MODELO NO-LINEAR INELSTICO PARA ANLISE DE ESTRUTURAS
METLICAS APORTICADAS EM CONDIES DE INCNDIO
Alexandre Landesmann
TESE SUBMETIDA AO CORPO DOCENTE DA COORDENAO DOS
PROGRAMAS DE PS-GRADUAO DE ENGENHARIA DA UNIVERSIDADE
FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS
NECESSRIOS PARA A OBTENO DO GRAU DE DOUTOR EM CINCIAS EM
ENGENHARIA CIVIL.
Aprovada por:
________________________________________________
Prof. Eduardo de Miranda Batista, D.Sc.
________________________________________________
Prof. Jos Luis Drummond Alves, D.Sc.
________________________________________________
Prof. Ronaldo Carvalho Battista, Ph.D.
________________________________________________
Prof. Ricardo Hallal Fakury, D.Sc.
________________________________________________
Prof. Paulo de Mattos Pimenta, Dr.-Ing.
________________________________________________
Prof. Valdir Pignatta e Silva, D.Sc.
RIO DE JANEIRO, RJ - BRASIL
DEZEMBRO DE 2003
ii
LANDESMANN, ALEXANDRE
Modelo No-Linear Inelstico para Anlise
de Estruturas Metlicas Aporticadas em
Condies de Incndio [Rio de Janeiro] 2003
XXIII, 295 p. 29,7 cm (COPPE/UFRJ,
D.Sc., Engenharia Civil, 2003)
Tese - Universidade Federal do Rio de
Janeiro, COPPE
1. Estruturas de ao 2. Incndio 3. Modelo
Computacional 3. Plasticidade
I. COPPE/UFRJ II. Ttulo ( srie )
iii
A DEUS por tudo,
Aos meus pais, Henry e Catharina, e minha irm, Miriam,
A Carol.
iv
Agradecimentos:
Ao meu orientador, Professor Eduardo de Miranda Batista, pela competncia,
dedicao, aconselhamento e amizade minha sincera gratido.
Ao Professor Jos Luis Drummond Alves, pela valiosa co-orientao e pelo importante
estmulo nas diversas etapas do desenvolvimento deste trabalho de pesquisa.
Ao Professor Francisco Claudio Pereira de Barros da Comisso Nacional de Energia
Nuclear CNEN, pelo constante incentivo, apoio e amizade.
Ao Engenheiro Artur Correa Filho da CNEN, pelo apoio e compreenso, demonstrados
durante todas as etapas deste estudo.
A todos meus colegas de trabalho na CNEN, em especial, aos Engenheiros Ricardo
Colosimo, Humberto Teixeira e Ronaldo Pollis, pelo apoio e amizade no decorrer desta
jornada.
Comisso Nacional de Energia Nuclear, pelo apoio institucional, que viabilizou o
desenvolvimento deste trabalho de pesquisa.
A todos meus colegas da COPPE/UFRJ, especialmente, Hisashi Inoue, Santigo
Venncio, Danilo Fernandes, Maurcio Alves e Tiago de Oliveira, pela amizade,
companheirismo e diversas colaboraes neste perodo de convivncia.
COPPE/UFRJ, em particular, ao Programa de Engenharia Civil, representado por
todos seus Professores e Funcionrios, o meu sincero agradecimento.
Aos professores Roger Plank e Ian Burgess da Universidade de Sheffield (UK) pela
precisa orientao durante minha estadia naquela instituio.
Ao Professor Jean-Marc Franssen da Universidade de Lige (Blgica) pela permisso
de utilizao do Programa de Anlise Estrutural SAFIR, largamente utilizado nesta
pesquisa para fins de validao dos nossos resultados.
Coordenao de Aperfeioamento de Pessoal de Nvel Superior CAPES, pelo
auxlio financeiro, que possibilitou a realizao do Programa de Doutorado no Brasil
com Estgio no Exterior (PDEE), durante o perodo de Novembro/2002 a
Fevereiro/2003 junto a Universidade de Sheffield (UK).
v
Resumo da Tese apresentada COPPE/UFRJ como parte dos requisitos necessrios
para a obteno do grau de Doutor em Cincias (D.Sc.)
