
²µ±ª
©ªª
¯µ³ª =
²µ³ª
©ªª
¯µ´ª
Where lt are the coefficient vectors for each translational degree of freedom, lr are the
coefficient vectors for each rotational degree of freedom according to:
¶ª = «ª ® ·
where r is the influence vector which represents the displacements resulting from a static
unit ground displacement in the direction of the corresponding translational or rotational
degree of freedom. Finally, the effective masses meff.i for the translational degrees of
freedom in global X , Y , and Z direction are given by:
¸¹,°±ª =
¶º°±ª
©ªª
¸¹,°³ª =
¶º°³ª
©ªª
¸¹,°´ª =
¶º°´ª
©ªª
The data files of the procedure analysis here described are detailed in ANNEX 4.
A.A. 2008-09
71
4.2
ANALYSIS RESULTS
In the following tables the eigenvalue analysis output results above detailed are presented
separately for single strut model 1, single strut model 2, triple strut model 1 and triple
strut model 2. Just an marginal note is necessary: how explained in §3.5.2, because of the
system unit, for the time depending variables (in this case frequencies) there is a factor of
√1000 that should be considered; so, frequency and period values shown in the tables are
the real ones already modified.
Below the tables the two main mode shapes are presented: the deformed shapes gave back
by DIANA are normalized so that the top displacement is equal to 1.
Table 4-1. Eigenvalue analysis: single strut model 1 results
Mode
Participat.
Eigen
Period
Factor
frequency
Ti [s]
fi [Hz]
γi [ - ]
1
2
3
4
5
6
7
8
9
10
7.01
16.73
17.20
21.07
48.46
58.30
58.88
79.42
106.93
132.78
Mode
1
2
3
4
5
6
7
8
9
10
72
0.143
0.060
0.058
0.047
0.021
0.017
0.017
0.013
0.009
0.008
1.06E+02
4.71E+01
-1.83E+01
5.68E+01
-1.27E+01
8.62E+01
-1.07E+01
2.26E+01
1.42E-01
9.24E+00
General.
Mass
mii [ - ]
[t]
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.27E+04
3.90E-05
5.06E+02
9.55E-09
1.22E+00
5.50E-07
5.94E+00
9.17E-01
8.43E-07
1.54E+01
Direct. Dependent Particip. Factor
ΓtXi
[-]
ΓtYi
[-]
ΓrZi
[-]
1.12E+02
-6.25E-03
-2.25E+01
-9.77E-05
-1.11E+00
-7.41E-04
2.44E+00
-9.58E-01
9.18E-04
3.92E+00
8.24E-04
4.79E+01
-8.97E-03
5.70E+01
-2.66E-03
8.58E+01
2.95E-02
-1.69E-03
7.60E-01
9.81E-04
1.71E+00
-7.99E-01
2.22E+00
-1.37E-01
-1.17E+01
4.08E-01
-1.55E+01
1.66E+01
-6.18E-01
-1.94E+00
Effective Mass
meff,tXi
[%] [%cum]
95.2
0.0
3.8
0.0
0.0
0.0
0.0
0.0
0.0
0.1
95.2
95.2
99.0
99.0
99.0
99.0
99.0
99.0
99.0
99.2
[t]
meff,tYi
[%] [%cum]
6.79E-07
2.29E+03
8.05E-05
3.25E+03
7.06E-06
7.37E+03
8.72E-04
2.85E-06
5.78E-01
9.62E-07
0.0
17.3
0.0
24.4
0.0
55.4
0.0
0.0
0.0
0.0
0.0
17.3
17.3
41.7
41.7
97.1
97.1
97.1
97.1
97.1
A.A. 2008-09
Figure 4-1. Mode shapes of the single strut model 1: mode 1 (left) and mode 2 (right)
Table 4-2. Eigenvalue analysis: single strut model 2 results
Mode
1
2
3
4
5
6
7
8
9
10
Participat. Direct. Dependent Particip. Factor
Eigen
Period
Factor
frequency
ΓtYi
ΓrZi
ΓtXi
Ti [s]
fi [Hz]
γi [ - ]
[-]
[-]
[-]
7.79
0.128 1.07E+02 1.13E+02 -1.27E-03 1.95E+00
18.99
0.053 -1.73E+01 -2.20E+01 -1.09E-03 2.96E+00
38.86
0.026 8.66E+00 -3.66E-02 3.85E-01 8.07E+00
41.54
0.024 1.06E+02
3.17E-03 1.09E+02 -2.62E+00
46.89
0.021 1.26E+01 -1.37E+00 1.09E-01 1.43E+01
58.30
0.017 -3.31E+01 1.20E-03 -2.71E+01 -6.03E+00
62.52
0.016 -5.25E+00 -3.35E-03 -7.89E+00 2.65E+00
65.06
0.015 -2.20E+01 9.98E-01 -1.55E-02 -2.11E+01
116.54
0.009 2.43E+01 -2.02E-02 1.97E+01 4.59E+00
121.79
0.008 2.37E+00 1.63E+00 1.46E-01 -2.22E-01
Mode
General.
Mass
mii [ - ]
1
2
3
4
5
6
7
8
9
10
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
A.A. 2008-09
Effective Mass
meff,tXi
meff,tYi
[t]
[%] [%cum]
[t]
[%] [%cum]
1.27E+04 95.4
95.4
1.61E-06 0.0
0.0
4.83E+02 3.6
99.0
1.18E-06 0.0
0.0
1.34E-03 0.0
99.0
1.49E-01 0.0
0.0
1.00E-05 0.0
99.0
1.18E+04 89.0
89.0
1.88E+00 0.0
99.0
1.20E-02 0.0
89.0
1.45E-06 0.0
99.0
7.34E+02 5.5
94.5
1.12E-05 0.0
99.0
6.23E+01 0.5
95.0
9.96E-01 0.0
99.0
2.39E-04 0.0
95.0
4.09E-04 0.0
99.0
3.89E+02 2.9
97.9
2.65E+00 0.0
99.1
2.13E-02 0.0
97.9
73
Figure 4-2. Mode shapes of the single strut model 2: mode 1 (left) and mode 2 (right)
Table 4-3. Eigenvalue analysis: triple strut model 1 results
Mode
Participat.
Eigen
Period
Factor
frequency
Ti [s]
fi [Hz]
γi [ - ]
1
2
3
4
5
6
7
8
9
10
7.31
17.02
17.92
21.50
48.91
58.38
59.75
79.67
107.31
133.86
Mode
1
2
3
4
5
6
7
8
9
10
74
General.
Mass
mii [ - ]
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.137
0.059
0.056
0.047
0.020
0.017
0.017
0.013
0.009
0.007
1.07E+02
4.37E+01
-1.97E+01
5.88E+01
-1.46E+01
8.65E+01
-1.03E+01
2.09E+01
2.38E+00
1.40E+01
ΓtYi
[-]
ΓtXi
[-]
1.12E+02
-7.00E-01
-2.25E+01
3.37E-03
-6.49E-01
-1.01E-02
2.23E+00
-8.77E-01
1.00E-02
3.90E+00
ΓrZi
[-]
1.22E-02 1.47E+00
4.55E+01 -1.08E+00
-1.34E+00 2.08E+00
5.95E+01 -7.12E-01
-1.28E-02 -1.40E+01
8.55E+01 1.16E+00
4.29E-01 -1.52E+01
6.96E-03 1.60E+01
9.24E-01 1.40E+00
-5.55E-02 2.85E+00
Effective Mass
meff,tXi
meff,tYi
[t]
[%] [%cum]
[t]
[%] [%cum]
1.27E+04 95.2
95.2
1.49E-04 0.0
0.0
4.90E-01 0.0
95.2
2.07E+03 15.5
15.5
5.05E+02 3.8
99.0
1.80E+00 0.0
15.6
1.13E-05 0.0
99.0
3.53E+03 26.6
42.2
4.21E-01 0.0
99.0
1.63E-04 0.0
42.2
1.03E-04 0.0
99.0
7.31E+03 55.0
97.1
4.97E+00 0.0
99.1
1.84E-01 0.0
97.1
7.70E-01 0.0
99.1
4.84E-05 0.0
97.1
1.00E-04 0.0
99.1
8.54E-01 0.0
97.1
1.52E+01 0.1
99.2
3.08E-03 0.0
97.1
A.A. 2008-09
Figure 4-3. Mode shapes of the triple strut model 1: mode 1 (left) and mode 3 (right)
Table 4-4. Eigenvalue analysis: triple strut model 2 results
Mode
Participat.
Eigen
Period
Factor
frequency
Ti [s]
fi [Hz]
γi [ - ]
1
2
3
4
5
6
7
8
9
10
8.08
19.69
39.21
41.61
47.03
58.65
62.80
65.54
117.00
122.27
Mode
1
2
3
4
5
6
7
8
9
10
A.A. 2008-09
General.
Mass
mii [ - ]
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.124
0.051
0.026
0.024
0.021
0.017
0.016
0.015
0.009
0.008
1.09E+02
-1.63E+01
5.80E+00
1.09E+02
1.56E+01
-2.51E+01
-1.00E+01
-2.51E+01
1.62E+01
-3.47E+00
ΓtXi
[-]
ΓtYi
[-]
ΓrZi
[-]
1.13E+02
-2.21E+01
-1.19E-01
4.49E-03
-1.43E+00
3.17E-04
-5.62E-03
1.05E+00
-2.18E-02
1.35E+00
-4.26E-04
5.39E-03
4.97E-01
1.09E+02
1.79E-01
-2.62E+01
-8.43E+00
-1.65E-02
1.96E+01
1.60E-01
2.45E+00
3.91E+00
5.37E+00
5.94E-01
1.76E+01
1.12E+00
-1.48E+00
-2.48E+01
-2.66E+00
-5.61E+00
Effective Mass
meff,tXi
meff,tYi
[t]
[%] [%cum]
[t]
[%] [%cum]
1.27E+04 95.4
95.4
1.81E-07 0.0
0.0
4.88E+02 3.7
99.0
2.91E-05 0.0
0.0
1.41E-02 0.0
99.0
2.47E-01 0.0
0.0
2.02E-05 0.0
99.0
1.19E+04 89.3
89.3
2.03E+00 0.0
99.0
3.20E-02 0.0
89.3
1.00E-07 0.0
99.0
6.88E+02 5.2
94.5
3.16E-05 0.0
99.0
7.11E+01 0.5
95.0
1.11E+00 0.0
99.1
2.73E-04 0.0
95.0
4.73E-04 0.0
99.1
3.86E+02 2.9
97.9
1.83E+00 0.0
99.1
2.57E-02 0.0
97.9
75
Figure 4-4. Mode shapes of the triple strut model 2: mode 1 (left) and mode 2 (right)
How expected, there is no contribution in Z direction translation, in X direction rotation
and in Y direction rotation neither in term of direction dependent participation factor,
neither in terms of effective mass: but it is logical because the model are planar and
created in a X-Y reference system. Rotational modes in XZ plane are obviously not
available to be catch with these models, even if in a regular building is possible that such
modes occur; however, because of the regularity in plan of the structure this fact
shouldn’t occur, and is also for that reason that the code consent planar analysis in such
these cases.
Looking at the previous tables and considering not representative of the building the
mode shapes referring to vertical (Y) direction, is possible to see that there are two
fundamental modes, translational in X direction: first and third modes for models 1, first
and second modes for models 2; it’s evident looking at the effective masses and at the
direction dependent participation factor (its sign in not relevant). These modes got almost
the totality of the effective translational mass in X direction, being the first one always the
prevalent with more than 95%.
Once more, both from results presented in the tables and from the figures representing the
mode shapes, is possible to notice the correspondence between single and triple strut
models: differences are very small: single strut models are just a little less stiffen than
triple strut ones (have 3-4% longer fundament periods), whereas for the other parameters
the divergences are less significant.
76
A.A. 2008-09
At last, an annotation already emerged in analyzing the capacity curves and now
confirmed by the eigenvalue analysis: the bigger stiffness of the frame with two spans and
three column (model 2), both in single and triple strut models.
A.A. 2008-09
77
5 PUSHOVER ANALYSIS
On the same models employed in the previous chapter, nonlinear static (pushover)
analysis have been carried out, both on single and triple strut models. It’s opportune to
remember that two planar models have been realized to represent the structure and that
two load patterns have to be taken into account for a pushover analysis, so that 4 analysis
are necessary to be performed for the single strut model and 4 for the triple strut model.
In the first paragraph of this chapter, the analysis procedure implemented in the code is
detailed, giving some information on the principal aspects, like load step, iteration
method, convergence criterion, etc.
Then the results (expressed in terms of capacity curves, target displacements, interstorey
drifts and solicitations on the structural element of the building) are commented, focusing
on some interesting aspects studied in this thesis and evaluating the accurateness, the
benefits and the deficits of the proposed strut models.
Sensitive analysis with the purpose to calibrate computational and mechanical parameters
have been successively carried out: it’s so possible to decide the most suitable model to
adopt in further dynamic nonlinear analysis in terms of accurateness and required
computational time. Sensitive analysis on infill strength have been also carried out to
analyze its relevance on the global behaviour of the structure.
Finally some safety assessment of primary importance, like limitation of interstorey drift
and brittle shear failure of columns, have been checked.
5.1
ANALYSIS PROCEDURE
After the calling of the data files and the verify of its correctness, the type of analysis has
be chosen: in that case structural nonlinear. The structural nonlinear settings are divided
in four blocks: the evaluation of the finite element model, the specification of the
nonlinearities to be take taken into account, the step execution and the selection of the
analysis results wanted in output.
78
A.A. 2008-09
The first module has been left unaltered, because all the primary flags (selected by
default) are necessary: evaluation of geometric and material properties for elements and
reinforcements; assembly of the elements to create an appropriate system degree of
freedom; setup of the element stiffness matrices; setup of the load vectors.
In the next panel the type of nonlinearities have to be chosen: obviously, just physically
nonlinear effects have been switched on in pushover analysis, while the other options
(geometrically nonlinear, transient and linear stress/strain effects) have been neglected.
In the step execution part details on the load step and on the procedure the code adopt to
carry out the analysis have to been decided. In a static structural problem, the governing
equilibrium relationship, obtained by the principle of virtual displacement, is:
=
where NxN is the known sparse NxN system stiffness matrix with a symmetric structure
(N is the number of equations in the problem), f1xN is the known right-hand-side nodal
forces vector and uNx1 the unknown solution vector of the degrees of freedom to be
computed. In a linear elastic problem, this provides a set of linear simultaneous equations
which can be solved in a direct or indirect way:
= O9
In nonlinear analysis the relation between a force vector and displacement vector is no
longer linear and the displacements often depend on the displacements at earlier stages.
To determine the state of equilibrium is necessary not only to make the problems discrete
in space (with finite elements) but also in time (with increments); to achieve equilibrium
at the end of the increment, iterative solution algorithm has to been used. The
combination of both is called an incremental-iterative solution procedure. A vector of
displacement increments that yields an equilibrium between internal and external forces,
and a stiffness matrix relating internal forces to incremental displacements are employed:
to reach an equilibrium state the internal force vector must equal the external force vector,
satisfying boundary conditions (fint = fext ,
ui = ui0). The system described above is
already discretized in space. To enable a numerical solution, a time discretization is
A.A. 2008-09
79
performed as well (time can have a real physical meaning or can only to describe a
sequence of situations). Starting at time t with an approximated solution tu , a solution
t+∆t
u is searched. Within the time-increment,
time increment, only the displacements at start and end are
known. The internal force vector is calculated from the situation at time t , the time
increment ∆t and the displacement increment ∆u.. The external forces only depend on the
t
current geometry. Considering
ing only one increment, the time increment and the situation at
the start of the increment (history) are fixed. The equilibrium equation within the
increment then only depends on ∆u . Introducing the out-of-balance force vector g (the
residual forces), the
he nonlinear problem can be rewritten as find ∆u such that:
that
t+∆t
u = tu + ∆u
g(∆u) = fext(∆u) - fint(∆u)) = 0
Starting the analysis at time tbegin is possible increment the time with a number of
increments, until the desired end value tend is reached. The pushover analysis have been
carried out in force control: at the beginning 10 steps for vertical loads (10% each step);
then 1200 steps for horizontal loads
load (each one 500 times the starting load).
