6th European Workshopon the seismicbehaviour of Ir-regular and
Complex Structures (6EWICS)O. Lavan, M. De Stefano (eds.)Haifa,
Israel, 1213 September 2011CARACTERIZACIN DE LA RESPUESTA SSMICA
DINMICA EN EDIFICIOS DE VARIOS NIVELES TORSIONALMENTE ACOPLADOSA.
G. Ayala1, O. Garca 21 Instituto de Ingeniera, UNAMCiudad
Universitaria, Mxico D.F. [email protected] Facultad de
Ingeniera, UNAMCiudad Universitaria, Mxico D.F. 04510
[email protected]: Center of torsion, dynamic
amplification, torsional seismic response, design
eccen-tricity,lateral load distribution, international code
provisions.Abstract. A large share of the buildings damaged during
recent destructive earthquakes has been attributed to ill torsional
behaviour; e.g., 42% of the buildings damaged by the 1985 Michoacn
earthquake in Mexico. One com-mon characteristic of the buildings
damaged by torsion was their asymmet-ricin-plandistributionsof
masses, stiffnessesand/orstrengths, modified when some of the
resisting elements exceeded their capacity, changing, of-ten in an
important way, the conception of the original design and therefore
the performance expected.A revision of current design codes shows
that the basis of the recommenda-tions for torsion design were
derived from investigations carried out on sim-plified models of
buildings, frequently single storey and linear elastic. These
results show evident differences when compared with those obtained
using more general and representative models which involve less
restrictive as-sumptionsandmorerealisticbehaviour modelsfor their
structural ele-ments. Particularly, most of the research carried
out in Mexico has been fo-cused on having a better understanding of
the problem and on finding the ways of improving the current
design. practice.Even though, many of the research results so far
obtained still have as limi-tation the models used, they still have
provided valuable information that has been used as basis for more
recent research at UNAM which considers more realistic models and
behaviors congruent with the limit states aimed to be satisfied
through performance based seismic design. In these investi-gations
several parametric studies have been carried out involving the
vari-ation of the most important seismic and structural
characteristics affecting the seismic performance of asymmetric
buildings, particularly, the in-plan distributions of masses, and
stiffnesses, generally defined by structural ec-centricity;
fundamental period, uncoupled frequencies ratio, in-plan aspect A.
G. Ayala and O. Garca ratio, and number of resisting planes, all
characteristics of interest in the linear range of behaviour; and,
the in-plan distribution of strengths, lateralstrengths and
over-strength, when they respond in the nonlinear range
ofbehaviour. The results obtained have increased our knowledge on
the per-formance of asymmetric buildings of different structural
layouts and seismic demands. The present paper studies the
influence of lateral load distribution in the location of the
center of torsion in multistory buildings taking different settings
of live loads as a func-tion of use structures. For these ones, it
makes an evaluation of the dynamic amplification com-paring them
with those used in international building codes. The scopes and
limitations of these codes to characterize of appropriate way the
torsional seismic response are discussed.2A. G. Ayala and O.