MODELO NO-LINEAR INELSTICO PARA ANLISE DE ESTRUTURAS
METLICAS APORTICADAS EM CONDIES DE INCNDIO
Alexandre Landesmann
Dezembro/2003
Orientadores: Prof. Eduardo de Miranda Batista
Prof. Jos Luis Drummond Alves
Programa: Engenharia Civil
Este trabalho dedicado ao desenvolvimento de um modelo computacional
para anlise no-linear elastoplstica de estruturas de ao, planas e aporticadas, sob
condies de incndio. A primeira etapa do processo de anlise, traduzida pela
determinao da variao do campo de temperaturas de sees-transversais expostas ao
fogo, realizada por meio de procedimento numrico no-linear transiente de
transferncia de calor, desenvolvido com base na formulao geral do Mtodo dos
Elementos Finitos (MEF). O comportamento estrutural numericamente investigado
por meio de princpios de plasticidade concentrada, que fazem uso de modelos refinados
de rtulas plsticas, funes de estabilidade, mdulos tangentes e superfcies inelsticas
de reduo de resistncia, permitindo-se assim, estimar o tempo crtico de resistncia ao
fogo, associado formao de mecanismos de colapso estrutural. Os resultados obtidos,
para um grupo selecionado de estruturas aporticadas, so examinados tomando-se por
base o Programa SAFIR e recomendaes previstas pela normatizao nacional e
internacional, vigente.
vi
Abstract of Thesis presented to COPPE/UFRJ as a partial fulfillment of the
requirements for the degree of Doctor of Science (D.Sc.)
SECOND-ORDER INELASTIC MODEL FOR THE ANALYSIS OF STEEL-
FRAMED STRUCTURES UNDER FIRE CONDITIONS
Alexandre Landesmann
December/2003
Advisors: Prof. Eduardo de Miranda Batista
Prof. Jos Luis Drummond Alves
Department: Civil Engineering
This work is dedicated to the development of a computational model for the
inelastic second-order analysis of plane steel-framed structures under fire conditions.
The first step of the analysis process, represented by the determination of the variation
of the transversal temperature field, is performed by a numerical transient nonlinear
heat transfer procedure, that was developed on the general basis of the Finite Element
Method (FEM). The structural behavior is numerically tracked by the concept of
concentrated plasticity, making use of refined plastic hinges models, stability functions,
tangent modulus models and gradual inelastic plastic surfaces, allowing the estimation
of the fire-resistance critical time, associated with the development of the structural
collapse mechanism. The numerical results, for a selected group of framed structures,
are examined in contrast with the SAFIR computational program results as well as
recommendations proposed by national and international standards.
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Sumrio
Captulo 1: INTRODUO
1.1 Motivao ................................................................................................... 1
1.2 Importncia da anlise estrutural no contexto da engenharia de
incndio....................................................................................................... 3
1.3 Pesquisa bibliogrfica sobre a anlise de estruturas de ao sob
fogo .............................................................................................................. 9
1.4 Mtodo das rtulas plsticas..................................................................... 16
1.5 Organizao deste trabalho ...................................................................... 19
Captulo 2: ANLISE TRMICA 2.1 Introduo ..................................................................................................22
2.2 Curvas de incndio ....................................................................................24
2.3 Modelo trmico simplificado segundo EC-3 ...........................................27
2.3.1 Elementos estruturais sem proteo contra incndio...................................27
2.3.2 Elementos estruturais com material de proteo contra incndio ...............33
2.4 Elemento unidimensional de transferncia de calor ..............................35
2.4.1 Elementos estruturais sem proteo trmica................................................35
2.4.2 Elementos estruturais protegidos contra incndio .......................................44
2.5 Verificao dos modelos trmicos implementados .................................48
2.5.1 Elementos estruturais sem proteo trmica................................................50
2.5.2 Elementos estruturais protegidos contra incndio .......................................54
2.6 Considerao da variao de temperatura na seo-transversal pelo
mtodo de anlise avanada......................................................................57
2.6.1 Seo-transversal equivalente......................................................................57
2.6.2 Limites equivalentes de resistncia plstica ................................................61
2.6.3 Esforos de engastamento perfeito devido variao de temperatura ........62
2.6.4 Temperatura de referncia ...........................................................................64
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Captulo 3: ANLISE ESTRUTURAL