The option arc-length
length control has been switched on: it’s a method that can adapt the step
size depending on the results in the current step and may be useful in case of local snapsnap
through or snap-back
back behaviour;
behavio in fact, the loading doesn’tt have to be restricted to load
control, displacement control or time increments, but they can be combined; in an
ordinary iteration process (with fixed load increment prescribed) the predictions for the
displacement increments can become very large,
large especially if the load-displace
displacement curve
is subhorizontal: the
he problem can be overcome with the use of an arc-length
length method, that
constrains the norm of the incremental displacements to a prescribed value,
value adapting the
size of the increment.
Figure 5-1. Arc-length control
80
A.A. 2008-09
The regular Newton-Raphson
Raphson iteration method has been set with a maximum of 50
iterations and a convergence criterion
criteri for the equilibrium iteration process based both on
load and on displacement: in fact, besides
esides stopping the iteration in case of convergence,
the iteration process is also stopped if a specified maximum number of iterations has been
reached or if the iterationn obviously leads to divergence; the
he detection of divergence is
based on the same norms as the detection of convergence.
Figure 5-2. Regular Newton-Raphson
Newton
iteration
Figure 5-3. Convergence criterion
criteri
Another option that has been switched on is the line search:: it’s an algorithm that scales
the incremental displacements in the iteration process automatically, with the purpose to
stabilize the convergence
ence behaviour or increase the convergence speed; all iteration
methods described are based on a reasonable prediction,, so that the iteration process
converges to the exact numerical solution, but if the prediction is too far from equilibrium
the iteration process will not converge: this easily takes place in structures with strong
nonlinearities, for instance cracking. Line Search algorithm can increase the convergence
rate and are especially useful if the ordinary iteration process fails; it uses a prediction
predic
of
the iterative displacement increment δu as obtained by one of the ordinary iteration
algorithms and scales this vector by a value to minimize the energy potential: while the
local minimum of the energy potential represents the equilibrium, the minimum
mini
in the
line search direction can be regarded as the best solution in the predicted direction.
Finally, all the datas essential to represent the capacity curve, to get informations about
interstorey drift and to display the solicitations have been required
required in the output window.
The data files of the analysis procedure here described
escribed are detailed in ANNEX 5.
5
A.A. 2008-09
81
5.2
ANALYSIS RESULTS
The main results of a pushover analysis are the capacity curve of the structure and the
localization on it of the target displacement, that represent the seismic demand. Hence
more emphasis will be focus on dealing with these topics; however, also the interstorey
drifts and the solicitations on the structural element will be analyzed.
5.2.1 Capacity Curve
The capacity curves are expressed like base reaction vs. displacement of the control point
(one of the points at the top of the structure has been considered); in the next two figures
Force [kN]
the capacity curves of the single strut models are presented:
90
80
70
60
50
40
30
20
10
0
uniform load pattern
modal load pattern
0
10
20
30
40
50
displacement [mm]
60
70
Force [kN]
Figure 5-4. Capacity curve: single strut model 1
90
80
70
60
50
40
30
20
10
0
uniform load pattern
modal load pattern
0
10
20
30
40
50
60
70
displacement [mm]
Figure 5-5. Capacity curve: single strut model 2
82
A.A. 2008-09
Some conclusion can be argued by analyzing the capacity curves shown in the previous
figures. At first is evident the very strong similarity in results between the curves obtained
by applying an uniform load pattern and those obtained by applying a modal load pattern:
this fact could be explained by considering that the global resultant force of the patterns
generates in the two cases a similar value of bending moment (the distance where it is
applied is similar, see Fig.3-23).
Another aspect to remark is the bigger stiffness of the model 2, but also this feature is
quite obvious, because in that direction all the structure is stiffener: in fact in that frame
there are three columns (in the other frame just two) and moreover the spans are littler
(2.075 m vs. 3.650 m).
Finally, is interesting to notice the shape of the curves: there is an ascending and a
descending branch (creating something like to a parabola), and after the curve stabilizes
on a subhorizontal branch; how will be presented just now, this is in total accord with the
materials constitutive law adopted. The capacity curves above displayed are compared
with the components of the bare frame and of the infills: it’s so possible to analyze the
resistance of each element of the complete structure:
90
Complete Structure
Lower Infill
Upper Infill
Bare Frame
80
Force [kN]
70
60
50
40
30
20
10
0
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-6. Capacity curve and its components: single strut model 1 – uniform load pattern
A.A. 2008-09
83
90
80
Complete Structure
Lower Infill
Upper Infill
Bare Frame
Force [kN]
70
60
50
40
30
20
10
0
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-7. Capacity curve and its components: single strut model 1 – modal load pattern
100
90
80
70
60
50
40
30
20
10
0
Force [kN]
Complete Structure
Bare Frame
Lower Left Infill
Lower Right Infill
Upper Left Infill
Upper Right Infill
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-8. Capacity curve and its components: single strut model 2 – uniform load pattern
100
90
80
70
60
50
40
30
20
10
0
Force [kN]
Complete Structure
Lower Left Infill
Bare Frame
Lower Right Infill
Upper Left Infill
Upper Right Infill
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-9. Capacity curve and its components: single strut model 2 – modal load pattern
84
A.A. 2008-09
Looking at the diagrams above presented, is evident how much infills affect the capacity
curve of the infilled frame in the first part, adding stiffness and resistance respect to the
capacity curve of the bare frame (got by a separate pushover analysis); then, after that the
lower infill fails, the curve stabilize on the final plastic branch of the bare frame curve;
the upper infill doesn’t reach his failure strength and maintain a low residual resistance.
These results show once more the benefit brought by the infills to the structure and the
importance of take them into account in structural analysis.
The capacity curves of the triple strut models, compared with the relative ones of the
Force [kN]
single strut models, are now presented:
90
80
70
60
50
40
30
20
10
0
triple struts
single strut
0
10
20
30
40
50
60
70
displacement [mm]
Force [kN]
Figure 5-10. Capacity curve: single and triple strut model 1 – uniform load pattern
90
80
70
60
50
40
30
20
10
0
triple struts
single strut
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-11. Capacity curve: single and triple strut model 1 – modal load pattern
A.A. 2008-09
85
Force [kN]
90
80
70
60
50
40
30
20
10
0
triple struts
single strut
0
10
20
30
40
50
displacement [mm]
60
70
Force [kN]
Figure 5-12. Capacity curve: single and triple strut model 2 – uniform load pattern
90
80
70
60
50
40
30
20
10
0
triple struts
single strut
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-13. Capacity curve: single and triple strut model 2 – modal load pattern
The two type of models are very similar in terms of capacity curves: both in the shape
(triple strut ones seem just a little stiffener) and maximum force values: the single strut
model underestimates the force values, but this difference is quite negligible.
Table 5-1. Capacity curves: maximum force values
SINGLE STRUT
MODEL 1
MODEL 2
UNIFORM
MODAL
UNIFORM
MODAL
FMAX
FMAX
FMAX
FMAX
FMAX
FMAX
FMAX
FMAX
[kN]
[kN]
[kN]
[kN]
[kN]
[kN]
[kN]
[kN]
83.46
83.00
82.51
82.29
85.21
84.71
84.57
84.36
86
UNIFORM MODAL
TRIPLE STRUTS
MODEL 1
MODEL 2
UNIFORM MODAL
A.A. 2008-09
5.2.2 Target Displacement
Among the methods proposed by various authors, the C.S.M. (capacity spectrum method,
Freeman S.A. [1998]) and N2 method (Fajfar P., Gaspersič P. [1996]) are those that
found more success. In both methods there are two main phases: the first consisting in
defining the capacity curve of the structure through a static incremental analysis, and the
second consisting in determining on that curve a point corresponding to the inelastic
response of the frame consequent to the assigned P.G.A.: this is obtained by the study of
S.D.O.F. (single degree of freedom) equivalent to the real M.D.O.F. (multi degree of
freedom).
In fact, in the European Standards, the target displacement is defined as the seismic
demand derived from the elastic response spectrum in terms of displacement of an
equivalent single-degree-of-freedom system [EN 1998-1:2003 – 4.3.3.4.2.6]; the N2
procedure for the determination of the target displacement from the elastic response
spectrum is adopted, even if it is slightly modified [EN 1998-1:2003 – ANNEX B].
Starting from the M.D.O.F. capacity curve obtained by a pushover analysis, the S.D.O.F.
is get by dividing shear and displacement values for the participation factor Γ.
∗ =
:
Γ
/∗ =
/6
Γ
where Fb and dn are respectively the base shear force and the control node displacement
of the M.D.O.F. system, whilst F* and d* are the corresponding variables of the S.D.O.F.
system. The participation factor can be calculated as:
Γ = ∑ F ¾¿
F∗
3 3
with
&∗ = ∑ &5 À5
and À5 is the displacement in the i-th storey normalized in such a way that À6 =1 (n is the
where m* is the mass of the equivalent S.D.O.F. system, mi is the mass in the i-th storey
control node). The curve is thus bi-linearized with an elasto-perfectly plastic relationship,
based on the equal energy criterion applied up to the formation of the plastic mechanism:
A.A. 2008-09
87
/d =
∗
2 /F
∗
KF
− ∗
d
Figure 5-14. N2 method: bi-linearization of the S.D.O.F. capacity curve
The remaining part of the procedure is not explained here because this method has been
considered not suitable to be used in the present work: represent the capacity curve with
an elasto-perfectly plastic relationship is here supposed to be a rough approximation for
r.c. infilled structures, because they are characterized by a substantial strength
degradation: the original N2 method has been conceived for r.c. structures where infills
are not considered resistant elements in structural analysis.
Figure 5-15. Hysteretic behaviour of the equivalent S.D.O.F. system
The N2 method has been extended by Dolšek M., Fajfar P. [2004] in order to make it
applicable to infilled r.c. frames. Compared to the simple basic variant of the N2 method,
two important differences are apply: a multi-linear idealization of the pushover curve,
which takes into account the strength degradation which occurs after the infill fails, and
specific reduction factors (R-µ-T relation), developed in a companion paper (Dolšek M.,
Fajfar P. [2004]) for the determination of inelastic spectra.
So, the first step is to define an idealization of the pushover curve that take into account
also the degrading brunch: the authors propose a multi-linear based on equal energy
criterion and on two characteristic points of the capacity curve, those corresponding to the
maximum and minimum force (DF_max, Fmax) and (DF_min, Fmin):
88
A.A. 2008-09
Figure 5-16. Idealization of the capacity curve in the N2 extended method
The displacement at yielding and at the start of the degradation are so given by:
Ád = 2 ÂÁU_F; −
ÁÆ =
KÄ,U_F;
Å
F;
2
ÇK
− KÄ,U_F56 + F; ÁU_F; − 0.5ÁU_F56 F; − F56 È
F; − F56 Ä,U_F;
The M.D.O.F. idealized curve above defined must be divided for the participation factor
Γ to get the equivalent S.D.O.F. curve; then, its ordinates must be divided for the mass of
the equivalent S.D.O.F. system, m*, so that is possible to plot it in the ADRS format
together with the elastic response spectrum. Now, the R-µ-T relation has to be introduced
to get the inelastic spectrum, whose intersection with the extension of the horizontal yield
plateau of the capacity diagram define the demand point. In the previous relation, R is the
reduction factor and µ is the ductility demand: the inelastic spectrum is get by dividing
the elastic spectral acceleration for the reduction factor R and multiplying the elastic
spectral displacements by the ratio µ / R.
The R-µ-T relation has been determined by the authors on the base of an extensive
parametric study employing a S.D.O.F. mathematical model composed of structural
elements representing the r.c. frame and the infill: different ground motion sets have been
used as seismic input in the nonlinear dynamic analysis. To get the relation, the following
parameters have been varied in the study: the normalized initial ratio T / TC , the ultimate
reduction factor ru = Fmax / Fmin , the ductility at the beginning of the strength degradation
µs = Ds / Dy and the ductility corresponding at the infill zero strength µu = DF_min / Dy .
A.A. 2008-09
89
Analysis results are expressed in terms of maximum displacement and corresponding
ductility µ , defined as the ratio between the maximum displacement and the displacement
at the beginning of the nonlinear behaviour: results show a negligible dependence on the
ductility µu and this parameter has been thus not considered in the formulation.
Expressing µ as a function of R, the relation is:
É=
where
=
Ï
Í
Í
Í
Í
Í
0.7
1
Ë − Ë + É
Ê
Ê
p
p>
Ë ≤ ËÉÆ , p ≤ p>
Ë ≤ ËÉÆ , p> < p ≤ px ∗
0.7 + 0.3 ∆p
p
0.7 ÑD Â Å
p>
9
}Ò
Î
Í
Í
Í0.7 ÑD 1 − ∆p + ∆p
Í
Í
Ì
1
Ë > ËÉÆ , p> < p ≤ px ∗
p > px ∗
with
É = Õ
px ∗ = px 2 − ÑD
Ë ≤ ËÉÆ 1
Ë > ËÉÆ ÉÆ
where
ËÉÆ =
90
Ï
Í
Ô
Ô
Ë > ËÉÆ , p ≤ p>
,
∆p =
p
0.7 Â Å ÉÆ − 1 + 1
p>
p − p`
px ∗ − p`
Ë = Õ
Î 0.7 + 0.3 ∆pÉÆ − 1 + 1
Í
Ì
ÉÆ
1
ËÉÆ p ≤ p>
Ë ≤ ËÉÆ Ë > ËÉÆ p> < p ≤ px ∗
p > px ∗
Ô
Ô
A.A. 2008-09
Otherwise, changing the intervals for the reduction factors (µ ≤ µs and µ > µs instead of R
≤ R(µs) and R > R(µs)) the basic equation can be rewritten in the inverted relation to
determine the reduction factor and plot the inelastic spectrum for constant ductility:
Ë = ÊÉ − É + Ë
In applying this method, ductility demand has to be found for a constant reduction factor
R (obtained by the ratio between the elastic spectral acceleration at the initial period T of
the given system Sae and the yield acceleration of the inelastic system Say); successively,
the inverted relation can be used to obtain the reduction factor R and so define the
inelastic spectrum.
Figure 5-17. Elastic and inelastic spectra versus capacity diagram
The method above explained has been adopted to get the target displacement of the
building objective of the present thesis, considering the triple strut model capacity curves
detailed in the previous paragraph. Because of the strong similarity in the pushover curves
obtained with the two load pattern, just one has been taken into account: that got applying
the uniform load pattern; of course the target displacement won’t be too much different if
the other curve has been considered.
Table 5-2. Parameters defining the idealized pushover curve in the N2 extended method
Model 1
Model 2
m*
[t]
Γ
[-]
DFmax
[mm]
Fmax
[kN]
DFmin
[mm]
Fmin
[kN]
10.26
10.29
1.23
1.23
11.44
9.05
85.21
84.57
17.43
14.46
55.76
50.93
A.A. 2008-09
Dy
[kN*mm] [kN*mm] [mm]
EFmax
EFmin
650.95
511.31
1099.13
913.61
7.60
6.01
Ds
[mm]
13.20
11.19
91
pushover curve
idealised curve
80
70
60
50
40
30
20
10
90
Base shear force FB [kN]
Base shear force FB [kN]
90
pushover curve
idealised curve
80
70
60
50
40
30
20
10
0
0
0
20
40
60
Control node displacement dn [mm]
0
20
40
60
Control node displacement dn [mm]
Figure 5-18. Idealization of the capacity curve: model 1 (left) and model 2 (right)
The capacity curves have been idealized taking into account not their minimum value but
a value slightly higher that consent a better fitting of the multi-linear with the original
curve.
Considering for this type of building a reasonable value of 2% for the viscous damping
ratio (Faria R. [1994]), the elastic response spectrum to be drown in the spectral domain
has been created. In the next table the parameters necessary to get the inelastic spectrum
and the resulting target displacements are presented for the two models and for the two
types of spectrum: note that target displacements shown in the table are those of the
M.D.O.F. system, obtained by multiplying the ones of the S.D.O.F. system by the
participation factor Γ.
Table 5-3. Parameters of R-µ-T relation in the N2 extended method
Spectrum
Type
Type 1
Type 2
92
Model
µs
[-]
ru
[-]
T*
[s]
µ
[-]
T.D.