GarcaINTRODUCTIONWhileadvancesinscienceandtechnologyhaveadvancedenormouslyandthestructural
field, there are new design trends based on the nonlinear behavior
of the structure, which most closely represent the actual behavior
of a structure,the static remains an alternative seismic analysis
the vast majority of international design codes.To consider the
effects of torque on a structure, the seismic method static,
outcomes need to calculate the degree of asymmetry, which has
traditionally been measured by static or structural eccentricity,
given by the relative position of the shear center (CC) compared to
CT, the latter defined from the rigidities of the elements of
mezzanine. However, in previous approaches, [1] has been de-show
that for asymmetric rigidities structures, the location of CT in
the height of the building, is a direct function of the lateral
distribution of loads.CT torsion center of a building is defined as
the locus in their levels or mezzanines, in which they must apply
the seismic shear, so that there is only translation. In other
words, the center of torque is the point at which sits the result
of the resistance of a mezzanine where there is only
translation.Given the complexity of the phenomenon of torque
represents and in order to under-stand the seismic behavior of
asymmetric structures and identify the structural parameters
involved in major seismic torsional response, several
investigations have been conducted both in the range elastic and
the inelastic, initially using the simplified model shear level,
[p] Ayala and Garcia (1992). Recently, investigations have been
conducted using models of buildings of various lev-els that have
compared the elastic and inelastic responses of structures
torsionally coupled, [p,q] Ortega (2001), Chipol (2001).Results of
studies on seismic torsion, [p] Chipol (2001) indicate that the
position of the CT and hence the level of asymmetry of a building
have variations with height, despite maintaining the same proposed
structure. Analysis of the results of a number of building models
of various levels, has been evident that there is a tendency to
decrease the level of asymmetry in the case of relatively large
structures, there have been levels of asymmetry such that they are
able re-verse the direction of twist in the last mezzanines.It is
known that various international regulations seismic design force,
considered as an op-tion for elastic analysis, the static seismic
method, which requires rigorous application of the correct estimate
of structural asymmetry to determine the effects of torque.To
determinethe position of CTexist some approximate procedures based
on the useof mezzanine rigidities, however there is also a rigorous
procedure ([p] Damy and Alcocer, 1987) that can be applied with the
help of existing structural analysis programs. From them, it is
rela-tively easy to use three-dimensional models, whereby the
center of torque can be obtained by analysis in two orthogonal
directions and restricting the rotation of the levels on a vertical
axis. The center coordinates of torque (CT) are obtained by static
shear forces produced in the mez-zanine, [p] Chipol (2001) and [p]
Zrate (2002).In this work, a first group of models examines the
most critical condition that results in spa-tial variation of CT in
buildings of medium and high altitudes, although the RCDF currently
limits the application of the static method at 20 and 30 meters
high regular and irregular struc-tures respectively. Since the CT
is a function not only of the distribution and structural
rigidi-ties, but also the lateral distribution of loads, which in
turn depend directly on the charges on the levels of the building,
this work explores a range of conditions loads.On the one hand, the
variation of lateral load in each of the levels of a building,
according to the static method is a direct function of weight
levels, which consists of dead load plus live load. Assuming that
for a given structure, the dead load remains constant, then the
variation of the load vector depends only on the live loads
regulations.3A. G. Ayala and O. Garca According to RCDF, the live
load varies depending on the destination floor or deck. In re-gard
to building type structures, we find structures to use
was-seasonally, such as residential houses, apartments, houses,
bedrooms, hotel rooms, in-school suits, among others; Offices,
of-fices and laboratories; sites meeting as stadiums, libraries,
cinemas, theaters, etc.. In this paper, we consider the extreme
values of live load to see how it affects their participation in
the form presented by the spatial variation of
CT.Sincemostadverseeffectsinasymmetricstructures,
arecausedbythepresenceofstiff walls, such items were added to
structural systems of buildings 4 and 8 levels placed at its
pe-riphery.Given that the purpose of this approach is to observe
the variation with the height of the CT, it was proposed to keep a
constant value of base shear for each of the distributions
considered arbitrary lateral load.On the other hand, to
characterize the seismic torsional response, the majority of
inter-na-tional codes used dynamic amplification factors of the
static eccentricity, so in this work, and a second group of models
of buildings, the specified expressions are evaluated by the 2001
UBC, COVENIN 2001,Eurocode 8-1994 and RCDF NTCDS-2004 compared with
FADs obtained spectral modal analysis.The results show the
sensitivity of the location of the CT to the lateral distribution
of loads, in addition to the effect that the RCDF considering
different values of live load by use of the structure and the
inability of the expressions to characterize the seismic response
dynamically to the limited or no use of parameters upon which the
real answer.1 CONSIDERACIONES INTERNACIONALES EN EL DISEO POR
TORSIN SSMICAWhen used to analyze a static method to take into
account the effects of torque, most of the international code
established for each direction of analysis, an amplification factor
of the static eccentricity which controls the design of the flat
resilient the flexible area of the plant. Similar-ly, for the
design of the rigid zone of the plant is considered a controlling
factor to modify the static eccentricity, which is omitted is 1.