3.1 Anlise avanada de estruturas................................................................69
3.2 Considerao de efeitos no-lineares geomtricos..................................71
3.2.1 Restries e consideraes gerais para elemento de viga-coluna................71
3.2.2 Funes de estabilidade para elemento de viga-coluna...............................73
3.2.3 Relao de rigidez tangente .........................................................................77
3.2.4 Aplicaes com modelos de funes de estabilidade ..................................80
3.2.5 Fatores de amplificao de momentos fletores............................................83
3.3 Conceito de mdulo tangente....................................................................91
3.3.1 Adaptao do conceito de anlise avanada s prescries da NBR-8800 .94
3.3.2 Considerao do efeito de temperatura........................................................101
3.3.3 Resistncia de barras comprimidas, segundo EC-3/Parte-2 (2001) ............102
3.3.4 Modelo de mdulo tangente segundo o Eurocdigo ...................................105
3.3.5 Estudos com modelos de mdulos tangentes...............................................107
3.4 Modelo inelstico de reduo de rigidez flexional ..................................115
3.5 Considerao de ligaes semi-rgidas.....................................................123
3.5.1 Modelo de ligao semi-rgida KISHI e CHEN (1990) ..............................125
3.5.2 Modificao da rigidez do elemento devido presena de ligaes...........127
Captulo 4: RESULTADOS
4.1 Introduo .................................................................................................. 131
4.2 Vigas isoladas em condies de incndio ................................................. 134
4.3 Pilares isolados sob ao de incndio ....................................................... 143
4.4 Prtico plano sob ao de incndio .......................................................... 155
4.5 Prtico plano industrial sob ao de incndio......................................... 166
4.6 Edifcio industrial sob ao de incndio .................................................. 169
4.7 Anlise comparativa dos resultados......................................................... 177
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Captulo 5: CONSIDERAES FINAIS 5.1 Breve resumo do presente trabalho ......................................................... 176
5.2 Concluses .................................................................................................. 178
5.3 Sugestes para trabalhos futuros ............................................................. 181
6. Referncias bibliogrficas ......................................................................... 185
Anexo A: IMPLEMENTAO COMPUTACIONAL A.1 Introduo .................................................................................................. 203
A.2 Entrada de dados para o programa de anlise trmica ......................... 204
A.3 Entrada de dados para o programa de anlise estrutural ..................... 209
A.4 Procedimentos de solues numricas ..................................................... 224
Anexo B: RESULTADOS COM MODELO DE ANLISE TRMICA B.1 Introduo .................................................................................................. 231
B.2 Variao do campo de temperaturas ....................................................... 233
B.3 Seo-transversal equivalente .................................................................. 245
Anexo C: VERIFICAES ESTRUTURAIS EM TEMPERATURA
AMBIENTE C.1 Introduo .................................................................................................. 268
C.2 Prtico plano tipo portal ........................................................................ 271
C.3 Prtico plano tipo industrial .................................................................. 273
C.4 Edifcio de seis andares ............................................................................ 275
C.5 Parmetros adimensionais padronizados para ligaes com
cantoneiras.................................................................................................. 277
C.6 Prtico de oito andares.............................................................................. 281
C.7 Exemplo de aplicao em projeto............................................................. 289
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ndice de figuras:
Captulo 1: Introduo
Figura 1.1: Fases de um incndio natural, comparadas com curva
padronizada temperatura-tempo (ISO 834-1, 1999). ......................... 4
Figura 1.2: Medidas de segurana contra incndio em edificaes...................... 6
Figura 1.3: Principais etapas seguidas pelo procedimento computacional
desenvolvido....................................................................................... 20
Captulo 2: Anlise Trmica
Figura 2.1: Comparao entre diferentes curvas de incndio, previstas
pelo EC-1/Parte-2 (2001).................................................................27