[mm]
Model 1
Model 2
Model 1
Model 2
1.737
1.861
1.737
1.861
0.654
0.602
0.654
0.602
0.190
0.170
0.190
0.170
1.000
1.018
1.014
1.197
7.4
5.9
7.5
7.5
A.A. 2008-09
7
6
5
4
3
capacity curve
elastic spectrum
inelastic spectrum
2
1
0
0,00
Type 1 specreum - Model 2
Spectral acceleration Sa [m/s2]
Type 1 spectrum - Model 1
8
8
7
6
5
4
3
2
1
0
0,01 0,02 0,03 0,04 0,05
Spectral displacement Sd [m]
0,00
capacity curve
elastic spectrum
inelastic spectrum
0,00
0,01 0,02 0,03 0,04 0,05
0,01 0,02 0,03 0,04 0,05
8
7
6
5
4
3
2
1
0
capacity curve
elastic spectrum
inelastic spectrum
8
7
6
5
capacity curve
elastic spectrum
inelastic spectrum
4
3
2
1
0
0,00
0,01 0,02 0,03 0,04 0,05
Figure 5-19. Individuation of the target displacement in the N2 extended method
Standing at results obtained with the N2 extended method, even if the authors stated that
this method is quite conservative, target displacements found in the present case are
small: the structure remain in the elastic branch or quite near to the peak value of the
capacity curve; it means that infills shouldn’t fail.
Understanding the relevance and the innovation of the proposed method in dealing with
target displacement of infilled r.c. structures, is important to state that probably more
researches and assessments are needed to improve the method because in some cases it
can provide unreliable results: i.e., in the present case doesn’t convince the fact that, for
type 2 spectrum, the procedure leads to the same result for both models, whereas lower
value are expected for model 2 (as in the case of type 1 spectrum) because of its higher
stiffness. Studies carried out by the authors are mostly focused on higher level of ductility
and reduction factor; moreover it’s important to remember that the N2 extended method
A.A. 2008-09
93
is rather recent (2004), and for that reason has been applied in quite a few cases of study
(an interesting application carried out by the same authors is present in Dolšek M., Fajfar
P. [2008]). Hence, to determine the target displacement of the structure objective of the
present thesis, also the capacity spectrum method has been taken into account.
The capacity spectrum method is a static nonlinear procedure to get the maximum
displacement of a structure subjected to a seismic event: seismic action is represented by
the elastic response spectrum, whereas structure behaviour is represented by the capacity
curve of the equivalent S.D.O.F. system. As in the previous method, demand and capacity
are plotted in the ADRS format: once again, the target displacement is individuated as the
intersection of the demand curve with the capacity curve. The procedure is iterative
because the elastic spectrum needs to be reduced to take into account the effective
damping (different from that used to get the elastic response spectrum), and it depend on
the target displacement (Albanesi T. and Nuti C. [2007]).
After defining the elastic response spectrum (2% damping in the present case) and the
capacity curve, and after transforming these curves to plot them in the spectral domain, a
first attempt point must be selected on the capacity curve, i.e. that whose displacement
correspond to the displacement of an elastic system approximating the S.D.O.F. system.
Figure 5-20. Choose of the displacement of first attempt
Then, the spectral capacity curve is approximate with a bilinear adopting the equal energy
criterion; the parameters that define the bilinear curve are: the elastic circular frequency
ωe (proceedings of the elastic period), the yielding spectral acceleration ay, and the
degrading factor p defined as the ratio between the post-elastic and the elastic stiffness:
94
A.A. 2008-09
Ö
1E />
Bd + × 1E Ø/> − /d Ù
/ > ≤ /d
/ > > /d
Ô
where dy is the yielding displacement defined as: /d = W Û \ Bd .
Ú
Ü
Figure 5-21. Bilinear representation of the capacity spectrum corresponding to dC,i
It’s supposed that the response of the bilinear system corresponding to the generic
displacement dC is equal to the response of an equivalent linear system characterized by a
the following period and viscous damping:
pEÝ =
2Þ
/`
= 2Þß
1EÝ
B`
àEÝ = à + á àÄ = 2% + á
B d /` − B ` /d
B ` /`
Figure 5-22. Equivalent viscous damping associated to energy hysteretic dissipation
A.A. 2008-09
95
The factor κ depends on the hysteretic behaviour of the system, i.e. on the behaviour
category of the structure, defined both by the quality of the element that form the seismic
resistant mechanism and by the duration of the earthquake. ATC-40 defines three
behaviour categories: type A indicates a structure with ample and stable hysteretic cycles
(almost ideal behaviour); type C indicates a structure with degraded hysteretic cycles
(pinching phenomena); type B indicates a structure with an intermediate behaviour
between type A and type B. Supposing a pinched structural behaviour with negligible
hysteretic dissipation, type C values has been considered (κ = 0.33); this hypothesis could
be justified by considering that, according to results got by N2 extended method, the
building doesn’t entry in an advanced plastic or degrading phase; however, only dynamic
nonlinear analysis results can give a certain validation of that hypothesis.
Figure 5-23. κ factor diagram
It’s so possible now to calculate the factor necessary to reduce the elastic response spectra
for considering hysteretic energy dissipated by the system:
10
{=ß
5 + àEÝ
The target displacement dC,j is obtained as the intersection of the reduced response
spectrum with the capacity spectrum. If this value is not so far from the previous attempt
dC,i (i.e. 5% tolerance), the target displacement could be considered determined;
otherwise, this value is considered to be the starting point of another iteration.
96
A.A. 2008-09
Finally, once the procedure converge (in the present cases 4 or 5 iterations are necessary),
the target displacement of the M.D.O.F. system is get by multiplying that of the S.D.O.F.
system by the participation factor Γ.
Figure 5-24. Determination of the target displacement
In the next tables the datas calculated with the purpose to determine the target
displacements are presented; as done for the N2 extended method, triple strut models
have been considered and, because of the strong similarity between the capacity curves in
the case of uniform and modal pattern load, just one of this two cases is analyzed, that
regarding the uniform pattern load.
Target displacement results are shown for both frames (model 1 and 2) and for both types
of spectrum (type 1 and 2):
Table 5-4. Model 1 type 1 spectrum: determination of the T.D. with CSM
1
2
3
4
dC,i
aC,i
[mm] [m/s2]
3.2
3.91
6.0
6.07
5.2
5.54
5.4
5.71
A.A. 2008-09
Te
[s]
0.137
0.137
0.137
0.137
aY
dY
p
Teq
[m/s2] [mm] [-]
[s]
0.695 0.3 0.532 0.180
1.820 0.9 0.389 0.198
1.430 0.7 0.434 0.192
1.540 0.7 0.421 0.194
T.D. [mm]
νeq
[%]
3.6
5.3
4.7
4.8
dC,J
aC,J
η
[-] [mm] [m/s2]
1.080 6.0
6.07
0.985 5.2
5.54
1.017 5.4
5.71
1.008 5.3
5.66
6.6
97
1
2
3
4
aC,i
dC,i
[mm] [m/s2]
2.6
3.92
4.8
5.94
4.1
5.45
4.3
5.61
Te
[s]
0.124
0.124
0.124
0.124
aY
dY
p
Teq
2
[m/s ] [mm] [-]
[s]
0.895 0.3 0.533 0.160
1.975 0.8 0.386 0.178
1.615 0.6 0.431 0.172
1.710 0.7 0.418 0.174
T.D. [mm]
νeq
[%]
3.9
5.6
5.0
5.1
aC,J
dC,J
η
2
[-] [mm] [m/s ]
1.058 4.8
5.94
0.971 4.1
5.45
1.000 4.3
5.61
0.933 4.2
5.58
5.2
1
2
3
4
5
dC,i
aC,i
[mm] [m/s2]
3.6
4.28
8.2
6.81
5.4
5.71
6.8
6.43
6.1
6.09
Te
[s]
0.137
0.137
0.137
0.137
0.137
aY
dY
p
Teq
2
[m/s ] [mm] [-]
[s]
0.830 0.4 0.512 0.182
2.935 1.4 0.271 0.218
1.540 0.7 0.421 0.194
2.180 1.0 0.350 0.204
1.835 0.9 0.388 0.199
T.D. [mm]
νeq
[%]
3.8
7.5
4.8
5.9
5.3
dC,J
aC,J
η
[-] [mm] [m/s2]
1.068 8.2
6.81
0.895 5.4
5.71
1.008 6.8
6.43
0.957 6.1
6.09
0.984 6.1
6.11
7.5
1
2
3
4
5
dC,i
aC,i
[mm] [m/s2]
3.0
4.43
6.1
6.63
4.5
5.79
5.3
6.25
5.0
6.07
Te
[s]
0.124
0.124
0.124
0.124
0.124
aY
dY
p
Teq
2
[m/s ] [mm] [-]
[s]
1.105 0.4 0.501 0.164
2.780 1.1 0.295 0.191
1.855 0.7 0.401 0.176
2.250 0.9 0.354 0.182
2.065 0.8 0.375 0.179
T.D. [mm]
νeq
[%]
4.2
7.1
5.4
6.1
5.7
aC,J
dC,J
η
2
[-] [mm] [m/s ]
1.040 6.1
6.63
0.908 4.5
5.79
0.981 5.3
6.25
0.950 4.9
6.05
0.965 5.0
6.07
6.1
From the previous tables is possible to check that the parameters necessary to determine
the target displacement, and also the same target displacement, converge quickly at the
fourth or at the fifth iteration.
As example of the iterative procedure, ADRS format graphics for any iteration is shown
just for one of this four cases (model 1 – type 1 spectrum) in the next figures:
98
A.A. 2008-09
iteration 1
[m/s2]
6
6
4
elastic spectum
capacity spectrum
elastic capacity spectrum
bilinear capacity spectrum
reduced spectum
2
0
0,000
0,002
0,004
0,006
0,008
4
0
0,000
0,002
0,004
0,006
0,008
iteration 3
iteration 4
8
6
4
elastic spectum
capacity spectrum
reduced spectum
2
0
0,000
0,002
0,004
0,006
0,008
Spectral acceleration Sa
[m/s2]
[m/s2]
8
elastic spectum
capacity spectrum
reduced spectum
2
iteration 2
8
8
6
4
elastic spectum
capacity spectrum
reduced spectum
2
0
0,000
0,002
0,004
0,006
0,008
Figure 5-25. Individuation of the target displacement in the CSM: model 1, type 1 spectrum
Three aspects are remarkable: first of all, on the contrary of what happen using the N2
extended method, the target displacement is lower respect for the model 2 respect to that
calculated for the model 1 considering both types of spectra; furthermore, in all cases, it
belongs to the horizontal branch of the spectrum and to the elastic pre-peak branch of the
capacity curve, and it means that infills still haven’t collapsed; the latter fact means also
that target displacement calculated in the case of type 2 spectrum are higher than those
calculated for type 1 spectrum, because this type of spectrum is supposed to be critical for
stiff structures.
A.A. 2008-09
99
Table 5-8. T.D.: N2 extended method vs. CSM
T.D. [mm]
Spectrum
Type
Type 1
Type 2
Model
N2 extended method
Model 1
Model 2
Model 1
Model 2
Model
7.4
5.9
7.5
7.5
Prototype
11.1
8.8
11.2
11.2
CSM
Model
6.6
5.2
7.5
6.1
Prototype
9.9
7.8
11.3
9.1
How is possible to see from the previous table, target displacements calculated with the
two methods don’t differ too much for type 1 spectrum: the C.S.M. gives values a little
bit lower. Regarding to type 2 spectrum, has been already said that probably N2 extended
method doesn’t give reliable results, so comparisons with C.S.M. is senseless.
5.2.3 Interstorey Drift
Regarding the analysis step related to the target displacement above calculated, storey
displacements have been checked to compute the interstorey drift dr , evaluated as the
r
difference of the average lateral displacement at the top and at the bottom of the storey.
Table 5-9. T.D.: interstorey drift values: triple strut model, uniform load pattern. CSM
dr,0-1 / h
dr,1-2 / h
Type 1 Spectrum
Model 1 Model 2
0.25%
0.20%
0.08%
0.06%
Type 2 Spectrum
Model 1 Model 2
0.29%
0.24%
0.09%
0.06%
Figure 5-26. T.D. def. shape: type 1 spectrum - triple strut model 1 (left) and 2 (right). CSM
100
A.A. 2008-09
Figure 5-27. T.D. def. shape: type 2 spectrum - triple strut model 1 (left) and 2 (right). CSM
5.2.4 Solicitations
In the following figures, solicitations on the structure at the step corresponding to the
target displacement are shown for the uniform load pattern case in the triple strut model 1
and type 1 spectrum; to check all the cases, look at ANNEX 6.
Figure 5-28. T.D.: bending moment MZ (left); NX and NY force resultant (right). CSM
A.A. 2008-09
101
Figure 5-29. T.D.: NX force (left); NY force (right). CSM
5.3
SENSITIVE ANALYSIS
Starting from the model described in chapter 3 and whose capacity curves have been
detailed in the previous paragraph (CLII_7-5_7: class II element both for beam and truss
elements, with 7 integration points along the bar axis for beam elements, 5 integration
points along the section height for beam elements and 7 integration points along the bar
axis for truss elements) sensitive analysis have been carried out with to calibrate some
model’s parameters. The models analyzed are:
CLII_7-7_7: class II element both for beam and truss elements, with 7 integration
points along the bar axis for beams, 7 integration points along the section height for
beams and 7 integration points along the bar axis for trusses;
CLIII_4-7_7: class III element both for beams and trusses, with 4 integration points
along the bar axis for beams (maximum available in DIANA), 7 integration points
along the section height for beams and 7 integration points along the bar axis for
trusses;
CLIII-II_4-7_7: class III element for beams and class II element for trusses, with 4
integration points along the bar axis for beams, 7 integration points along the
section height for beams and 7 integration points along the bar axis for trusses;
102
A.A. 2008-09
CLIII-II_4-7_7_0.05: class III element for beams and class II element for trusses,
with 4 integration points along the bar axis for beams, 7 integration points along the
section height for beams and 7 integration points along the bar axis for trusses; the
shear retention factor has been assumed equal to 0.05;
CLIII-II_4-7_7_0.25: class III element for beams and class II element for trusses,
with 4 integration points along the bar axis for beams, 7 integration points along the
section height for beams and 7 integration points along the bar axis for trusses; the
shear retention factor has been assumed equal to 0.25;
CLII_7-7_7_def: class II element both for beams and trusses, with 7 integration
points along the bar axis for beams, 7 integration points along the section height for
beams and 7 integration points along the bar axis for trusses; the default value has
been assumed for the crack bandwidth;
CLII_7-7_bare-frame: class II element for beams, with 7 integration points along
the bar axis and 7 integration points along the section height.
In this way is possible to study the influence of:
o the number of integration points along the section height (CLII_7-5_7 vs. CLII_77_7 and CLIII-II_4-7_7 vs. CLIII-II_4-5_7);
o the element class (CLII_7-7_7 vs. CLIII_4-7_7 vs. CLIII-II_4-7_7);
o the number of integration points along the bar axis of truss elements (CLIII-II_47_7 vs. CLIII-II_4-7_3);
o the shear retention factor (CLIII-II_4-7_7_0.05 vs. CLIII-II_4-7_7_0.25);
o the crack bandwidth (CLII_7-7_7 vs. CLII_7-7_7_def);
o the presence of the struts (CLII_7-7_7 vs. CLII_7-7_bare-frame).
A.A. 2008-09
103
In the next diagram the results of these analysis in terms of capacity curves are presented:
the analyzed model is the single strut model 1, with the uniform load pattern applied.
Force [kN]
CLII_7-5_7
CLIII-II_4-7_7
CLII_7-7_7
CLIII-II_4-5_7
120
110
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
displacement [mm]
CLII_7-5_7
CLIII-II_4-7_7_0.25
Force [kN]
CLIII_4-7_7
CLIII-II_4-7_3
60
70
CLIII-II_4-7_7_0.05
CLII_7-7_7_def
120
110
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-30. Sensitive analysis results: capacity curves
From what shown in the previous capacity curves, it’s possible to asses that increasing the
number of integration point along the section height from 5 to 7 the results (that should be
more accurate), in both cases, are just a little greater, but not in a relevant quantity.
104
A.A. 2008-09
Looking at the class element we can see that class III truss elements are not suitable to
model struts, because they are curved elements with more d.o.f. than the class II ones, and
for this reason they are mostly used to model, i.e., prestress cables; on the other side, the
models with class III element for beams and columns and class II elements for the struts
give results just a little lower because the shear deformation is now taken into account.