Addition is added to or subtracted from the flexible zone for a
rigid zone called accidental eccentricity, which is set as a
percentage of the width of the floor, perpendicular to the
direction of analysis. This will generate two design torsional
mo-ments having the form:
1 1 d d sM Ve ( e b ) + (1)
1 1 d d sM Ve ( e b ) + (2)The use of eccentric design results
in an increase in the forces shear resistant elements due to
torsional moment generated in an asymmetric building. Due to the
different amplification fac-tors involved in the ex-centric design
in different codes, for torsional effects overstrength pro-vided
may vary from one approach to another.When performing a spectral
modal dynamic analysis can be obtained for each mezzanine, a
torsional moment referred to the CM, which, when divided by the
dynamic inter-cutting floor, you get a dynamic eccentricity,. From
this dynamic eccentricity, it is possible to obtain the dy-namic
amplification factor of static eccentricity, defined according to
Bazan and Meli (200) ass dse ee+(3)4A. G. Ayala and O. Garca1.1
Reglamento del Distrito Federal NTCDS-2004According to the RCDF in
their NTCDS-2004, you can use the static method to analyze reg-ular
structures up to 30 m high and irregular structures up to 20 m
tall.The torsional moments are taken at least equal to the shear
mezzanine times the eccentricity for each frame or wall is more
favorable of the following two expressions:b e es d1 . 0 5 . 11+
(4)b e es d1 . 02 (5)where the factors of 1.5 and 1.0 consider
effects of dynamic amplification factor and the ampli-fication
effects 0.1 considers the effects of accidental eccentricity. 1.2
Cdigo Venezolano COVENIN-2001 The code Venezuelan COVENIN 2001,
proposed expressions to determine di-rights dynamic amplification
factors, which involve the structural eccentricity, the turning
radius and the ratio of uncoupled frequencies. For accidental
effects considered a value of = 0.06.[ ]41 4 16 0 5 11 4 16 2 2 1
21 2( ) .( ) ( ) + + ' (6)
6 1 0 6 1 1 ( ) . (7)1.3 Normativas Amricanas, UBC y ASCE7 -
2002The UBC and ASCE7 - 2002 consider the torsional moment due to
the structural eccentricity plusatorsionalmomentduetoaccidental
displacementofCM. Definedasirregular when torsionally
mezzanine>1.2, where y are the maximum and average dis-tortion
of a mezzanine in particular. In this case, calculate an
amplification factor A, from the maximum displacements maxand
average promthe end points at each level of the building and for
each direction, under the application of shear forces of floor time
with torsional 1 and 05 . 0 . 32 . 112max
,_
promA (8) The structure is designed by applying to expressions
with factors1 ,1 and A 05 . 0 1.4 Eurocode 8In the Eurocode (E[p]
C8-1994), it has an eccentricity e2 which takes into account the
cou-pling of the dynamic effects of translation and torsion, which
should be taken as the lesser of the following equations:22 2 2 2 2
2 2 2 2100 1 0 1142ss s s s sse. ( L B) . ( L B)Le mnl e r ( l e r
) e re+ +'1 + + + ] (9)where e= the eccentricity between CT and the
CM5A. G. Ayala and O. Garca 2 2212sL Bl+square of the gyration
ratio r2 = ratio of torsional and lateral stiffnessessThe
additional eccentricity e2 may be omitted if2 2 25s sr ( l e ) >
+In this paper, we evaluate dynamic amplification factors in
building models of 4 and 8 levels of asymmetric masses and
rigidities, which were studied above-[p,q] Ortega (2003) and De La
Rosa (2008).2 DESCRIPCIN DE LOS MODELOS DE EDIFICIOSFor the first
group of buildings studied in this work, group I, was taken as the
base models stud-ied by [p] Chipol (2001). We analyzed two
buildings, a 4 and one 8 lev-els, asymmetric rigidi-ties whose
asymmetry is provided by the addition of particular walls at its
periphery. In these models, we study the effect of lateral load
distribution in the spatial location of the CT. We con-sidered
arbitrary lateral load distributions and some others associated
with the variation in the weights of the levels produced by live
load values of maximum and minimum recommended standards. Models of
buildings are formed of reinforced concrete frames and correspond
to structures of group B, located in Zone IIIc based on seismic
zoning of the Valley of Mexico in accordance with the Building Code
of Federal District. Its structure is based on reinforced con-crete
frames. The plant is square and is divided into three bays.Figure
1: Vista en planta de los modelos de edificios estudiados, grupo I
Figure 2: Vista en elevacin de los modelos de edificios estudiados,
grupo IIn a second group of models, group II, the different factors
of dynamic amplification used in different international
codesobtainedcomparedwiththosetakenfromathree-dimensional spectral
modal analysis are analyzed. Building models that are based at this
stage were studied by [p,q] Ortega (2001) and De La Rosa (2008).