Figura 2.2: Diviso da seo-transversal de perfis I ou H para
utilizao do modelo trmico simplificado......................................28
Figura 2.3: Calor especfico do ao (ca) em funo da temperatura
(EC-3/Parte-2, 2001)........................................................................30
Figura 2.4: Elemento finito trmico unidimensional (1D) com funes de
interpolao lineares (Ni e Nj) ..........................................................36
Figura 2.5: Discretizao da seo-transversal por meio de elementos
unidimensionais (1D) para anlise trmica......................................37
Figura 2.6: Condutividade trmica do ao a em funo da temperatura,
segundo modelo recomendado pelo EC-3/Parte-2 (2001) ...............38
Figura 2.7: Balano trmico em cada elemento 1D; contribuio do fluxo
de calor para anlise trmica............................................................39
Figura 2.8: Esquema de integrao temporal pelo mtodo dos trapzios
para soluo do sistema de equaes transientes de
temperatura ......................................................................................42
Figura 2.9: Aplicao do elemento trmico unidimensional para anlise
de perfis metlicos envolvidos por material de proteo
contra incndio.................................................................................44
xi
Figura 2.10: Simulao de perfis metlicos protegidos por material de
revestimento trmico por meio de elementos trmicos
unidimensionais ...............................................................................46
Figura 2.11: Comparao entre variao de temperatura para o grupo de
perfis selecionados, assumindo-se exposio em 3 faces ................51
Figura 2.12: Comparao entre variao de temperatura para o grupo de
perfis selecionados, assumindo-se exposio em 4 faces ................52
Figura 2.13: Variao de temperatura para perfis selecionados, protegidos
por material de revestimento, expostos ao fogo em 3 faces ............54
Figura 2.14: Variao de temperatura para perfis selecionados, protegidos
por material de revestimento, expostos ao fogo em 3 faces ............55
Figura 2.15: Segmentao da seo-transversal em funo do aumento de
temperatura; (a) sistema de coordenadas dos segmentos.................58
Figura 2.16: Variao dos fatores de reduo do ao em funo da
temperatura (EC-3/Parte-2, 2001)....................................................58
Figura 2.17: Alongamento do ao () em funo da temperatura, segundo
modelo sugerido pelo EC-3/Parte-2 (2001). ....................................63
Captulo 3: Anlise Estrutural
Figura 3.1: Elemento de viga-coluna submetido a foras axiais e
momentos de extremidade. ..............................................................73
Figura 3.2: Deslocamentos nodais do elemento viga-coluna, para os ns
sistemas local e global. ....................................................................77
Figura 3.3: Sistemas de foras equivalentes para o elemento de viga-
coluna...............................................................................................79
Figura 3.4: Curvas de deslocamento para diferentes condies de
imperfeio geomtrica inicial.........................................................81
Figura 3.5: Modelo de pilar isolado adotado nas avaliaes do efeito P-
delta..................................................................................................85
Figura 3.6: Comparao entre os fatores de amplificao de momento
obtidos pelo modelo de funes de estabilidade (PNL-F),
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soluo terica e especificaes do LRFD (1999), para
relaes P/Pe inferiores a 0,4. ..........................................................86
Figura 3.7: Comparao entre os fatores de amplificao de momento
obtidos pelo modelo de funes de estabilidade (PNL-F),
soluo terica e especificaes do AISC-LRFD (1999), para
relaes P/Pe entre 0,5 e 0,9. ...........................................................87
Figura 3.8: Amplificao de momento obtidos pela soluo terica,
especificaes do AISC-LRFD (1999) e NBR-8800 (1986). ..........88
Figura 3.9: Comparao entre curvas de flambagem de pilares (a-d)
adotadas pela NBR-8800 (1986) e pelo AISC-LRFD (1999)..........96
Figura 3.10: Comparao entre curvas de flambagem de pilares (a-d)
adotadas pela NBR-8800 (1986) e as curvas pseudo-elsticas........97
Figura 3.11: Redues inelsticas de rigidez devido ao efeito da fora
axial, obtidos a partir das curvas de resistncia da
NBR-8800 (1986) e AISC-LRFD (1999). .......................................100
Figura 3.12: Curvas de resistncia de barras comprimidas para diferentes
nveis de temperatura, segundo NBR-14323 (1999). ......................102
Figura 3.13: Curvas de resistncia de barras comprimidas para diferentes
nveis de temperatura, segundo EC-3/Parte-2 (2001). .....................104
Figura 3.14: Comparao entre as curvas de flambagem de Euler, EC-
3/Parte-2 (2001) e NBR-14323 (1999), em condies de
temperatura ambiente (20oC). ..........................................................104
Figura 3.15: Reduo inelstica de rigidez devido ao efeito da fora axial,