About the number of integration point along the bar axis of truss elements the results are
exactly the same, the two curves are equivalent: increase them doesn’t bring to any effort.
Also the shear retention factor has no effects in the capacity curves results.
On the other side, the crack bandwidth influence a lot the response: DIANA default value
for beam elements is defined as the length of the element (200 mm, like the crack
bandwidth value adopted in other analysis), whereas for truss elements the default value
is 1 (in other analysis it correspond to the length of the trusses). Since compressive
constitutive models are governed by the ratio between compressive fracture energy and
crack bandwidth, it’s supposed to expect that changing in one of these parameters led to
relevant modifies in pushover curves.
Finally the presence of infills bring to the structure a remarkable increase of stiffness and
resistance, how confirmed by lot of studies and also already stated in the present work.
What said above can also be supported by the following table where the peak values of
the capacity curves are presented:
Table 5-10. Sensitive analysis results: capacity curves peak values
PEAK
VALUES
[kN]
CLII
7-5_7
83.46
CLIII-II
4-7_3
83.13
CLII
CLIII
CLIII-II
CLIII-II
7-7_7
4-7_7
4-7_7
4-5_7
83.37
54.94
83.13
83.22
CLIII-II
CLIII-II
CLII
CLII
4-7_7_0.05 4-7_7_0.25 7-7_7_def 7-7_bare-fr.
83.10
83.14
113.74
58.05
At the end of this sensitive analysis has been decided to choose for the further analysis the
base model (CLII_7-5_7), the one with class II element (both for beam and truss
elements), with 7 integration points along the bar axis (both for beam and truss elements)
A.A. 2008-09
105
and 5 integration points along the section height. This is the simplest model of the
analyzed ones, but has been chosen because increasing the accuracy of the model, for
example with more integration point or with superior class element, doesn’t bring to
significant changes in results; also the chosen values of the shear retention factor is good,
because this parameters doesn’t affect in a significant way the results, whereas the crack
bandwidth is a relevant parameter.
Now some sensitive analysis on the infill’s compression strength are presented, just to
understand how much infills influence the global response of the structure. The following
results are referred to single strut model 1 (the same which the previous diagrams are
Force [kN]
referred) with the uniform load pattern applied.
240
220
200
180
160
140
120
100
80
60
40
20
0
fc,E
2fc,2E
2fc,E
3fc,3E
3fc,E
0
10
20
30
40
50
displacement [mm]
60
70
Figure 5-31. Influence of the infill’s compressive strength: capacity curves
Table 5-11. Influence of the infill’s compressive strength: capacity curves peak values
PEAK
VALUES
[kN]
fC,E
2fC,2E
2fC,E
3fC,3E
3fC,E
83.46
137.60
158.54
193.15
222.56
Analysis have been carried out both increasing just the compressive strength and also the
elasticity module (according the Eurocode it is related to the compressive strength with
the following formula: E=1000⋅fC). How expected, stiffener models reach lower peak
values than ductile ones, but anyway infills contribution is very important.
106
A.A. 2008-09
5.4
SAFETY ASSESSMENTS
European Standards provide safety verification criteria regarding to the relevant limit
states: no-collapse requirement (ultimate l.s.) and damage limitation requirement. In the
first case, assessments regarding to resistance, ductility, equilibrium, foundation stability
and seismic joints have to be carried out, whereas in the second case limitation on
interstorey drift have to been satisfied [EN 1998-1:2003 – 4.4]. In the present work two of
these aspects have been analyzed: limitation of interstorey drift and shear resistance.
5.4.1 Limitation of Interstorey Drift
The damage limitation requirement, for buildings having non-structural elements of
brittle materials attached to the structure, is considered to have been satisfied if the
following limit is observed:
/} à ≤ 0.005 ℎ
where dr is the interstorey drift, h is the storey height (2.00 m in the DIANA models), ν
is the reduction factor which takes into account the lower return period of the seismic
action associated with the damage limitation requirement and whose recommended value
is 0.50. Values of ν related to the seismic action (0.40 and 0.55 respectively for type 1
and type 2 spectrum) can be found in the nation annex [NP EN 1998-1:2006 – NA.4.4.3.2
l]; because pushover analysis doesn’t take into account the spectrum type, the EC8
recommended valued has been here adopted.
Table 5-12. Interstorey drift safety assessment
dr,0-1 / h
dr,1-2 / h
Type 1 Spectrum
Model 1 Model 2
OK
OK
OK
OK
Type 2 Spectrum
Model 1 Model 2
OK
OK
OK
OK
Considering that the European Standards limit value of the ratio dr / h , as computed by
the previous formula, are 1.25% and 0.91% respectively for type 1 and type 2 spectrum,
and considering the obtained values shown in Table 5.9, the damage limitation
requirement is adequately satisfied for the considerate limit state.
A.A. 2008-09
107
5.4.2 Shear Resistance
It is of primary importance to prevent a brittle failure of the structure: the most common
cause of such a collapse mechanism is due to brittle shear failure of columns. In this
paragraph shear values (proceedings of pushover analysis) in the most critical zones of
the building have been compared with shear strength, both in according to the European
Standards formula [EN 1992-1-1:2004 – 6.2.3] and in according to a more refined
relationship proposed by Priestley et al. [1994] that take into account also the flexural
ductility of the structure.
Triple strut models have been obviously used in this assessment, because compared to
single strut models they are more suitable to predict the real shear behaviour of columns:
in that models in fact, the shear contribution given by infills is also considered. The
critical zones where the shear values have been analyzed are beam-column joints in
correspondence of the compressed lower struts (nodes 4 and 7 for model 1; nodes 3, 5, 7,
and 9 for model 2); upper nodes have been neglected because shear values are not
significant in that zones.
Figure 5-32. Mesh nodes of the models
Shear that have been compared with shear strength formulas has been considered not just
as the value gave back by DIANA (that take into account the contribution of the lateral
strut ending on the column), but also considering a percentage of the shear bore by the
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A.A. 2008-09
central strut: that choice proceeds of the observation that in the finite element model this
contribution is completely absorbed by the joint, whereas maybe that, considering the real
contact area of the central strut, a part of it could be reasonably assigned to the column.
Adopting a conservative value of this percentage (80%) the shear verification have been
carried out in according to the two formulas previously point out.
European Standards’ shear strength of member requiring design shear reinforcement is
evaluated considering a truss model to represent the shear resistant mechanism:
Figure 5-33. Truss model proposed by the EC 2 to represent the shear resistant mechanism
For member with vertical shear reinforcement, the shear resistance is evaluated as the
smaller value between the resistance offered by the reinforcement contribution VRd,s and
that offered by the concrete contribution VRd,max :
âãä,Æ =
where:
)Æ
-dä cot y
âãä,F; = å> à9 ->ä /cot + tan Asw is the cross sectional area of the shear reinforcement;
s
is the spacing of the stirrups;
fywd is the design yield strength of shear reinforcement;
ν1 is a strength reduction factor for concrete cracked in shear (recommended
bbbbbbbbbbvalue: 0.6);
αcw is coefficient considering the state of the stress in the compression chord;
bw
is the minimum width between tension and compression chord;
z is the inner level arm related to the bending moment in the considered element;
θ is the angle between shear reinforcement and the beam axis orthogonal to the
kkkkkkkkishear force (1 ≤ cotθ ≤ 2.5).
A.A. 2008-09
109
Note that, because in carrying out nonlinear analysis characteristic values must be
adopted, in this assessment the design strength used in the proposed formula have bee
replaced by characteristic strength. The limit value of cotθ = 2.5 has been assumed.
Codified shear strength methods of design cannot be considered as predictive equations
since they are intended to provide a conservative and safe lower bound to strength and
also because the scatter between predicted and measured values is rather high; for this
reason Priestley et al. [1994] proposed a formula to predict shear resistance of columns,
considering it as consisting of three independent components: a concrete component VC,
whose magnitude depends on the level of ductility; an axial load component VP, whose
magnitude depends on the column aspect ratio; a truss component VS, whose magnitude
depends on the transverse reinforcement content. Thus:
â6 = â` + âç + âè
The concrete component, clearly reduce with increasing ductility:
â` = Q -`é )E
where k depends on the member displacement ductility level, fC’ is the compressive
cylinder strength of concrete and Ae = 0.8Agross is the effective shear area of the column.
uniaxial ductility
biaxial ductility
0,40
k
0,30
0,20
0,10
0,00
0
1
2
3
4
member displacement ductility
5
Figure 5-34. Degradation of concrete shear strength with ductility
Concerning the axial-load component VP, it is considered that the column axial force
enhance the shear strength by arch action forming an inclined strut: for a column in
double bending, as those of in question, the inclination of the strut is found from the line
joining the centre of flexural compression at the top and bottom of the column. The
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A.A. 2008-09
enhancement to shear strength is the horizontal component of the diagonal compression
strut, since this component directly resists the applied shear force. So:
âç = A tan å =
Á−Ê
A
2B
where D is the overall section depth, c is the depth of the compression zone, a = L/2 for a
column in reverse bending and P is the axial load acting on member. This equation
implies that as the aspect ratio of the column decrease, the axial-load contribution to shear
strength will increase, and that for very slender columns the axial-load contribution may
be rather minimal. Since shear force is likely to be low in such cases, this may not be
significant. Moreover, as the axial load increases, the effectiveness of the axial-load
contribution to column shear strength will decrease since the depth of the compression
zone increase.
Não é possív el apresentar a imagem. O computador pode não ter memória suficiente para abrir a imagem ou a imagem pode ter sido danificada. Reinicie o computador e, em seguida, abra o ficheiro nov amente. Se o x v ermelho continuar a aparecer, poderá ter de eliminar a imagem e inseri-la nov amente.
Figure 5-35. Column shear strength due to axial force: reverse (a) and single bending (b)
The contribution of transverse reinforcement VS, is based on a truss mechanism using a
30° angle between the compression diagonals and the column axis.
âè =
)| -dÄ Á′
cot 30°

where Av is the total transverse reinforcement area per layer, fyh is the yield strength of
transverse reinforcement, D’ is the distance between centres of peripheral hoops, and s is
the spacing of transverse reinforcement along member axis.
A.A. 2008-09
111
Priestley approach has been validated by experimental tests on columns; a fitting and
reasonable prediction is get by applying a strength reduction factor ΦS = 0.85, that affect
the components of the formula in the following way: the upper and lower limits to k
become 0.25 and 0.085 (rather than 0.29 and 0.1), the axial-load contribution is multiplied
by 0.85, and the truss mechanism would be based on an angle of θ = 35° (against 30°).
Shear in the critical zones is compared with Eurocode and Priestley shear strength.
80
70
70
60
60
Shear [kN]
Shear [kN]
node 4
80
50
40
30
10
50
40
30
Vd
Vres (EC2)
Vres (Priestley et al.)
20
Vd
Vres (EC2)
20
node 7
10
0
0
0
50
100
top displacement [mm]
150
0
50
100
150
Figure 5-36. Shear verification: model 1 – uniform load patter
node 7
80
70
70
60
60
Shear [kN]
Shear [kN]
node 4
80
50
40
30
Vd
Vres (EC2)
20
10
0
0
50
100
150
50
40
30
Vd
Vres (EC2)
20
10
0
0
50
100
150
Figure 5-37. Shear verification: model 1 – modal load patter
112
A.A. 2008-09
node 5
node 3
80
60
50
40
30
50
40
30
20
20
10
10
0
0
0
20
40
Vd
Vres (EC2)
70
Shear [kN]
60
Shear [kN]
80
Vd
Vres (EC2)
70
60
0
node 7
60
node 9
80
80
60
50
40
30
Vd
Vres (EC2)
70
60
Shear [kN]
Vd
Vres (EC2)
70
Shear [kN]
20
40
50
40
30
20
20
10
10
0
0
0
20
40
60
0
20
40
60
Figure 5-38. Shear verification: model 2 – uniform load patter
node 3
Vd
Vres (EC2)
70
60
Shear [kN]
node 5
50
40
30
80
60
50
40
30
20
20
10
10
0
Vd
Vres (EC2)
70
Shear [kN]
80
0
0
A.A. 2008-09
20
40
60
0
20
40
60
113
node 7
80
Vd
Vres (EC2)
70
60
50
40
30
60
50
40
30
20
20
10
10
0
Vd
Vres (EC2)
70
Shear [kN]
80
Shear [kN]
node 9
0
0
20
40
60
0
20
40
60
Figure 5-39. Shear verification: model 2 – modal load patter
Some aspects can be put on evidence looking at the previous diagrams: the lower values
of shear bore by the column of the model 2 if compared to those of model 1, probably
because in a frame with one column more shear can redistribute in a better way; the
strong dependence on the ductility level shown in model 1 Priestley shear strength dued
to the concrete component, whereas in the model 2 this dependence is attenuated because
the ductility doesn’t reach levels so high as in the other frame; the constant EC2 shear
strength, dued to the fact that the minimum value is that concerning the shear
reinforcement contribution (that is constant).
To end this paragraph two more diagrams are shown, concerning design shear values and
resistances in the node 4 of model 1 with uniform load pattern applied: in the first, design
shear values is decomposed in the contribution acting on the column and gave back in
DIANA output solicitation diagrams and in the contribution of the central strut; in the
second, Priestley shear strength is decomposed in its components:
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A.A. 2008-09
node 4
Vd
Vd,col
80
Vd,inf_0
Vres (EC2)
Vd,inf_l
70
Shear [kN]
60
50
40
30
20
10
0
0
50
100
150
Figure 5-40. Components of shear design values
node 4
Vd
Vres,c (Pr.)
80
Vres (EC2)
Vres,p (Pr.)
Vres,s (Pr.)
70
Shear [kN]
60
50
40
30
20
10
0
0
50
100
150
Figure 5-41. Components of Priestley shear strength
A.A. 2008-09
115
6 TIME-HISTORY ANALYSIS
From eigenvalue and pushover analysis came out that single and triple strut models bring
to quite similar results: the only remarkable difference is in a better evaluation of shear in
the column gave by triple strut models for the reason already explained: thus, because of
their elevate computational time, dynamic nonlinear analysis have been carried out just
on one kind of models, i.e. triple strut models. Analysis has been realized both with
artificial and recorded accelerograms, with the purpose to check their vantages and
disadvantages and analyze differences in results.
At the beginning, as done in the previous chapters, the analysis procedure adopted is
explained, focusing mainly on the features implemented in the code characteristics of
time-history analysis: transient effects and time step; other features are the same used also
in pushover analysis, hence here they are dealt quickly.
Next, a paragraph is dedicated to damping, introduced in DIANA models just for these
analysis to consider the effects of viscous phenomena, that have a relevant importance in
dynamic analysis.
Results (expressed now in terms of time-history curves, maximum displacements, and
interstorey drifts) are commented, making a comparison with results obtained in pushover
analysis.
Then safety assessments on interstorey drift in according to the European Standards have
been performed.
With the aim to understand the rule played by important parameters like damping and
concrete tension strength, sensitive analysis have been finally carried out; moreover
analysis with increasing level of seismic input have been realized to test the general
strength level of the building.
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A.A. 2008-09
6.1
ANALYSIS PROCEDURE
The analysis procedure to be implemented for dynamic nonlinear analysis presents many
aspects analogous (or equal in some cases) to those described in the previous chapter in
dealing with static nonlinear analysis procedure. Once models have been called in
DIANA environment, the correctness verified, and the type of analysis chosen (structural
nonlinear), the four blocks of the settings must be fulfilled.
As in the pushover analysis, the first module (regarding the evaluation of geometric and
material properties for elements and reinforcements, the assembly of the elements to
create an appropriate system degree of freedom, the setup of the element stiffness
matrices and the setup of the load vectors) has been left unaltered.
In the next panel the type of nonlinearities have to be chosen: now, a part physically
nonlinear effects, also transient effects have been switched on, while the other options
(geometrically nonlinear and linear stress/strain effects) have been neglected once more.