Dian study is building models with different numbers of levels, for
the cases considered in mass asymmetry (0.1B and 0.2B) and
stiffness (one and two concrete walls in one corner). For
simplicity in the use of models, using a nomen-clature the like
that used by [p] De La Rosa. where i = 4 and 8Mi - M10 Model i
levels, eccentric masses 0.10b6A. G. Ayala and O. GarcaMi - M20
Model i levels, eccentric masses 0.20bMi - R10 Model i levels,
eccentric rigidities, 1 wall in each directionMi - R20 Model i
levels, eccentric rigidity, 2 walls in each directionFigure 3:
Vista en planta de los modelos asimtricos en masas, [p] Ortega
(2001) Figure 4: Vista en planta de los modelos asimtricos en
rigideces R10, [p] Ortega (2001)Figure 5: Vista en planta de los
modelos asimtricos en rigideces R20, [p] Ortega (2001)7A. G. Ayala
and O. Garca Figure 6: Vista en elevacin de los modelos estudiados,
[p] Ortega (2001)3 CONSIDERACIONES EN EL ANLISIS Y DISEOModels of
buildings are formed of reinforced concrete frames and correspond
to structures of group B, located in Zone IIIc based on seismic
zoning of the Valley of Mexico in accordance with the Building Code
of Federal District. We selected a value of Q = 4. The load
analysis was conducted based on the Complementary Technical Norms
on Criteria and Actions for the struc-tural design of buildings,
2004.3.1 MaterialsThe concrete into the design of reinforced
concrete buildings is a Class I, with the following properties:
Elasticity modulus of Compressive strength Volumetric Weight Shear
Modulus The properties of reinforcing steel are considered: Yield
stress Mpa 0 . 20 4 fy Elasticity modulus Mpa 0 . 00 000 2 E 3.2
Diseo de los elementos estructuralesTables1and2present
thegeometriccharacteristicsofthestructural elementsforthetwo groups
of models studied.Building Columns Beams Walls(m) (m) (m)4 Leveles
0.60 x 0.60 0.25 x 0.50 0.081 - 5 Niv120 x 120 0.35 x 0.90 0.158
Leveles 5 - 8 Niv 110 x 110Table1 Dimensiones de los elementos
estructurales, modelos grupo IBuilding Columns(m)Main
Beams(m)Secondary Beams(m)Walls(m)4 Leveles 0.70 X 0.70 0.70 X 0.30
0.60 X 0.25 0.168 Leveles 0.80 X 0.80 0.80 X 0.40 0.60 X 0.25
0.16Table2 Dimensiones de los elementos estructurales, modelos
grupo II8A. G. Ayala and O. Garca4 CONSIDERACIONES EN LA MODELACIN
DE LOS
EDIFICIOSIthasbeenconsideredafoundationcapableenoughtowithstandtheeffectsofstructure
without rotating basis to dismiss and accidental torsional effects
such co-mo increments of dis-placements in the structure. As the
supports are modeled as embedded.Floor systems that work have been
considered as a rigid floor diaphragms in the background, well
enough to do the analysis based on three degrees of freedom per
floor are two orthogonal translations in the horizontal plane and a
rotation around a vertical axis.To calculate the various factors
involved in the seismic behavior of buildings, taken as a
ref-erence design recommendations set out in the RCDF-2004 and its
Complementary Technical Norms.For the analysis of models of
buildings as well as computing centers have been using the
tor-sional analysis program (training type) TOR3D (Garcia and
Islas, 2003) and the commercial program SAP2000, . [p]5 PRESENTACIN
Y ANLISIS DE RESULTADOS5.1 Efecto de una distribucin arbitraria de
carga lateral en la ubicacin espacial del Centro de Torsin.Although
considered several lateral load distributions, given the wealth of
information for these models, only 3 cases are arbitrary, but it is
clear that the position of the CT did not show significant
variation. The charge distributions considered and the variation of