obtido a partir da curva de resistncia do EC-3/Parte2 (2001). .......107
Figura 3.16: Modelo de barra isolada empregada nas anlises numricas
para comparao entre o modelo de mdulo tangente e as
curvas de resistncia do AISC-LRFD (1999) e da
NBR-8800 (1986). ...........................................................................108
Figura 3.17: Comparao entre resultados de viga-coluna isolada sob
temperatura ambiente, para curva de flambagem a
(NBR-8800, 1986). ..........................................................................109
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Figura 3.18: Comparao entre resultados de viga-coluna isolada sob
temperatura ambiente, para curva de flambagem b
(NBR-8800, 1986). ..........................................................................109
Figura 3.19: Comparao entre resultados de viga-coluna isolada sob
temperatura ambiente, para curva de flambagem c
(NBR-8800, 1986). ..........................................................................110
Figura 3.20: Comparao entre resultados de viga-coluna isolada sob
temperatura ambiente, para curva de flambagem d
(NBR-8800, 1986). ..........................................................................110
Figura 3.21: Comparao entre resultados de viga-coluna isolada sob
temperatura ambiente, para curva de flambagem original do
AISC-LRFD (1999). ........................................................................111
Figura 3.22: Comparao entre os resultados obtidos com o modelo de
mdulo tangente e as curvas de resistncia do EC-3/Parte-2,
para valores de esbelteza entre 0,1 e 0,9..........................................113
Figura 3.23: Comparao entre os resultados obtidos com o modelo de
mdulo tangente e as curvas de resistncia do EC-3/Parte-2,
para valores de esbeltez entre 1,1 e 1,9............................................114
Figura 3.24: Modelos inelsticos para reduo de rigidez flexional
propostos por LIEW e WHITE (1993) em condies de
temperatura ambiente.......................................................................116
Figura 3.25: Curvas de resistncia plstica e de incio de plastificao
obtidas em funo das prescries do EC-3 (2003) e do
AISC-LRFD (1999). ........................................................................117
Figura 3.26: Relao tenso-deformao para o ao em condies de
temperatura elevada, segundo o EC-3/Parte-2 (2001). ....................119
Figura 3.27: Modificao da relao tenso-deformao do ao em funo
da temperatura, segundo o modelo proposto pelo
EC-3/Parte-2 (2001).........................................................................120
Figura 3.28: Modelos polinomial de 4o grau proposto para o fator de
reduo de rigidez flexional para as temperaturas no ao ()
entre 100oC e 1200oC.......................................................................122
xiv
Figura 3.29: Comportamento de curvas momento-rotao, para ligaes
semi-rgidas segundo o modelo de KISHI e CHEN (1990).............126
Figura 3.30: Elemento viga-coluna, modificado devido a presena de
ligaes de extremidade semi-rgidas. .............................................128
Captulo 4: Resultados
Figura 4.1: Modelo estrutural de viga isolada em condies de incndio;
(a) seo-transversal do perfil exposto ao incndio nas trs
faces inferiores. ................................................................................ 134
Figura 4.2: Comparao entre os deslocamentos verticais em funo do
tempo de incndio normalizado, para o modelo de viga
isolada. ............................................................................................. 135
Figura 4.3: Comparao entre os deslocamentos verticais elsticos em
funo do tempo de incndio normalizado. ..................................... 138
Figura 4.4: Variao da configurao deformada do modelo de viga
simples, sob fator de carga =0,6. ................................................... 139
Figura 4.5: Comparao entre temperaturas para o perfil IPE-360
exposto ao fogo em 3 faces: (a) temperaturas na seo a 600
segundos; (b) idem para 1800 segundos; (c) idem para 3600
segundos........................................................................................... 140
Figura 4.6: Modelo estrutural de pilares isolados; (a) perfil exposto ao
fogo em 3 faces: mesa inferior e alma; (b) idem para 4 faces:
alma e mesas. ................................................................................... 143
Figura 4.7: Deslocamentos horizontais em funo do tempo de incndio,
para o modelo de pilar isolado formado pelo perfil IPE-360. ......... 145
Figura 4.8: Deslocamentos horizontais em funo do tempo de incndio,
para o modelo de pilar isolado formado pelo perfil W-360............. 145
Figura 4.9: Distribuio de temperaturas para o perfil IPE-360 exposto
ao fogo em 4 faces; (a) temperaturas ao longo da seo-
transversal no tempo de 600 segundos; (b) idem para 1800
segundos; (c) idem para 3600 segundos. ......................................... 147
xv
Figura 4.10: Distribuio de temperaturas para o perfil W-360 exposto ao
fogo em 3 faces; (a) temperaturas a 600 segundos; (b) 1800