In the transient effect option one of the proposed time integration methods (Newmark,
Euler, Hilbert-Hughes-Taylor, Wilson, Runge-Kutta) is required to be selected; in the
specific case, the third one (HHT method) has been adopted: HHT method is an extension
of Newmark method, which propose to determine the time integration solution with the
following equations:
Gì∆G
Gì∆G
where
= G + Ç1 − í G + í Gì∆G È∆c
1
= G ∆c + Â − Å G + Gì∆G ∆c 2
Gì∆G
=
1
1 G
1 − 2 G
∆ −
−

∆c
∆c
2
The parameters β and γ define the accuracy of the method and take into account also the
numerical dissipation in a reverse proportional relationship (β is a function of γ); HHT
method manage to solve this problem, adopting the same equations above presented, but
expressing both β and γ as a function of a parameter α. The scheme is in this way second
A.A. 2008-09
117
order accurate and unconditionally stable: decreasing α means increasing the numerical
damping; the damping is low for low-frequency modes and high for high-frequency
modes. The default value of α = -0.1 has been setup.
In the step execution window load steps for vertical forces and time steps for horizontal
forces have been setup. In a dynamic structural problem, the governing equation of
motion for a transient dynamic problem at time t can be written as:
c + c + ªî° , , e, ï, c … = ¹ñ° c
right-hand-side vector of forcing functions, ü, and u are the resulting acceleration,
where M is the mass matrix, C the damping matrix and fext the external force vector or
velocity and displacement vectors, ε and σ are the strain and stress fields, and the vector
fint is the internal set of forces opposing the displacements. For the transient response of a
nonlinear analysis, the solution of the previous second order differential equation is
obtained by direct time integration techniques (i.e. HHT method). The solution will be
determined at a number of discrete time points: t0, t1, t2, ... , t - ∆t, t, t + ∆t, ... , T.
Assuming to have the solution at time t, the equation of motion holds to:
G + G + Gªî° , , e, ï, c … = G¹ñ°
Then, with the implicit time integration procedure used, Gì∆G is obtained from:
Gì∆G + Gì∆G + Gì∆Gªî° , , e, ï, c … =
that, considering the HHT method, is modified as:
Gì∆G
¹ñ°
Gì∆G + 1 + å Gì∆G − å G + 1 + å Gì∆Gªî° − å Gªî° =
Gì9ìò∆G
¹ñ°
After 10 steps for vertical loads (10% each step), there are the steps for horizontal loads:
the number of these steps depends on the time-history length. The time step has been
fixed as to fit three points within two consecutive values of the accelerogram, and amount
to 0.0025s: this value needs to be scaled to consider both the model scaling factor and the
adopted system unit; the time step so defined is in according with the proposal of Chopra
A.K. [1995], that suggest ∆t / Tn ≤ 0.01 (Tn natural period of the nth significant vibration
mode of the undamped structure) to ensure adequate accuracy in the numerical results.
118
A.A. 2008-09
The regular Newton-Raphson iteration method has been used with a maximum of 50
iterations (as in pushover analysis) and a convergence criterion for the equilibrium
iteration process based both on energy.
In the end, the datas necessary to plot the time-history curve and to get the interstorey
drift values have been asked in the output panel.
The data files of the analysis procedure here described are detailed in ANNEX 7.
6.2
DAMPING EFFECTS
The input of damping is only appropriate for dynamic and transient analysis. There are
various forms of damping input in DIANA: viscous damping for all structural elements
and the point mass/damping elements, structural damping for all structural elements,
continuous damping via dashpots or point elements, and strain energy based element
damping for all structural elements. In practice the presence of damping reduces the
steady-state response and damps out the transient response. Dynamic nonlinear analysis
assume the application of proportional viscous damping and that the damping matrix C
satisfies the orthogonality condition: modal damping can be employed for this and his
magnitude has to be specified as a percentage of the critical damping factor:
«ª ó = 2ôª õª öªó
where ωi is the natural angular frequency, and ξi the damping ratio. The critical damping
factor is:
Ê>}5G = 2√Q &
where k is the generalized stiffness «ª ó öªó and m is the generalized mass
«ª ó öªó . Thus, it’s necessary to evaluate the damping matrix C explicitly and usually
viscous damping effects can be included by assumption of Rayleigh damping which is of
the form (Rayleigh J.W.S. [1945]):
A.A. 2008-09
= B + B9
119
where a0 and a1 are constants to be determined from given damping ratio. In fact,
would be in the form ÷ = B © or ÷ = B9 ø. In both cases the matrix C is diagonal by
considering
mass-proportional or stiffness-proportional damping, the damping matrix
virtue of the modal orthogonality property and therefore these are classical damping
matrices; nevertheless, physically they represent the damping models below shown for a
multi-storey, that are not suitable for practical applications:
Figure 6-1. Mass-proportional damping (left); stiffness-proportional damping (right)
Relating the modal damping ratios for systems with mass or stiffness-proportional
damping to the constants, the generalized damping and the modal damping ratio for the
nth mode become:
î = B î
õî =
B
2ôî
î = B9 ôºî î
õî =
B9 ôî
2
In this way the coefficients a0 and a1 can be selected to obtain a specified value of the
damping ratio in any one mode:
B = 245 15
B9 =
245
15
Neither of the damping matrices so defined are appropriate for practical analysis of
M.D.O.F. systems: the variations of modal damping ratios with natural frequencies they
represent are not consistent with experimental data that indicate roughly the same
damping ratios for several vibration modes of a structure, as reported in Chopra A.K.
[1995].
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A.A. 2008-09
Rayleigh suggested to use a combination of the two damping as defined earlier:
÷ = B © + B9 ø
that led to a damping ratio for the nth mode for such a system of:
õî =
B
B9 ôî
+
2ôî
2
Figure 6-2. Variation of modal damping ratios with natural frequencies
The coefficients a0 and a1 can be determined imposing for the ith and the jth frequencies
specific damping ratios ξi and ξj, and solving the a system of two algebraic equations
(usually it’s supposed that both modes have the same damping ratio, as confirmed also by
experimental data):
B = 4
1 1⁄15
2 1⁄1
215 1
15 + 1
1 5 B
4
ùB ú = û ü
1
49
9
B9 = 4
2
15 + 1
The damping matrix is then known from and also the damping ratio for any other mode,
that varies with natural frequency. In the case objective of this thesis, just two modes are
relevant, as shown in eigenvalue analysis. The viscous damping ratio imposed, as stated
in §5.2.2, is 2%.
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121
6.3
ANALYSIS RESULTS
In analyzing results of a time-history analysis this work point out in studying the
behaviour of the time-history curve, the maximum displacement recorded on it, and the
maximum interstorey drift evidenced. On these topics a comparison with the analogous
results proceeding of the pushover analysis will be carried out.
6.3.1 Time-history Curve
The time-history curves obtained with a dynamic nonlinear analysis are the equivalent of
the capacity curve obtained with a static nonlinear analysis: it represent the development
during the analysis of the base reaction vs. the displacement of the control point. An
example of such a curve for the triple strut model 1, alone and compared with the two
capacity curve (uniform and modal load pattern), is plotted in the next diagrams.
The curve shows a structural behaviour characterized by the phenomenon of pinching,
with a quite low hysteretic dissipation: thus the hypothesis done in determining the target
displacement with the capacity spectrum method could be considered correct. Another
salient aspect of the curve is its tendency in conforming to the capacity curve of the static
nonlinear analysis: this fact suggest the correctness of the model adopted for the time-
Force [kN]
history analysis.
80
70
60
50
40
30
20
10
0
-10 -9 -8 -7 -6 -5 -4 -3 -2 -10
-1 0 1 2 3 4 5 6 7 8 9 10
-20
-30
-40
-50
-60
-70
-80
displacement [mm]
Figure 6-3. Example of a time-history curve
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100
Force [kN]
Time history
Pushover uniform
Pushover modal
80
60
40
20
0
-40
-30
-20
-10 -20 0
10
20
30
40
-40
-60
-80
-100
displacement [mm]
Figure 6-4. Comparison between the time-history and the capacity curve
The shape of the curve isn’t affected by the kind of seismic input loaded: in all cases
(artificial, recorded scaled, recorded scaled accelerograms) the shape is the same,
changing just in its peak value and maximum displacement. All the time-history curves
are available in ANNEX 8.
6.3.2 Maximum Displacement
Maximum displacements of the time-history curves for analysis carried out with artificial,
recorded unscaled and recorded scaled accelerograms are displayed in the next tables:
Table 6-1. Maximum displacements: artificial accelerograms
ACC 1
ACC 2
ACC 3
ACC 4
ACC 5
ACC 6
ACC 7
AVER.
A.A. 2008-09
Type 1 Spectrum
Model 1
Model 2
dMAX FMAX dMAX FMAX
[mm] [kN] [mm] [kN]
2.85
35.85 2.11 35.18
2.55
32.41 1.71 29.81
2.66
32.86 1.98 32.45
2.73
33.85 1.95 32.93
2.77
35.08 1.96 33.29
2.94
37.00 2.13 35.68
2.40
32.72 1.54 27.57
2.70
34.25 1.91 32.42
Type 2 Spectrum
Model 1
Model 2
dMAX
FMAX
dMAX FMAX
[mm]
[kN]
[mm] [kN]
2.17
28.20
1.67
28.02
2.82
34.13
2.03
32.73
2.51
30.77
1.79
29.59
2.38
28.93
1.82
30.30
2.63
34.10
1.76
30.93
2.69
32.32
2.01
32.68
2.17
26.44
1.64
27.76
2.48
30.70
1.82
30.29
123
Table 6-2. Maximum displacements: recorded unscaled accelerograms
ACC 1
ACC 2
ACC 3
ACC 4
ACC 5
ACC 6
ACC 7
AVER.
Type 1 Spectrum
Model 1
Model 2
dMAX FMAX dMAX FMAX
[mm] [kN] [mm] [kN]
6.55
71.58 4.51 64.65
2.92
35.01 2.23 36.02
2.73
35.49 2.06 35.01
3.79
47.28 2.75 44.45
2.34
30.30 1.62 28.02
2.00
25.26 1.31 22.74
1.27
16.63 0.93 16.50
3.09
37.36 2.20 35.34
Table 6-3. Maximum displacements: recorded scaled accelerograms
ACC 1
ACC 2
ACC 3
ACC 4
ACC 5
ACC 6
ACC 7
AVER.
Type 1 Spectrum
Model 1
Model 2
dMAX FMAX dMAX FMAX
[mm] [kN] [mm] [kN]
1.02
12.88 0.68 12.36
2.46
27.90 2.03 31.81
4.02
46.49 0.96 13.53
3.02
39.23 2.29 37.79
1.30
17.74 0.90 16.27
1.31
17.49 0.90 16.06
1.68
22.64 1.14 19.91
2.11
26.34 1.27 21.10
Analyzing results some general conclusion that confirm what saw from pushover analysis
can be argued: in all cases displacements are such that the structure remain in the prepeak elastic branch: infills are probably cracked but not yet collapsed; the biggest
stiffness (related to lower displacements) of the frame with two spans is confirmed again.
Specific considerations proceeding of an examination of time-history analysis results can
be got comparing results obtained by using recorded unscaled and scaled accelerograms:
in the latter case maximum displacements (and also peak values of the force) are quite
lower than those obtained by using unscaled accelerograms (that are 30% - 40% higher.
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Considering results obtained by adopting artificial accelerograms is possible to see that
there is a good fitting with results obtained by adopting recorded unscaled accelerograms
(that are 13% higher). This detail could be view as a confirm that the selected set of
scaled accelerograms is not suitable and leads to unreliable results: however, it doesn’t
mean that scaled accelerograms cannot be used in dynamic nonlinear analysis; use them
probably is also a better solution, but the choice of the set is a very important step both
regarding the scale factor to adopt and the intensity of the accelerograms, and hence it
needs good accuracy and knowledge of the particular situation.
Finally, the target displacement calculated by using the two procedures for static
nonlinear analysis is compared in the following table with the maximum displacement
achieved in dynamic nonlinear analysis: both N2 extended method and C.S.M. seems to
be conservative if results found out with time-history analysis are supposed to be the most
accurate. In fact with these methods, on the contrary of what happen in time-history
analysis, the target displacement correspond lays almost at the peak value of the capacity
curve, in a branch that is no more the initial elastic one.
Table 6-4. Target displacement (pushover) vs. Maximum displacement (time-history)
Spectrum
Type
Type 1
Type 2
Model
Model 1
Model 2
Model 1
Model 2
T.D. vs. Maximum displacement [mm]
Pushover
Time-history
N2 ext.
recorded recorded
CSM
artificial
method
unscaled
scaled
7.4
6.6
2.7
3.1
2.1
5.9
5.2
1.9
2.2
1.3
7.5
7.5
2.5
7.5
6.1
1.8
-
6.3.3 Interstorey drift
Differently from what done for pushover analysis, where interstorey drift has been
calculated at the step corresponding to the target displacement (because the lateral load
increase monotonically), here interstorey drift has been calculated along all the steps of
the analysis and then the maximum value has been evaluated.
A.A. 2008-09
125
Table 6-5. Maximum interstorey drifts of time-history analysis
Type 1 Spectrum
Type 2 Spectrum
Model 1
Model 2
Model 1 Model 2
recorded recorded
recorded recorded
artificial
artificial
artificial
unscaled scaled
unscaled scaled
dr,0-1 / h 0.09% 0.11% 0.00% 0.07% 0.08%
0.00% 0.08% 0.06%
dr,1-2 / h 0.04% 0.05% 0.00% 0.03% 0.03%
0.00% 0.04% 0.03%
How is logical to be expected considering the lower values of maximum displacement
faced to the target displacement, interstorey drifts also are largely lower than those
calculated in the previous chapter.
Table 6-6. Maximum interstorey drifts of pushover analysis
dr,0-1 / h
dr,1-2 / h
6.4
Type 1 Spectrum
Model 1 Model 2
0.25%
0.20%
0.08%
0.06%
Type 2 Spectrum
Model 1 Model 2
0.29%
0.24%
0.09%
0.06%
SAFETY ASSESSMENTS
Damage limitation requirement in according to the European Standards has been verified
in the same way as done in the previous chapter.
6.4.1 Limitation of Interstorey Drift
The damage limitation requirement, for buildings having non-structural elements of
brittle materials attached to the structure, is considered to be satisfied if:
/} à ≤ 0.005 ℎ
where dr is the interstorey drift, h is the storey height (2.00 m in the DIANA models), ν
is the reduction factor which takes into account the lower return period of the seismic
action associated with the damage limitation requirement and whose recommended value
is 0.50. Values of ν related to the seismic action (0.40 and 0.55 respectively for type 1
and 2 spectrum) can be found in the nation annex [NP EN 1998-1:2006 – NA.4.4.3.2 l].
126
A.A. 2008-09
Table 6-7. Interstorey drift safety assessment
Type 1 Spectrum
Model 1
Model 2
recorded recorded
recorded recorded
artificial
artificial
unscaled scaled
unscaled scaled
dr,0-1 / h
OK
OK
OK
OK
OK
OK
dr,1-2 / h
OK
OK
OK
OK
OK
OK
Type 2 Spectrum
Model 1 Model 2
artificial
OK
OK
OK
OK
Considering that the European Standards limit value of the ratio dr / h , as computed by
the previous formula, are 1.25% and 0.91% respectively for type 1 and type 2 spectrum,
and considering the obtained values shown in Table 6.6, the damage limitation
requirement is adequately satisfied for the considerate limit state.
6.5
SENSITIVE ANALYSIS
In order to comprehend the influence of parameters like concrete tensile strength and
damping, the results of sensitive analysis are now presented: they have been carried out
on the model 1, and considering just the artificial accelerogram 4 (type 1 spectrum).
Concerning the first variable, it has been set to one half and to the double of the value
adopted in all the analysis; concerning the other variable, it has been set to 0.5%, 1% and
3% whereas the value adopted in all the analysis is 2%.
-3
-2
40
30
20
10
0
-10
-1
0
-20
-30
-40
ft
1
displacement [mm]
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2
3
Force [kN]
Force [kN]
0.5 ft
-3
-2
40
30
20
10
0
-10
-1
0
-20
-30
-40
1
2
3
displacement [mm]
127
Force [kN]
2 ft
-3
-2
40
30
20
10
0
-10
-1
0
-20
-30
-40
1
2
3
displacement [mm]
Figure 6-5. Influence on results of concrete tensile strength
It’s clear from these time-history curves that concrete tensile strength doesn’t affect in
any way the results; maximum displacements, forces and initial stiffness: all the
parameters are more or less the same if concrete tensile strength is the double or one half.