CT for the 4-lev-els are shown in Figs. 7 to 9.12340.00 100.00
200.00 300.00 400.00NivelFuerza en kNDistribucin
deFuerzasC11234200.00 300.00 400.00 500.00 600.00
700.00EntrepisoXct(cm)Centrode TorsinC1CT CCFigure 7 Distribucin de
Fuerzas C1 y posicin del CT, Modelo de 4 Niveles12340.00 50.00
100.00 150.00 200.00NivelFuerza en kNDistribucin
deFuerzasC21234200.00 300.00 400.00 500.00 600.00
700.00EntrepisoXct(m)Centrode TorsinC2CT CCFigure 8 Distribucin de
Fuerzas C2 y posicin del CT, Modelo de 4 Niveles9A. G. Ayala and O.
Garca 12340.00 100.00 200.00 300.00 400.00NivelFuerza en
kNDistribucin deFuerzasC31234200.00 300.00 400.00 500.00 600.00
700.00EntrepisoXct(cm)Centrode TorsinC3CT CCFigure 9: Distribucin
de Fuerzas C3 y posicin del CT, Modelo de 4 NivelesUnlike models
based on stiffness of mezzanine of the above results, the
structural eccentrici-ty is not constant and tends to decrease with
height.Another aspect observed is that the distribution of applied
load has little effect on the spatial variation of CT, except for
the inverted triangular distribution in which there is a strong
varia-tion for the last mezzanine, however this distribution is
unreal .In Figs 10 to 12 have three lateral load distributions
applied to the models of 8 levels, in these shows that, regardless
of the distribution of forces considered, the location of the CT is
very similar in the mezzanine and lower on the other hand the
eccentricity is diminished greatly, so much invested in the last
mezzanine. 123456780.00 50.00 100.00 150.00 200.00 250.00 300.00
350.00NivelFuerza en kNDistribucin de FuerzasC112345678500.00
520.00 540.00 560.00 580.00 600.00 620.00
640.00EntrepisoXct(cm)Centro de Torsin C1CT CCFigure 10:
Distribucin de Fuerzas C1 y posicin del CT, Modelo de 8
Niveles123456780.00 100.00 200.00 300.00 400.00 500.00NivelFuerza
en kNDistribucin deFuerzasC212345678500.00 520.00 540.00 560.00
580.00 600.00 620.00EntrepisoXct(cm)Centro de Torsin C2CT CCFigure
11: Distribucin de Fuerzas C2 y posicin del CT, Modelo de 8
Niveles10A. G. Ayala and O. Garca123456780.00 100.00 200.00 300.00
400.00 500.00 600.00NivelFuerza en kNDistribucin de Fuerzas
C312345678500.00 520.00 540.00 560.00 580.00 600.00
620.00EntrepisoXct(cm)Centro de Torsin C3CT CMFigure 12:
Distribucin de Fuerzas C3 y posicin del CT, Modelo de 8
NivelesFigures 11 and 12 can also be observed that the asymmetry
given by the U-turn in the rota-tion of the mezzanine, tends to be
less critical if a force of greater magnitude in the last
mezza-nine, which would imply that the weight the roof is higher
than the lower levels.In general, for buildings 4 and 8 levels
studied, it appears that the eccentricity is not constant and
decreases with height. Moreover, the variation of CT is little
affected by the distribution of lateral forces applied. This does
not imply that the CT-defined do this only by the configuration of
rigidities in plant and this would lead to the CT locations for the
same structure constant, which is shown is incorrect.The results of
model 8 levels also were calibrated with the program SAP2000,
conducting in-dependent analysis and restricting rotation about a
vertical axis. In Figs. 13 to 15, shows three distributions of
lateral forces used and the variation of CT.DISTRIBUCINES DE
FUERZAS123456780.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00Fuerza
en tonNivelD_UnifD_MesttD_TinvFigure 13: Distribucin de fuerzas
laterales utilizadas COMPARACIN DEL CT, SAP200012345678400.00
500.00 600.00 700.00 800.00 900.001000.00 1100.00Xct en
cmEntrepisoXct_UnifXccXct_TinvXct_MesttFigure 14:CT para las
diferentes distribuciones de fuerzas, obtenidos con SAP200011A. G.