seg.; (c) 3600 seg. ............................................................................ 148
Figura 4.11: Distribuio de temperaturas para o perfil W-360 exposto ao
fogo em 4 faces(a) temperaturas a 600 segundos; (b) 1800
seg.; (c) 3600 seg. ............................................................................ 148
Figura 4.12: Superfcies de resistncia plstica da seo-transversal do
perfil IPE-360 exposta ao fogo em 3 e 4 faces. ............................... 149
Figura 4.13: Superfcies de resistncia plstica da seo-transversal do
perfil W-360 aquecido em 3 e 4 faces. ............................................ 150
Figura 4.14: Mdulos elsticos equivalentes: EA e EI em funo do
tempo de incndio para os perfis IPE-360 e W-360, expostos
em 3 e 4 faces................................................................................... 151
Figura 4.15: Variao de esforos axiais equivalentes: Py e P, em funo
do tempo de incndio para o perfil IPE-360, exposto ao fogo
em 3 e 4 faces................................................................................... 152
Figura 4.16: Variao de esforos axiais equivalentes: Py e P, em funo
do tempo de incndio para o perfil W-360, exposto ao fogo
em 3 e 4 faces................................................................................... 153
Figura 4.17: Momentos equivalentes normalizados: Mp e M, em funo
do tempo de incndio para o perfil IPE-360, exposto ao fogo
em 3 e 4 faces................................................................................... 154
Figura 4.18: Momentos equivalentes normalizados: Mp e M, em funo
do tempo de incndio para o perfil W-360, exposto ao fogo
em 3 e 4 faces................................................................................... 154
Figura 4.19: Modelo de prtico plano tipo portal adaptado de VOGEL
(1985), sob condies de incndio normalizado; (a) perfis
expostos ao fogo nas trs faces internas: alma e mesa inferior. ...... 156
Figura 4.20: Comparao entre deslocamentos horizontais do modelo de
prtico plano em funo do tempo de incndio, obtidos pelos
programas PNL-F e SAFIR (FRANSSEN et al., 2000), para
diferentes nveis de carregamento aplicado ()............................... 157
xvi
Figura 4.21: Configurao deformada do prtico plano para diferentes
intervalos de tempo de incndio, =0,4........................................... 159
Figura 4.22: ndices plsticos associados flexo (*) para a estrutura do
prtico plano deformada sob fator de carga de 0,4; no
instante de 960s................................................................................ 160
Figura 4.23: Distribuio de temperaturas para o perfil HEA-340 exposto
ao fogo em 3 faces; (a) distribuio de temperaturas ao longo
da seo-transversal no instante de 600 segundos; (b) idem
para 1800 segundos; (c) idem para 3600 segundos. ........................ 161
Figura 4.24: Distribuio de temperaturas para o perfil HEB-300 exposto
ao fogo em 3 faces; (a) 600 segundos; (b) 1800 seg.;
(c) 3600 seg...................................................................................... 162
Figura 4.25: Curvas de resistncia plstica para os perfis HEA-340 e
HEB-300, para diferentes instantes do incndio padronizado......... 163
Figura 4.26: Variao normalizada da resistncia axial e esforo axial de
engastamento, em funo do tempo de incndio. ............................ 164
Figura 4.27: Variao do momento plstico e momento de engastamento
perfeito em funo do tempo de incndio para os perfis
HEA-340 e HEB-300. ...................................................................... 164
Figura 4.28: Reduo dos mdulos elsticos equivalentes: EA e EI em
funo do tempo de incndio para os perfis HEA-340 e
HEB-300 adotados no modelo de prtico plano. ............................. 165
Figura 4.29: Configurao geomtrica inicial e carregamento externo do
modelo de prtico plano industrial; (a) seo-transversal do
perfil IPE-360. ................................................................................. 166
Figura 4.30: Comparao entre os deslocamentos horizontais em funo
do tempo de incndio normalizado, obtidos pelos programas
PNL-F e SAFIR, para o modelo de prtico plano industrial,
em regime elstico. .......................................................................... 167
Figura 4.31: Desenvolvimento da configurao deformada do prtico
plano industrial para diferentes instantes do incndio, obtido
pelo programa PNL-F ...................................................................... 168
xvii
Figura 4.32: Modelo de edifcio de 4 andares sob incndio; (a) pilares
expostos em 3 faces; (b) idem para 4 faces; (c) vigas expostas
em 3 faces; (d) carregamentos vertical; (e) 50% do vento
proposto LEON et al. (1996). .......................................................... 169
Figura 4.33: Relao momento-rotao segundo o modelo tri-linear
proposto por LEON et al. (1996) para ligaes semi-rgidas. ......... 171
Figura 4.34: Distribuio de ligaes semi-rgidas para o edifcio de
4 andares proposto por LEON et al. (1996)..................................... 172
Figura 4.35: Variao de deslocamento horizontal do edifcio de 4 andares