ξ=1%
60
40
40
20
0
-6 -5 -4 -3 -2 -20
-1 0 1 2 3 4 5 6
Force [kN]
60
20
0
-6 -5 -4 -3 -2 -20
-1 0 1 2 3 4 5 6
-40
-40
-60
-60
displacement [mm]
displacement [mm]
ξ=2%
ξ=3%
60
60
40
40
20
0
-6 -5 -4 -3 -2 -20
-1 0 1 2 3 4 5 6
Force [kN]
Force [kN]
Force [kN]
ξ=0.5%
20
0
-6 -5 -4 -3 -2 -20
-1 0 1 2 3 4 5 6
-40
-40
-60
-60
displacement [mm]
displacement [mm]
Figure 6-6. Influence on results of viscous damping
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A.A. 2008-09
Damping play a key rule in the seismic response of the frames, maybe even more of what
expected: displacements are very sensitive to variations of this parameter. It suggest that
the chose of the viscous damping value to be considered for the structure is a critical step
that must be analyzed very accurately case by case: an incorrect or superficial decision in
this phase could lead to uncorrected results. Otherwise, the procedures used to get the
target displacement probably should be less sensitive to the selected damping, but more
researches are necessary to confirm that supposition.
To end this chapter, time-history analysis have been carried out with increasing level of
seismic input to check if and when infills fail, and how is the seismic response of the
frame in such a case. Once again the analysis have been carried out on the model 1, just
considering the artificial accelerogram n°4 proceeding of the type 1 spectrum: it has been
increased of 1.5, 2, 3 and 5 times.
1.0 x
Force [kN]
Time history
Pushover uniform
Pushover modal
-40
-30
-20
-10
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
10
20
30
40
displacement [mm]
1.5 x
Force [kN]
Time history
Pushover uniform
Pushover modal
-40
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-30
-20
100
80
60
40
20
0
-20
-10
0
10
-40
-60
-80
-100
displacement [mm]
20
30
40
129
2.0 x
Force [kN]
Time history
Pushover uniform
Pushover modal
-40
-30
-20
-10
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
10
20
30
40
10
20
30
40
20
30
40
displacement [mm]
3.0 x
Force [kN]
Time history
Pushover uniform
Pushover modal
-40
-30
-20
-10
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
displacement [mm]
5.0 x
Force [kN]
Time history
Pushover uniform
Pushover modal
-40
-30
-20
-10
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
10
displacement [mm]
Figure 6-7. Influence on results of seismic intensity level
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From the time-history curves shown in the previous figure is possible to check that, until
a seismic input equal to three times the reference one (that used in the analysis), the frame
still remain in the elastic pre-peak branch: just for a seismic input triple than the basic one
the peak of the capacity curve is almost reached. Then, using an accelerogram five times
higher that adopted in the current analysis, infill fails (surely the lower one whereas the
upper still hasn’t reach its maximum strength), the structure lose stiffness until the timehistory curve lies on the subhorizontal branch of the pushover curve, that corresponding
to the bare frame.
A.A. 2008-09
131
7 SUMMARY AND CONCLUSIONS
Two planar models of the building, one for each main direction, have been realized
adopting the concept of strut models, namely models where concrete parts of the structure
are represented by beam elements, whereas infills are represented by diagonal
compression struts. Both models with one and three struts instead of each masonry panel
have been employed to analyze the building behaviour if subjected to lateral loads.
The two types of models return rather similar results: pushover analysis have been carried
out to point out this aspect, and differences in term of shape of the curve and in term of
maximum values are decidedly negligible; anyway an higher elastic initial stiffness in the
direction parallel to the frame with two spans is always shown.
Nevertheless, one relevant aspect must be studied paying particular attention: masonry
infills transfer to columns a shear contribute that isn’t taken into account with single strut
model, but just with triple strut models where it is simulated by the lateral strut: adopt
single strut models in structures where columns are very solicited to shear stresses could
lead to brittle failure mechanism not predicted by the analysis. Moreover, in a calculation
for the shear present in the columns, also a percentage of that bore by the central strut (not
considered by the code that transfer it totally to the joint) should be considered to take
into account also the real contact area of the central strut.
The target displacement has been evaluated for the two reference spectrum types, but just
for the two triple strut models and just for the mass proportional lateral load pattern:
because of the strong similarity in the capacity curves between single and triple strut
model and also between the mass proportional and modal proportional lateral load
pattern, results shouldn’t be very different for the other cases. An extended version of the
N2 method for infilled frames and the capacity spectrum method have been employed for
the purpose: results show that the target displacement belong to the elastic pre-peak
branch of the capacity curve (infills are probably cracked but not yet collapsed), with
lower values for the model 2 respect to the model 1 except for the type 2 spectrum in the
N2 extended method that maybe is not reliable in this circumstance.
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A.A. 2008-09
Some safety assessment prescribed by the Eurocode 8 have been verified, namely the
limitation of interstorey drift and the shear resistance: in the first case, the assessment is
largely verified, whereas in the latter is also verified but without such a big margin in a
few critical zones; regarding to the shear resistance, further than the European Standards
criteria, a more refined formulation considering the flexural ductility has been employed.
On the base of sensitive analysis on computational parameters, the model to be adopted
for time-history analysis has been chose, but improving the model quality model doesn’t
transfer in a perceptible changes: the basic model adopted up to now has been thus chose.
Dynamic nonlinear analysis with artificial and recorded, unscaled and scaled,
accelerograms have been then realized: time-history curves follows precisely the shape of
the capacity curves and the maximum displacement, to be compared with the target
displacement of pushover analysis, is still in the elastic pre-peak branch but with values
quite lower than those evaluated with the two static nonlinear procedures (also the
interstorey drift values are consequently lower and then satisfied plenty the European
Standards requirement): the low level of the seismic input probably doesn’t allow a really
good estimate with these procedure, but generally speaking they remain a good tool if
time-history analysis are supposed to be too much complex and time computationally
onerous. Another aspect which is necessary to pay attention is the employing of recorded
scaled accelerograms, whose mean scale factor needs to be chose carefully: in the present
case output results i.e. diverge respect to those registered for artificial or recorded
unscaled accelerograms.
From sensitive analysis has been checked how, on the contrary of concrete tensile
strength, damping affect in a strong way the results: it’s a characteristic to be evaluated
with very attention, also because maybe it doesn’t affect in a such way the static nonlinear
procedures and hence could lead to inconsistent comparisons between dynamic and static
analysis; maybe in the present case it has been underestimate and then differences in
results could be related to this parameter too. Tests have been carried out also raising the
seismic input, and it is shown that, to reach the peak value of the capacity curve, an input
three times the basic one is necessary.
A.A. 2008-09
133
In conclusion it’s possible to asses that strut model are becoming nowadays a rather
common tool in the analysis of r.c. infilled frames, and the current transposing by the
national codes of suggestions and rules present in the Eurocodes will make of them in the
next future a tool employed not just in the research field but also in the structural design;
moreover, considering infill panels in the modelling, improve the performances of the
structures and allow to take into account relevant phenomena otherwise underestimated,
like brittle shear failure of the columns.
In the particular case studied in the present thesis, the two-storey simple building
analyzed (representative of the common Mediterranean constructions), that should be
tested in this period at the L.N.E.C. of Lisbon, accomplish the European Standards for the
low seismic level input of the considered zone, but maybe for higher seismicity level
zones different solution can be useful to improve the seismic response of the structure.
134
A.A. 2008-09
ANNEXES
ANNEX 1 – STRUCTURAL DESIGN OF THE BUILDING .................................................... 136
ANNEX 2 – TIME-HISTORIES ................................................................................................ 148
ANNEX 3 – DATA FILES OF THE MODELS IMPLEMENTED IN DIANA ........................... 156
ANNEX 4 – DATA FILES OF THE EIGENVALUE ANALYSIS PROCEDURE ...................... 172
ANNEX 5 – DATA FILES OF THE PUSHOVER ANALYSIS PROCEDURE .......................... 173
ANNEX 6 – SOLICITATIONS AT TARGET DISPLACEMENT STEP ..................................... 175
ANNEX 7 – DATA FILES OF THE TIME-HISTORY ANALYSIS PROCEDURE .................... 178
ANNEX 8 – TIME-HISTORY CURVES .................................................................................... 180
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ANNEX 1 – STRUCTURAL DESIGN OF THE BUILDING
136
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137
138
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139
140
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141
142
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143
144
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145
146
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147
ANNEX 2 – TIME-HISTORIES
ATRIFICIAL ACCELEROGRAMS – TYPE 1
ACC. 1A
acceleration [ms2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
-1,0
-1,5
-2,0
time [s]
ACC. 2A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 3A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
4
6
8
10
-1,5
-2,0
148
2
-1,0
time [s]
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ACC. 4A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
-1,0
-1,5
-2,0
time [s]
ACC. 5A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 6A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 7A
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
A.A. 2008-09
time [s]
149
ATRIFICIAL ACCELEROGRAMS – TYPE 2
ACC. 1B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
-1,0
-1,5
-2,0
time [s]
ACC. 2B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 3B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
4
6
8
10
-1,0
-1,5
-2,0
150
2
time [s]
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ACC. 4B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
-1,0
-1,5
-2,0
time [s]
ACC. 5B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 6B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
time [s]
ACC. 7B
acceleration [m/s2]
2,0
1,5
1,0
0,5
0,0
-0,5 0
2
4
6
8
10
-1,0
-1,5
-2,0
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time [s]
151
UNSCALED RECORDED ACCELEROGRAMS – TYPE 1
IZMIT (1A)
acceleration [m/s2]
4
3
2
1
0
-1 0
10
20
30
acceleration [m/s2]
60
70
80
50
60
70
80
60
70
80
-3
time [s]
IZMIT (2A)
4
3
2
1
0
-1 0
10
20
30
40
-2
-3
-4
time [s]
CAMPANO-LUCANO (3A)
4
acceleration [m/s2]
50
-2
-4
3
2
1
0
-1 0
10
20
30
40
50
-2
-3
-4
152
40
time [s]
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KALAMATA (4A)
acceleration [m/s2]
4
3
2
1
0
-1 0
10
20
30
acceleration [m/s2]
60
70
80
50
60
70
80
50
60
70
80
50
60
70
80
-3
time [s]
VALNERINA (5A)
4
3
2
1
0
-1 0
10
20
30
40
-2
-3
-4
time [s]
SOUTH ICELAND aftershock (6A)
4
acceleration [m/s2]
50
-2
-4
3
2
1
0
-1 0
10
20
30
40
-2
-3
-4
time [s]
FRIULI (7A)
4
acceleration [m/s2]
40
3
2
1
0
-1 0
10
20
30
40
-2
-3
-4
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time [s]
153
SCALED RECORDED ACCELEROGRAMS – TYPE 1
acceleration [m/s2]
KALAMATA (1A)
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
50
60
70
80
60
70
80
60
70
80
time [s]
acceleration [m/s2]
UMBRIA MARCHE aftershock (2A)
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
50
time [s]
acceleration [m/s2]
MONTENEGRO aftershock (3A)
154
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
50
time [s]
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acceleration [m/s2]
SPITAK (4A)
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
50
60
70
80
60
70
80
50
60
70
80
50
60
70
80
time [s]
acceleration [m/s2]
FRIULI aftershock (5A)
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
50
time [s]
acceleration [m/s2]
UMBRIA MARCHE aftershock (6A)
4
3
2
1
0
-1 0
-2
-3
-4
10
20
30
40
time [s]
acceleration [m/s2]
VALNERINA (7A)
4
3
2
1
0
-1 0
-2
-3
-4
A.A. 2008-09
10
20
30
40
time [s]
155
ANNEX 3 – DATA FILES OF THE MODELS IMPLEMENTED IN DIANA
MODEL 1 SINGLE STRUT
FEMGEN MODEL
: MODEL1
ANALYSIS TYPE : Structural 2D
'UNITS'
LENGTH MM
TIME SEC
TEMPER KELVIN
FORCE N
'COORDINATES' DI=2
1
0.000000E+00 0.000000E+00
2
0.000000E+00 2.000000E+02
...
74
2.362093E+03 2.000000E+03
75
2.571450E+03 2.000000E+03
'ELEMENTS'
CONNECTIVITY
1 L7BEN 1 2
2 L7BEN 2 3
...
61 L7BEN 62 63
62 L7BEN 63 42
63 L4TRU 11 22
64 L4TRU 21 32
65 L7BEN 64 65
66 L7BEN 65 66
...
76 L7BEN 74 75
77 L7BEN 75 64
78 L4TRU 1 32
79 L4TRU 11 42
80 PT3T 11
81 PT3T 21
82 PT3T 32
83 PT3T 42
'REINFORCEMENTS'
LOCATI
32 BAR
LINE
0.896599E+03 0.187600E+04
0.278081E+04 0.187600E+04
ELEMEN 69-77 /
...
37 BAR
LINE -0.450000E+02 0.000000E+00
-0.450000E+02 0.200000E+04
ELEMEN 1-10 /
DATA
/ 1-62 65-77 / 1
/ 63 64 78 79 / 2
MATERIALS
/ 1-40 / 1
156
0.000000E+00
0.000000E+00
0.000000E+00
0.000000E+00
A.A. 2008-09
/ 63 64 78 79 / 2
/ 41-44 65-77 / 5
/ 45-62 / 6
/ 80 82 / 7
/ 81 83 / 8
/ 32-37 /
3
GEOMETRY
/ 41-62 65-77 / 6
/ 1-40 / 11
/ 63 64 78 79 / 12
/ 34-36 /
7
/ 32 /
8
/ 33 / 10
/ 37 / 13
'DATA'
1 NINTEG 7 5
NUMINT GAUSS GAUSS
2 NINTEG 7
NUMINT GAUSS
'MATERIALS'
1 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.500000E-06
“for time-history analysis”
RAYLEI 1.251190E+00 2.600000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
2 YOUNG 9.259000E+02
POISON 1.500000E-01
DENSIT 0.000000E+00
RAYLEI 1.251190E+00 2.600000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 1.000000E-10
GF1
1.000000E-02
CRACKB 4.162031E+03
COMCRV PARABO
COMSTR 9.299999E-01
GC
1.480000E+00
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 6.000000E-06
3 YOUNG 2.000000E+05
DENSIT 7.850000E-06
RAYLEI 1.251190E+00 2.600000E-04
YIELD VMISES
A.A. 2008-09
157
YLDVAL 4.000000E+02
THERMX 1.200000E-05
5 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.270000E-05
RAYLEI 1.251190E+00 2.600000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
6 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 1.130000E-05
RAYLEI 1.251190E+00 2.600000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
7 MASS
2.327290E+03 2.327290E+03 0.000000E+00
8 MASS
1.187810E+03 1.187810E+03 0.000000E+00
'GEOMETRY'
6 RECTAN 3.000000E+02 1.500000E+02
7 CROSSE 1.570800E+02
8 CROSSE 2.261900E+02
9 CROSSE 2.356200E+02
10 CROSSE 3.141600E+02
11 RECTAN 1.500000E+02 2.250000E+02
12 CROSSE 7.020000E+04
13 CROSSE 6.031900E+02
'TYINGS'
EQUAL TR 1
/11 43-46 68-75 64-67/ 32
/21 47-63/ 42
'SUPPORTS'
/ 1 22 / TR 1
/ 1 22 / TR 2
/ 11 21 32 42 / TR 3
/ 1 22 / RO 3
'LOADS'
CASE 1
WEIGHT
158
A.A. 2008-09
2 -9.81000
“for pushover analysis”
CASE 2
NODAL
11 FORCE 1 0.665000E+00
21 FORCE 1 0.335000E+00
CASE 3
NODAL
11 FORCE 1 0.498000E+00
21 FORCE 1 0.502000E+00
CASE 2
WEIGHT
1
1.00000
'TIMELO'
LOAD 2
TIMES 0.0:421.21538(0.21082) /
FACTOR IMPORT "accelerogram1A.dat"
skip 1
scale 1
'DIRECTIONS'
1 1.000000E+00 0.000000E+00 0.000000E+00
2 0.000000E+00 1.000000E+00 0.000000E+00
3 0.000000E+00 0.000000E+00 1.000000E+00
'END'
A.A. 2008-09
159
MODEL 2 SINGLE STRUT
FEMGEN MODEL
: MODEL2
'UNITS'
LENGTH MM
TIME SEC
TEMPER KELVIN
FORCE N
'COORDINATES' DI=2
1
0.000000E+00 0.000000E+00
2
0.000000E+00 2.000000E+02
...