Ayala and O. Garca COMPARACIN DEL CT, TOR3D12345678400.00 500.00
600.00 700.00 800.00 900.00Xct en
cmEntrepisoXct_UnifXccXct_TinvXct_MesttFigure 15.CT para las
diferentes distribuciones de fuerzas, obtenidos mediante el
TOR3DAlthough both procedures for the uniform distribution and the
corresponding static method, there are differences in the position
of maximum TC of 12%, mainly for intermediate mezza-nine, for the
inverted triangular distribution are reached at the last level
differences of the order 20%. However the way in which the position
of the CT varies with height for both procedures can be considered
the same, which reaffirms that the position of center of torsion is
not influ-enced strongly by the manner of distribution of lateral
forces.5.2 Efecto de la carga viva en la distribucin de carga
lateral y en la ubicacin espacial del Centro de Torsin.The
variation of lateral load in each of the levels of a building,
according to the static method is a direct function of weight
levels, which consists of dead load plus live load. Assuming that
for a given structure, the dead load remains constant, then the
variation of the load vector de-pends only on the live loads of the
code.According to the RCDF, the live load varies depending on the
destination floor or deck. In regard to building type structures,
we find structures to use was-seasonally, such as residential
houses, apartments, houses, bedrooms, hotel rooms, in-school suits,
among others; Offices, of-fices and laboratories; sites meeting as
stadiums, libraries, cinemas, theaters, etc..In this section we
consider the extreme values of live load to see how it affects
their partici-pation in the form presented by the spatial variation
of CT.The values of live load specified in the NTCDS, established
two surround-sales, a minimum load and peak loads according to the
destination floor. Thus, designating the lower value, mini-mum live
load (CVmin), which corresponds to the instantaneous live load for
residential use with a value of 0.90 kN/m2 (90 kg/m2), and
similarly, maximum live load (CVmax), the live load of larger
magnitude corresponding to venues with a value of 2.5 kN/m2 (250
kg/m2).The roof instant live load is independent of the use of the
building and the NTCDS specify a value of 0.70 kN/m2 (70 kg/m2) for
roofs with slopes less than 5% and a value of 0.20 kN/m2 (20 kg /
m2) for slopes larger than 5%.For the analysis, taking reference to
the model of eight levels studied in the previous section,
considering a roof slope less than 5%. From the live load envelopes
were established two vec-tors of lateral loads, which are denoted
as Fmin and Fmax and shown in Fig. 16.In Fig. 16 shows that there
is variation in the magnitude of lateral load due to instant live
load, except for the last level, which the same amount of live load
is considered, in accordance with the RCDF.For each of the lateral
load vector obtained from the analysis in the direction "Y", it was
deter-mined the coordinate "X" for the CT for each mezzanine TOR3D
with the program. Static ec-12A. G. Ayala and O. Garcacentricities
were compared and normalized to the width on the ground
perpendicular to the di-rection of analysis. Table 3 and Figs. 17
and 18 show the results.01234567890.00 100.00 200.00 300.00 400.00
500.00NivelFuerza en kNDISTRIBUCIONES DE FUERZAS CON CARGA VIVA
INSTANTNEAFmnFmxFigure 16:Envolvente de fuerzas laterales,
considerando el efecto de carga vivaCENTRE OF TORSION XCT CC STATIC
ECCENT.NORMALIZED ECCENT. LEVEL CT_Fmn (cm) CT_Fmax (cm) (cm) emn
(cm) emax (cm) emn/B (%) emax/B (%) %VAR1 557.10 557.08 600.00
42.90 42.92 3.58 3.58 0.002 539.07 539.00 600.00 60.93 61.00 5.08
5.08 0.013 540.38 540.26 600.00 59.62 59.74 4.97 4.98 0.014 543.74
543.59 600.00 56.26 56.41 4.69 4.70 0.015 547.51 547.29 600.00
52.49 52.71 4.37 4.39 0.026 549.90 549.83 600.00 50.10 50.17 4.17
4.18 0.017 567.94 569.03 600.00 32.06 30.97 2.67 2.58 0.098 651.54
668.40 600.00 -51.54 -68.40 4.29 5.70 1.41Table3 Centros de torsin
y excentricidades estticas012345678500.00 550.00 600.00 650.00
700.00EntrepisoCT en cmCENTRO DE TORSINCT_FmnCT_FmxCMFigure 17:
Posicin del CT para la envolvente de cargas mnimas y mximas13A. G.