(LEON et al., 1996), sob diferentes condies de incndio,
obtidos pelos programas PNL e SAFIR
(FRANSSEN et al., 2000). .............................................................. 172
Figura 4.36: Variao da configurao deformada do edifcio de 4 andares
adaptado de LEON et al. (1996) no tempo de incndio,
obtido pelo programa PNL-F........................................................... 175
Figura 4.37: Superfcies de resistncia plstica para diferentes instantes do
incndio padronizado para os perfis: W21x44 (3 e 4 faces) e
W14x82............................................................................................ 176
Figura 4.38: Reduo dos mdulos elsticos equivalentes: EA e EI em
funo do tempo de incndio para os perfis: W21x44 e
W14x82 (3 e 4 faces) adotados no modelo de 4 andares
adaptado de LEON et al. (1996). ..................................................... 176
xviii
ndice de tabelas
Captulo 2: Anlise Trmica
Tabela 2.1: Permetros expostos para elementos que compem a seo-
transversal de perfis metlicos I ou H. .......................................32
Tabela 2.2: Propriedades de elementos unidimensionais utilizados na
discretizao de sees-transversais de perfis metlicos I ou
H envolvidos por material de proteo contra incndio................47
Tabela 2.3: Fatores de massividade para o grupo de perfis metlicos
selecionados para anlise trmica entre os diferentes modelos
implementados. ................................................................................49
Tabela 2.4: Propriedades trmicas do material de proteo contra
incndio adotado nas anlises de comparao de variao de
temperatura. .....................................................................................50
Tabela 2.5: Diferenas percentuais entre resultados trmicos, para perfis
metlicos sem a presena de material de proteo contra
incndio............................................................................................53
Tabela 2.6: Diferenas percentuais entre resultados trmicos, para perfis
metlicos envolvidos por material de proteo contra
incndio............................................................................................56
Tabela 2.7: Fatores de reduo das propriedades mecnicas do ao para
diferentes nveis de temperatura ......................................................59
Tabela 2.8: Diferenas percentuais para propriedades de sees
equivalentes, de perfis metlicos desprotegidos, expostos ao
fogo em 3 faces. ...............................................................................66
Tabela 2.9: Diferenas percentuais para propriedades de sees
equivalentes, de perfis metlicos desprotegidos, expostos ao
fogo em 4 faces. ...............................................................................67
Tabela 2.10: Diferenas percentuais para propriedades de sees
equivalentes, de perfis metlicos protegidos, expostos ao
fogo em 3 faces. ...............................................................................67
xix
Tabela 2.11: Diferenas percentuais para propriedades de sees
equivalentes, de perfis metlicos protegidos, expostos ao
fogo em 4 faces. ...............................................................................68
Captulo 3: Anlise Estrutural
Tabela 3.1: Comparao entre valores de funes de estabilidade.....................76
Tabela 3.2: Diferenas entre os fatores mximos de amplificao de
momento para diferentes valores de carga axial e momento
aplicado. ...........................................................................................89
Tabela 3.3: Comparao entre os fatores mximos de amplificao de
momento, para momentos fletores com o mesmo sentido
(MA/MB 0).....................................................................................90
Tabela 3.4: Comparao entre fatores mximos de amplificao de
momento fletor, para momentos fletores opostos
(MA/MB < 0).....................................................................................90
Tabela 3.5: Curvas de flambagem para perfis tipo I ou H, fletidos
segundo seu eixo de maior inrcia (NBR-8800, 1986)....................95
Tabela 3.6: Fator de imperfeio para curvas de flambagem
(NBR8800, 1986).............................................................................95
Tabela 3.7: Fator de escala curvas de flambagem elsticas................................97
Tabela 3.8: Constantes para as expresses analticas do fator de reduo
de rigidez inelstico .........................................................................99
Tabela 3.9: Constantes para fator de reduo de rigidez inelstico
aproximados por polinmios de quarto-grau ...................................100
Tabela 3.10: Diferena entre as curvas de resistncia originais
(NBR8800, 1986 e LRFD-AISC, 1999) (A) e os resultados
obtidos com respectivo modelo de mdulo tangente (B). ...............112
Tabela 3.11: Diferena entre curvas de resistncia (EC-3/Parte-2, 2001)
(A) e resultados pelo modelo de mdulo tangente PNL-F
(B), para valores de esbeltez entre 0,1 e 1,9; temperatura no
ao entre 20oC e 1200oC. .................................................................115