99
3.788706E+03 2.000000E+03
100 3.969353E+03 2.000000E+03
'ELEMENTS'
CONNECTIVITY
1 L7BEN 1 2
2 L7BEN 2 3
...
84 L7BEN 85 86
85 L7BEN 86 63
86 L4TRU 11 22
87 L4TRU 32 43
88 L4TRU 21 32
89 L4TRU 42 53
90 L7BEN 65 87
91 L7BEN 87 88
...
104 L7BEN 99 100
105 L7BEN 100 53
106 L4TRU 1 32
107 L4TRU 22 53
108 L4TRU 11 42
109 L4TRU 32 63
110 PT3T 11
111 PT3T 21
112 PT3T 53
113 PT3T 63
'REINFORCEMENTS'
LOCATI
34 BAR
LINE
0.000000E+00 0.187600E+04
0.473588E+03 0.187600E+04
ELEMEN 61-62 /
...
41 BAR
LINE
0.203000E+04 0.000000E+00
0.203000E+04 0.200000E+04
ELEMEN 21-30 /
DATA
/ 1-85 90-105 / 1
/ 86-89 106-109 / 2
MATERIALS
/ 1-60 / 1
/ 86-89 106-109 / 2
/ 110 112 / 7
160
0.000000E+00
0.000000E+00
0.000000E+00
0.000000E+00
A.A. 2008-09
/ 111 113 / 8
/ 61-65 90-105 / 9
/ 66-85 / 13
/ 34-41 /
3
GEOMETRY
/ 86-89 106-109 / 4
/ 61-85 90-105 / 5
/ 1-60 / 6
/ 34 41 /
7
/ 37-39 /
8
/ 36 /
9
/ 35 / 10
/ 40 / 11
'DATA'
1 NINTEG 7 5
NUMINT GAUSS GAUSS
2 NINTEG 7
NUMINT GAUSS
'MATERIALS'
1 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.500000E-06
RAYLEI 1.388070E+00 2.400000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
2 YOUNG 9.259000E+02
POISON 1.500000E-01
DENSIT 0.000000E+00
RAYLEI 1.388070E+00 2.400000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 1.000000E-10
GF1
1.000000E-02
CRACKB 2.881948E+03
COMCRV PARABO
COMSTR 9.299999E-01
GC
1.480000E+00
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 6.000000E-06
3 YOUNG 2.000000E+05
DENSIT 7.850000E-06
RAYLEI 1.388070E+00 2.400000E-04
YIELD VMISES
YLDVAL 4.000000E+02
A.A. 2008-09
161
THERMX 1.200000E-05
7 MASS
1.940150E+03 1.940150E+03 0.000000E+00
8 MASS
9.626100E+02 9.626100E+02 0.000000E+00
9 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.410000E-05
RAYLEI 1.388070E+00 2.400000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
13 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 1.230000E-05
RAYLEI 1.388070E+00 2.400000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
'GEOMETRY'
4 CROSSE 4.590000E+04
5 RECTAN 3.000000E+02 1.500000E+02
6 RECTAN 1.500000E+02 1.500000E+02
7 CROSSE 2.261900E+02
8 CROSSE 1.570800E+02
9 CROSSE 2.356200E+02
10 CROSSE 3.141600E+02
11 CROSSE 4.021200E+02
'TYINGS'
EQUAL TR 1
/11 64-65 87-93 32 66-68 94-100/ 53
/21 69-77 42 78-86/ 63
'SUPPORTS'
/ 1 22 43 / TR 1
/ 1 22 43 / TR 2
/ 11 21 53 63 / TR 3
/ 1 22 43 / RO 3
'LOADS'
CASE 1
WEIGHT
2 -9.81000
162
A.A. 2008-09
CASE 2
NODAL
11 FORCE 1 0.665000E+00
21 FORCE 1 0.335000E+00
CASE 3
NODAL
11 FORCE 1 0.498000E+00
21 FORCE 1 0.502000E+00
CASE 2
WEIGHT
1
1.00000
'TIMELO'
LOAD 2
TIMES 0.0:421.21538(0.21082) /
skip 1
scale 1
'DIRECTIONS'
1 1.000000E+00 0.000000E+00 0.000000E+00
2 0.000000E+00 1.000000E+00 0.000000E+00
3 0.000000E+00 0.000000E+00 1.000000E+00
'END'
A.A. 2008-09
163
MODEL 1 TRIPLE STRUT
FEMGEN MODEL
: MODEL1
'UNITS'
LENGTH MM
TIME SEC
TEMPER KELVIN
FORCE N
'COORDINATES' DI=2
1
0.000000E+00 0.000000E+00
2
0.000000E+00 1.661601E+02
...
80
3.256418E+03 2.000000E+03
81
3.453209E+03 2.000000E+03
'ELEMENTS'
CONNECTIVITY
1 L7BEN 1 2
2 L7BEN 2 3
...
13 L7BEN 17 18
14 L7BEN 18 19
15 L4TRU 4 7
16 L4TRU 16 10
17 L7BEN 20 21
18 L7BEN 22 23
...
25 L7BEN 29 30
26 L7BEN 30 20
27 L4TRU 1 10
28 L4TRU 4 31
...
35 L4TRU 15 37
36 L4TRU 6 38
37 L7BEN 3 39
38 L7BEN 39 40
...
90 L7BEN 80 81
91 L7BEN 81 10
92 PT3T 4
93 PT3T 16
94 PT3T 10
95 PT3T 31
'REINFORCEMENTS'
LOCATI
32 BAR
LINE
0.896599E+03 0.187600E+04
0.278081E+04 0.187600E+04
ELEMEN 18-26 /
...
37 BAR
LINE -0.450000E+02 0.000000E+00
-0.450000E+02 0.332320E+03
ELEMEN 1-2 /
DATA
/ 1-14 17-26 37-91 / 1
/ 15 16 27-36 / 2
164
0.000000E+00
0.000000E+00
0.000000E+00
0.000000E+00
A.A. 2008-09
MATERIALS
/ 1-8 37-72 / 1
/ 15 16 27-36 / 2
/ 9-11 17-26 88-91 / 5
/ 12-14 73-87 / 6
/ 92 94 / 7
/ 93 95 / 8
/ 32-37 /
3
GEOMETRY
/ 9-14 17-26 73-91 / 6
/ 1-8 37-72 / 11
/ 29-36 / 16
/ 15 16 27 28 / 17
/ 34-36 /
7
/ 32 /
8
/ 33 / 10
/ 37 / 13
'DATA'
1 NINTEG 7 5
NUMINT GAUSS GAUSS
2 NINTEG 7
NUMINT GAUSS
'MATERIALS'
1 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.500000E-06
RAYLEI 1.304740E+00 2.500000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
2 YOUNG 9.259000E+02
POISON 1.500000E-01
DENSIT 0.000000E+00
RAYLEI 1.304740E+00 2.500000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 1.000000E-10
GF1
1.000000E-02
CRACKB 4.162031E+03
COMCRV PARABO
COMSTR 9.299999E-01
GC
1.480000E+00
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 6.000000E-06
3 YOUNG 2.000000E+05
DENSIT 7.850000E-06
A.A. 2008-09
165
RAYLEI 1.304740E+00 2.500000E-04
YIELD VMISES
YLDVAL 4.000000E+02
THERMX 1.200000E-05
5 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.270000E-05
RAYLEI 1.304740E+00 2.500000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
6 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 1.130000E-05
RAYLEI 1.304740E+00 2.500000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
7 MASS
2.327290E+03 2.327290E+03 0.000000E+00
8 MASS
1.187810E+03 1.187810E+03 0.000000E+00
'GEOMETRY'
6 RECTAN 3.000000E+02 1.500000E+02
7 CROSSE 1.570800E+02
8 CROSSE 2.261900E+02
9 CROSSE 2.356200E+02
10 CROSSE 3.141600E+02
11 RECTAN 1.500000E+02 2.250000E+02
13 CROSSE 6.031900E+02
16 CROSSE 2.095900E+04
17 CROSSE 3.510000E+04
'TYINGS'
EQUAL TR 1
/4 13-15 22-30 20-21 80-81/ 10
/16-19 67-77 38 78-79/ 31
'SUPPORTS'
/ 1 7 33 35 / TR 1
/ 1 7 33 35 / TR 2
/ 4 10 16 31 / TR 3
166
A.A. 2008-09
/ 1 7 / RO 3
'LOADS'
CASE 1
WEIGHT
2 -9.81000
CASE 2
NODAL
4 FORCE 1 0.665000E+00
16 FORCE 1 0.335000E+00
CASE 3
NODAL
4 FORCE 1 0.498000E+00
16 FORCE 1 0.502000E+00
CASE 2
WEIGHT
1
1.00000
'TIMELO'
LOAD 2
TIMES 0.0:421.21538(0.21082) /
skip 1
scale 1
'DIRECTIONS'
1 1.000000E+00 0.000000E+00 0.000000E+00
2 0.000000E+00 1.000000E+00 0.000000E+00
3 0.000000E+00 0.000000E+00 1.000000E+00
'END'
A.A. 2008-09
167
MODEL 2 TRIPLE STRUT
FEMGEN MODEL
: MODEL2
'UNITS'
LENGTH MM
TIME SEC
TEMPER KELVIN
FORCE N
'COORDINATES' DI=2
1
0.000000E+00 0.000000E+00
2
0.000000E+00 2.441408E+02
...
97
3.499441E+03 4.000000E+03
98
3.693527E+03 4.000000E+03
'ELEMENTS'
CONNECTIVITY
1 L7BEN 1 2
2 L7BEN 3 4
...
9 L7BEN 15 16
10 L7BEN 17 18
11 L4TRU 3 5
12 L4TRU 7 9
13 L4TRU 15 7
14 L4TRU 17 11
15 L7BEN 19 20
16 L7BEN 20 21
...
26 L7BEN 31 32
27 L7BEN 32 33
28 L4TRU 1 7
29 L4TRU 5 11
...
46 L4TRU 14 45
47 L4TRU 8 46
48 L7BEN 2 47
49 L7BEN 47 48
...
118 L7BEN 98 46
119 L7BEN 46 34
120 PT3T 3
121 PT3T 15
122 PT3T 11
123 PT3T 34
'REINFORCEMENTS'
LOCATI
34 BAR
LINE
0.000000E+00 0.187600E+04
0.265420E+03 0.187600E+04
ELEMEN 7 /
...
41 BAR
LINE
0.203000E+04 0.000000E+00
0.203000E+04 0.259597E+03
ELEMEN 3 /
DATA
168
0.000000E+00
0.000000E+00
0.000000E+00
0.000000E+00
A.A. 2008-09
/ 1-10 15-27 48-119 / 1
/ 11-14 28-47 / 2
MATERIALS
/ 1-6 48-96 / 1
/ 11-14 28-47 / 2
/ 120 122 / 7
/ 121 123 / 8
/ 7 8 15-27 97-101 / 9
/ 9 10 102-119 / 13
/ 34-41 /
3
GEOMETRY
/ 7-10 15-27 97-119 / 5
/ 1-6 48-96 / 6
/ 32-47 / 14
/ 11-14 28-31 / 15
/ 34 41 /
7
/ 37-39 /
8
/ 36 /
9
/ 35 / 10
/ 40 / 11
'DATA'
1 NINTEG 7 5
NUMINT GAUSS GAUSS
2 NINTEG 7
NUMINT GAUSS
'MATERIALS'
1 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.500000E-06
RAYLEI 1.439660E+00 2.300000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
2 YOUNG 9.259000E+02
POISON 1.500000E-01
DENSIT 0.000000E+00
RAYLEI 1.439660E+00 2.300000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 1.000000E-10
GF1
1.000000E-02
CRACKB 2.881948E+03
COMCRV PARABO
COMSTR 9.299999E-01
GC
1.480000E+00
SHRCRV CONSTA
BETA 1.500000E-01
A.A. 2008-09
169
THERMX 6.000000E-06
3 YOUNG 2.000000E+05
DENSIT 7.850000E-06
RAYLEI 1.439660E+00 2.300000E-04
YIELD VMISES
YLDVAL 4.000000E+02
THERMX 1.200000E-05
7 MASS
1.940150E+03 1.940150E+03 0.000000E+00
8 MASS
9.626100E+02 9.626100E+02 0.000000E+00
9 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 2.410000E-05
RAYLEI 1.439660E+00 2.300000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
13 YOUNG 3.000000E+04
POISON 2.000000E-01
DENSIT 1.230000E-05
RAYLEI 1.439660E+00 2.300000E-04
TOTCRK FIXED
TENCRV EXPONE
TENSTR 2.200000E+00
GF1
5.140000E-02
CRACKB 2.000000E+02
COMCRV PARABO
COMSTR 2.800000E+01
GC
2.422000E+01
SHRCRV CONSTA
BETA 1.500000E-01
THERMX 1.000000E-05
'GEOMETRY'
5 RECTAN 3.000000E+02 1.500000E+02
6 RECTAN 1.500000E+02 1.500000E+02
7 CROSSE 2.261900E+02
8 CROSSE 1.570800E+02
9 CROSSE 2.356200E+02
10 CROSSE 3.141600E+02
11 CROSSE 4.021200E+02
14 CROSSE 1.313900E+04
15 CROSSE 2.295000E+04
'TYINGS'
EQUAL TR 1
/3 13 19-26 7 14 84 27-33/ 11
/15-16 85-91 42 17-18 92-98 46/ 34
'SUPPORTS'
170
A.A. 2008-09
/ 1 5 9 36 39 41 43 / TR 1
/ 1 5 9 36 39 41 43 / TR 2
/ 3 11 15 34 / TR 3
/ 1 5 9 / RO 3
'LOADS'
CASE 1
WEIGHT
2 -9.81000
CASE 2
NODAL
3 FORCE 1 0.665000E+00
15 FORCE 1 0.335000E+00
CASE 3
NODAL
3 FORCE 1 0.498000E+00
15 FORCE 1 0.502000E+00
CASE 2
WEIGHT
1
1.00000
'TIMELO'
LOAD 2
TIMES 0.0:421.21538(0.21082) /
skip 1
scale 1
'DIRECTIONS'
1 1.000000E+00 0.000000E+00 0.000000E+00
2 0.000000E+00 1.000000E+00 0.000000E+00
3 0.000000E+00 0.000000E+00 1.000000E+00
'END'
A.A. 2008-09
171
ANNEX 4 – DATA FILES OF THE EIGENVALUE ANALYSIS PROCEDURE
SINGLE STRUT – MODEL 1
*FILOS
INITIA
*INPUT
READ FILE " C:/Documents and Settings/Francesco/Desktop/Uni/tesi/tesi magistrale/diana/eigenvalues
analysis/single strut/model1/model1.dat"
*EIGEN
BEGIN EXECUT
MAXITE 30
NMODES 10
END EXECUT
OUTPUT FILE M1-SINGLE_STRUT
BEGIN OUTPUT
TABULA
FILE M1-SINGLE_STRUT
END OUTPUT
*END
The files of the other models (single strut – model 2, triple strut – model 1, triple strut –
model 2) are exactly identical to this one.
172
A.A. 2008-09
ANNEX 5 – DATA FILES OF THE PUSHOVER ANALYSIS PROCEDURE
SINGLE STRUT – MODEL 1
*FILOS
INITIA
*INPUT
READ FILE "C:/Documents and Settings/Francesco/Desktop/Uni/tesi/tesi magistrale/diana/pushover
analysis/pushover_CLII_7-5_7/model1.dat"
*NONLIN
BEGIN EXECUT
BEGIN LOAD
BEGIN STEPS
BEGIN EXPLIC
ARCLEN
SIZES 0.1(10)
END EXPLIC
END STEPS
END LOAD
BEGIN ITERAT
BEGIN CONVER
BEGIN DISPLA
CONTIN
TOLCON 0.001
END DISPLA
BEGIN FORCE
CONTIN
TOLCON 0.001
END FORCE
END CONVER
LINESE
MAXITE 50
END ITERAT
TEXT LC1
END EXECUT
BEGIN EXECUT
BEGIN LOAD
“for uniform load pattern”
LOADNR 2
“for modal load pattern”
LOADNR 3
BEGIN STEPS
BEGIN EXPLIC
ARCLEN
SIZES 500(1200)
END EXPLIC
END STEPS
END LOAD
BEGIN ITERAT
BEGIN CONVER
BEGIN DISPLA
CONTIN
TOLCON 0.001
END DISPLA
BEGIN FORCE
A.A. 2008-09
173
CONTIN
TOLCON 0.001
END FORCE
END CONVER
LINESE
MAXITE 50
END ITERAT
“for uniform load pattern”
TEXT LC2
“for modal load pattern”
TEXT LC3
END EXECUT
BEGIN OUTPUT
FILE 1B_NONL
TEXT NONLIN
DISPLA INCREM TRANSL GLOBAL
DISPLA TOTAL TRANSL GLOBAL
FORCE REACTI ROTATI GLOBAL
FORCE REACTI TRANSL GLOBAL
STRESS TOTAL CAUCHY GLOBAL
STRESS TOTAL FORCE GLOBAL
STRESS TOTAL MOMENT GLOBAL
STRESS TOTAL TRACTI LOCAL INTPNT
END OUTPUT
BEGIN OUTPUT
TABULA
FILE 1B_DISPL
SELECT NODES 42 /
TEXT DISPLA
END OUTPUT
BEGIN OUTPUT
TABULA
FILE 1B_REACT
SELECT NODES 1 22 /
TEXT REACT
END OUTPUT
BEGIN OUTPUT
TABULA
FILE 1B_INFILL1
SELECT ELEMEN 63 /
TEXT INFILL
END OUTPUT
BEGIN OUTPUT
TABULA
FILE 1B_INFILL2
SELECT ELEMEN 64 /
TEXT INFILL
END OUTPUT
*END
The files of the other models (single strut – model 2, triple strut – model 1, triple strut –
model 2) are quite similar: just is necessary to change the nodes and the elements in the
last part of the files, the one concerning the output results.
174
A.A. 2008-09
ANNEX 6 – SOLICITATIONS AT TARGET DISPLACEMENT STEP
TRIPLE STRUT – MODEL 2, TYPE 1 SPECTRUM
A.A. 2008-09
175
176
A.A. 2008-09
A.A. 2008-09
177
ANNEX 7 – DATA FILES OF THE TIME-HISTORY ANALYSIS PROCEDURE
*FILOS
INITIA
*INPUT
READ FILE "C:/Documents and Settings/Francesco/Desktop/Uni/tesi/tesi magistrale/diana/time history
analysis/artificial accelerograms/accel_1/triple strut/model1_typeAspectrum/model1A.dat"
*END
*NONLIN
BEGIN EXECUT
LOAD STEPS EXPLIC SIZES 0.1(10)
BEGIN ITERAT
BEGIN CONVER
DISPLA OFF
ENERGY CONTIN
FORCE OFF
END CONVER
LINESE
MAXITE 50
END ITERAT
TEXT LC1
END EXECUT
BEGIN EXECUT
TIME STEPS EXPLIC SIZES 0.05270463(8000)
BEGIN ITERAT
BEGIN CONVER
DISPLA OFF
ENERGY CONTIN
FORCE OFF
END CONVER
MAXITE 50
END ITERAT
TEXT LC2
END EXECUT
BEGIN OUTPUT
FILE ACC1_M1_A
TEXT MODEL
DISPLA INCREM TRANSL GLOBAL
FORCE REACTI ROTATI GLOBAL
STRESS TOTAL CAUCHY GLOBAL
STRESS TOTAL MOMENT GLOBAL
STRESS TOTAL TRACTI LOCAL
END OUTPUT
BEGIN OUTPUT
TABULA
FILE DISPL
SELECT NODES 31 /
TEXT DISPL
178
A.A. 2008-09
END OUTPUT
BEGIN OUTPUT
TABULA
FILE DISPL_
SELECT NODES 10 /
TEXT DISPL_
END OUTPUT
BEGIN OUTPUT
TABULA
FILE REACT
SELECT NODES 1 7 33 35 /
TEXT REACT
END OUTPUT
BEGIN TYPE
BEGIN TRANSI
DYNAMI DAMPIN
METHOD HHT
END TRANSI
END TYPE
*END
The files of the other models (triple strut – model 1 type B spectrum, triple strut – model
2 type A spectrum, triple strut – model 2 type B spectrum) are quite similar: just is
necessary to change the time step at the beginning, and the nodes and the elements in the
last part of the files, the one concerning the output results.
A.A. 2008-09
179
ANNEX 8 – TIME-HISTORY CURVES
Force [kN]
-4
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
-5
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
4
5
5
4
5
4
5
Force [kN]
-4
-5
-4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
-5
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
ARTIFICIAL ACCELEROGRAMS – MODEL 1, TYPE 1 SPECTRUM
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
-5
-4
4
5
4
5
4
5
180
A.A. 2008-09
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2
-30
-40
-50
displacement [mm]
ACCELEROGRAM 2
3
4
5
Force [kN]
Force [kN]
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2
-30
-40
-50
displacement [mm]
3
4
5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
ACCELEROGRAM 6
3
4
5
3
4
5
Force [kN]
Force [kN]
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2
-30
-40
-50
displacement [mm]
5
ACCELEROGRAM 4
Force [kN]
Force [kN]
ACCELEROGRAM 3
4
-5
-4
-3
50
40
30
20
10
0
-10
-2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
ACCELEROGRAM 7
-5
-4
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2
-30
-40
-50
displacement [mm]
A.A. 2008-09
181
Force [kN]
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
5
4
5
4
5
Force [kN]
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
-5
-4
Force [kN]
Force [kN]
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
182
A.A. 2008-09
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
5
4
5
4
5
-5
-4
Force [kN]
Force [kN]
-4
Force [kN]
-5
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
Force [kN]
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
RECORDED UNSCALED ACCELEROGRAMS – MODEL 1, TYPE 1 SPECTRUM
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183
Force [kN]
-5 -4
-5 -4
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
Force [kN]
Force [kN]
-5 -4
-5 -4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
ACCELEROGRAM 1
80
70
60
50
40
30
20
10
0
-10
-20
-7 -6 -5 -4 -3 -2 -30
-1 0 1 2 3 4 5 6 7
-40
-50
-60
-70
-80
displacement [mm]
Force [kN]
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
RECORDED UNSCALED ACCELEROGRAMS – MODEL 2, TYPE 1 SPECTRUM
184
A.A. 2008-09
Force [kN]
-5
Force [kN]
-5
-5
-4
-4
-4
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
5
4
5
4
5
Force [kN]
4
-5
Force [kN]
Force [kN]
-5 -4
ACCELEROGRAM 1
70
50
30
10
-10
-3 -2 -1
-30 0 1 2 3
-50
-70
displacement [mm]
Force [kN]
Force [kN]
-4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
-5
-4
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
RECORDED SCALED ACCELEROGRAMS – MODEL 1, TYPE 1 SPECTRUM
A.A. 2008-09
185
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
-5 -4
4
5
5
4
5
4
5
Force [kN]
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
-4
Force [kN]
-5
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
RECORDED SCALED ACCELEROGRAMS – MODEL 2, TYPE 1 SPECTRUM
186
A.A. 2008-09
ACCELEROGRAM 5
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
-5 -4
Force [kN]
-5
-5
-4
-4
ACCELEROGRAM 7
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
5
4
5
4
5
Force [kN]
ACCELEROGRAM 3
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
ACCELEROGRAM 2
50
40
30
20
10
0
-10
-5 -4 -3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
Force [kN]
Force [kN]
-4
Force [kN]
-5
ACCELEROGRAM 1
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
ACCELEROGRAM 4
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
-5
Force [kN]
Force [kN]
-5
-4
-4
ACCELEROGRAM 6
50
40
30
20
10
0
-10
-3 -2 -1
-20 0 1 2 3
-30
-40
-50
displacement [mm]
4
5
4
5
4
5
REFERENCES
A.A. 2008-09
187
–. EN, 1990:2002. Eurocode: Basis of structural design. C.E.N., Brussels, Belgium.
–. EN, 1991-1-1:2001. Eurocode 1: Action on structures - Part 1-1: General actions Densities, self-weight, imposed loads for buildings. C.E.N., Brussels, Belgium.
–. EN, 1992-1-1:2004. Eurocode 2: Design of concrete structures - Part 1-1: General
rules and rules for buildings. C.E.N., Brussels, Belgium.
–. EN, 1996-1-1:2005. Eurocode 6: Design of masonry structures - Part 1-1: General Rules for reinforced and unreinforced masonry structures. C.E.N., Brussels, Belgium.
–. EN, 1996-3:2005. Eurocode 6: Design of masonry structures - Part 3: Simplified
calculation methods for unreinforced masonry structures. C.E.N., Brussels, Belgium.
–. EN, 1998-1:2003. Eurocode 8: Design of structures for earthquake resistance - Part 1:
General rules, seismic actions and rules for buildings. C.E.N., Brussels, Belgium.
–. EN, 1998-3:2003. Eurocode 8: Design of concrete structures - Part 3: Strengthening
and repair of buildings. C.E.N., Brussels, Belgium.
–. NP EN, 1998-1:2006. Eurocódigo 8: Projecto de estruturas para resistência aos sismos
- Parte 1: Regras gerais, acções sísmicas e regras para edifícios. Anexo Nacional NA.
Comissão Técnica Portuguesa de Normalização CT 115 – Eurocódigos Estruturais,
Lisbon, Portugal.
–. R.E.A.E. [1983]. Regulamento de estruturas de aço para edificios. Diário da
República, Lisbon, Portugal.
–. R.E.B.A.P. [1983]. Regulamento para estruturas de betão armado e pré-esforçado.
Diário da República, Lisbon, Portugal.
–. R.S.A. [1983]. Regulamento de segurança e acções para estruturas de edificios e
pontes. Diário da República, Lisbon, Portugal.
Albanesi T., Nuti C. [2007]. Analisi statica non lineare (pushover). Dipartimento di
Strutture, Università degli Studi di Roma Tre, Rome, Italy.
188
A.A. 2008-09
Ambraseys, N., Smit, P., Sigbjornsson, R., Suhadolc, P. and Margaris, B. [2002].
Internet-Site for European Strong-Motion Data. European Commission, ResearchDirectorate
General,
Environment
and
Climate
Programme.
http://www.isesd.cv.ic.ac.uk/ESD/
Bergami A.V. [2008]. Implementation and experimental verification of models for
nonlinear analysis of masonry infilled r.c. frames. Ph.D. Thesis, Università degli Studi di
Roma Tre, Rome, Italy.
Brazão Farinha J.S., Correia dos Reis, A. [1996]. Tabelas Tecnicas. Edições técnicas
E.T.L., Lisbon, Portugal.
CEB-FIP [1993], Model Code 1990: design code. Bulletin d’information 213/214.
Thomas Telford, London, England.
Chopra A.K. [1995]. Dynamic of structures. Theory and applications to earthquake
engineering. Prentice Hall, Englewood Cliffs (NJ), U.S.A.
Dolšek M., Fajfar P. [2004]. Inelastic spectra for infilled reinforced concrete structures.
Earthquake Engineering and Structural Dynamics 33:1395-1416, DOI 10.1002/eqe.410.
Dolšek M., Fajfar P. [2004]. Simplified non-linear seismic analysis of infilled reinforced
concrete frames. Earthquake Engineering and Structural Dynamics 34:49-66, DOI
10.1002/eqe.411.
Dolšek M., Fajfar P. [2008]. The effect of masonry infills on the seismic response of a
four-storey reinforced concrete frame – a deterministic assessment. Engineering
Structures 30(7):1991-2001, DOI 10.1016.
Elfgren L. [1989]. Fracture mechanics of concrete, from theory to applications. Report of
the Technical Committee 90 - FMA Fracture Mechanics to Concrete - Applications –
RILEM. Chapman and Hall, London and New York.
Fajfar P., Gaspersič P. [1996]. The N2 method for seismic damage analysis of R.C.
structures. Earthquake Engineering and Structural Dynamics, vol.25.
A.A. 2008-09
189
Fardis M.N. [1996]. Experimental and numerical investigations on the seismic response
of r.c. infilled frames and recommendations for code provisions. ECOEST-PREC8 Report
N°6, Laboratório Nacional de Engenharia Civil, Lisbon, Portugal.
Faria R. [1994]. Avaliação do comportamento sísmico de barragens de betão através de
um modelo de dano contínuo. Ph.D. Thesis, Universidade do Porto, Porto, Portugal.
Freeman S.A. [1998]. The capacity spectrum method as a tool for seismic design.
Proceeding of the 11th European Conference on Earthquake Engineering, Paris, France.
Gasparini D.A, Vanmarcke, E.H. [1976]. SIMQKE: a program for artificial motion
generation. User’s manual and documentation. Department of Civil Engineering,
Massachusetts Institute of Technology, Cambridge (MA), U.S.A.
Gasparini, D.A., Vanmarcke E.H., Nau R.F. et al. [1976]. Simulated earthquake motions
compatible with prescribed response spectra. Department of Civil Engineering, Research
report R76-4, Massachusetts Institute of Technology, Cambridge (MA), U.S.A.
Iervolino I., Cornell A. [2005]. Record selection for nonlinear seismic analysis of
structures. Earthquake Spectra 21(3):685-713
DOI 10.1193/1.1990199, E.E.R.I.,
Oakland (CA), U.S.A.
Iervolino I., Maddaloni G, Cosenza E. [2008]. Eurocode 8 compliant real record sets for
seismic analysis of structures. Journal of Earthquake Engineering 12(1):54-90, DOI
10.1080/13632460701457173, Lynnfield (MA), U.S.A.
Iervolino I., Galasso C., Cosenza E. [20]. REXEL 2.2 beta: Uno strumento per la
selezione di accelerogrammi naturali per le NTC e l’Eurocodice 8. Università degli Studi
di Napoli Federico II, Naples, Italy.
Iervolino I., Galasso C. [2010]. REXEL v2.6 beta: Selezione automatica di
accelerogrammi naturali per l’analisi dinamica non lineare delle strutture. Dipartimento
di Ingegneria Strutturale, Università di Napoli Federico II, Naples, Italy.
190
A.A. 2008-09
Iervolino I., Galasso C., Cosenza E. [2010]. REXEL: Computer aided record selection for
code-based seismic structural analysis. Bulletin of Earthquake Engineering 8:339-362,
DOI 10.1007/s10518-009-9146-1.
Leite J. [2009]. Avaliação da segurança baseada em deslocamentos: aplicação a
edificios de betão armado sem e com paredes de preenchimento. Tese de Mestrado,
Universidade do Minho, Guimarães, Portugal.
Lourenco P.B. [2008]. Structural masonry analysis: recent developments and prospects.
Proceeding of the 14th International Brick and Block Masonry Conference (14IBMAC),
Sydney, Australia.
Lourenço P.B. et al. [2009]. Masonry infills and earthquakes. Proceedings of the 11th
Canadian Masonry Symposium, Toronto, Canada.
Manie J., Wolthers A. [2008]. DIANA 9.3. Finite element analysis. User manual. TNODIANA BV, Delft, Netherlands.
Mauro A. [2008]. Behaviour of masonry infills under seismic loads: state of the art
review on the analysis technique. Report 08-DEC/E-30, Universidade do Minho,
Guimarães, Portugal.
Priestley N. [1994]. Seismic shear strength of reinforced concrete columns. Journal of
Structural Engineering (ASCE), vol.129, Paper n. 6588.
Rayleigh J.W.S. [1945]. The theory of sound. Vol.1. Dover Publications, New York (NY),
U.S.A.
Safina S. [2002]. Vulnerabilidad sísmica de edificaciones esenciales. Análisis de su
contribución al riesgo sísmico. Ph.D. Thesis, Universidad Politécnica de Cataluña,
Barcelona, Spain.
Stafford Smith B. [1966]. Behavior of square infilled frames. Journal of the Structural
Division (ASCE), vol.92, n. ST1.
A.A. 2008-09
191
–. http://dicata.ing.unibs.it/gelfi/software/programmi_studenti.html
–. http://nisee.berkeley.edu/elibrary/getpkg?id=SIMQKE1
–. http://www.lnec.pt
–. http://www.seismosoft.com
–. http://ww.tnodiana.com
192
A.A. 2008-09