Tabela 3.12: Limites de tenso-deformao para o ao em condies de
temperatura elevada, segundo o EC-3/Parte-2 (2001). ....................119
xx
Tabela 3.13: Coeficientes polinomiais para o fator de reduo de rigidez
flexional, em funo da temperatura do ao. ...................................121
Captulo 4: Resultados
Tabela 4.1: Tempo Crtico de Resistncia ao Fogo (TCRF) obtido pelos
programas PNL-F e SAFIR (FRANSSEN et al., 2000) para o
modelo de viga isolada. ................................................................... 137
Tabela 4.2: Comparao entre valores de flechas mximas elsticas
obtidos pelos programas PNL-F e SAFIR para diferentes
nveis de momento fletor, aps 1h de incndio. .............................. 139
Tabela 4.3: TCRF obtidos pelos programas PNL-F e SAFIR para
diferentes nveis de carregamento aplicado () para o prtico
plano tipo portal. .......................................................................... 158
Tabela 4.4: TCRF obtidos pelos programas PNL-F e SAFIR para
diferentes condies de aquecimento do prtico de 4 andares
(LEON et al. ,1996). ........................................................................ 173
xxi
Lista de smbolos:
Letras romanas
A rea da seo-transversal, rea do elemento
Ae superfcie da seo-transversal exposta ao fogo
Am rea da seo-transversal material de proteo trmica
A rea equivalente da seo-transversal
b caractersticas trmicas do material de fechamento do compartimento
B1 fator de amplificao de momento fletor
B2 fator de amplificao de momento fletor
bf largura da mesa de perfil metlico
ca calor especfico do ao
cm calor especfico do material de proteo contra incndio
Cm fator de homogeneizao de momentos fletores
dci deslocamento locais do elemento viga-coluna
dgi grau de liberdade em coordenadas globais
E mdulo elstico (mdulo de Young)
EA mdulo elstico equivalente associado rigidez axial
EI mdulo elstico equivalente associado rigidez flexional
Et mdulo tangente
fp, tenso limite proporcional
fy tenso de escoamento para temperatura ambiente
h altura total da alma de perfil metlico
Hcr matriz de transferncia de calor por conveco e por radiao
hw altura til da alma de perfil metlico
I momento de inrcia equivalente
k segmento bsico da seo-transversal
ka fator de correo emprico para anlises em condies de incndio
Kt matriz de conduo de calor
L comprimento do elemento
comprimento do elemento
M matriz de massa concentrada
M momento de extremidade
xxii
Mp20 momento plstico para temperatura ambiente (ou simplesmente Mp)
Mp momento plstico em funo da temperatura
n parmetro de forma de ligaes semi-rgidas
Ni funes de interpolao lineares
O fator de abertura para o compartimento
P esforo axial
p permetro exposto da seo-transversal
Pcr carga crtica de Euler, tambm adotado (Pe)
Py20 resistncia plstica axial para temperatura ambiente (ou simplesmente Py)
Py resistncia plstica axial em funo da temperatura
q fluxo de calor
qt,d densidade de carga de incndio acondicionada no compartimento
Rcr vetor de transferncia de calor por conveco e por radiao
Rkt rigidez tangente da ligao
rs comprimento de raio de solda de perfil metlico
S1 funes de estabilidade
S2 funes de estabilidade
tf espessura da mesa de perfil metlico
tlim taxa de crescimento do incndio
tm espessura do material de proteo contra incndio
tw espessura da alma de perfil metlico
u permetro efetivo da seo-transversal
u/A fator de massividade de elementos estruturais de ao sem proteo contra incndio
um permetro efetivo da seo-transversal envolvida por material de proteo
um/A fator de massividade para elementos com material de proteo contra fogo
Letras gregas:
j fluxo de calor por unidade de rea
la coeficiente de conduo de calor do ao
ra massa especfica do ao
ac coeficiente de transferncia de calor por conveco
jc fluxo de calor por unidade de rea associado conveco
qg temperatura do ambiente
xxiii
lm condutividade trmica do material de proteo contra incndio
qm temperatura na superfcie do ao
rm massa especfica do material de proteo
ar coeficiente de transferncia de calor por radiao
jr fluxo de calor por unidade de rea associado radiao
x comprimento qualquer do elemento de viga-coluna
estado de esforos combinados, momento fletor e esforo axial
fator de imperfeio
fator de forma especfico para anlises em condies de incndio
t intervalo de tempo
a,t elevao de temperatura do ao em funo do tempo t
res coeficiente de emissividade resultante
alongamento do ao em funo da temperatura
parmetros de reduo de rigidez sob temperatura ambiente
parmetros de reduo de rigidez em funo da temperatura
rotao de extremidade do elemento
temperatura
0 temperatura do ambiente antes do incio do aquecimento
a temperatura do ao
max temperatura mxima dada pela fase de aquecimento
r rotao relativa entre a viga e a coluna
ref temperatura de referncia
E, fator de reduo do mdulo elstico
p, fator de reduo do limite proporcional
y, fator de reduo do limite de escoamento
parmetro de esbeltez para barras comprimidas
armazenagem relativa de calor do material de proteo trmica
fator de correo
deformao
tenso
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