Ayala and O. Garca EXCENTRICIDADES NORMALIZADAS0123456780.00 1.00
2.00 3.00 4.00 5.00 6.00e/B en %EntrepisoFmnFmxFigure 18:
Excentricidades normalizadas con respecto al ancho en plantaThe
results can be seen that the effect of live load in the calculation
of the location of CT does not influence drastically and that the
maximum variation occurs in the last mezzanine, which is observed
in the curve of normalized eccentricity, Fig. 18.Based on the
results we can conclude that for a particular building, the
variation in the distri-bution of lateral forces applied due to the
regulatory live load used has little effect on the calcu-lation of
the location of the CT Mezzanine, regardless that the matrix
approach considered for obtaining the coordinates involving a
vector of lateral forces.Another important aspect is that by
increasing the load on the last level, there is a better
per-formance in building models studied, which is consistent with
studies in the inelastic range in which to consider a major force
in the last level decreases the ductility demands in recent
mez-zanines, De la Colina (2003). Based on the foregoing, it is
necessary to consider that the current regulation will incorporate
additional concentrated force at the last level, as included in
other international codes such as UBC 1997, among others.5.3
Dynamicamplification We present the results of dynamic
amplification factors (real) through a three-dimensional
spectralmodalanalysisandcomparedwiththoseobtainedby ex-pressuresset
design codes such as UBC, COVENIN, EC8 and RCDF. The results are
shown in the graphs below.For mass asymmetric models, we observe
that the amplification factor obtained had real, is always less
than that specified in the RCDF. The tendency to be constant in
height, because the proposed eccentricity constant at all levels
(0.1B y 0.20B).12340.5 1 1.5 2 2.5 3 3.5
4EntrepisoFADxM4-M10EUROCOD8REALESUBCCOVENINRDF12340.5 1 1.5 2 2.5
3EntrepisoFADxM4-M20EUROCOD8REALESUBCRDFFigure 19: Factores de
amplificacin dinmica, direccin x, modelos asimtricos en masas14A.
G. Ayala and O. Garca123456780 0.5 1 1.5
2EntrepisoFADxM8-M10EUROCOD8REALESUBCCOVENINRDF123456780 1 2
3EntrepisoFADxM8-M20EUROCOD8REALESUBCRDFFigure 20: Factores de
amplificacin dinmica, direccin x, modelos asimtricos en masasFor
the direction "X" in the asymmetric mass models shows that the
amplification factor ob-tained by using the term considered in the
UBC and the Eurocode 8 is closer to the actual values when the
level of asymmetry is very large (0.2B). For his part, COVENIN,
reports results with sufficient accuracy, but only in the range of
, interval for which the formula is applica-ble. 12340.5 1 1.5 2
2.5 3 3.5 4EntrepisoFADyM4-M10EUROCOD8REALESUBCCOVENINRDF12340.5 1
1.5 2 2.5 3EntrepisoFADyM4-M20EUROCOD8REALESUBCRDFFigure 21:
Factores de amplificacin dinmica, direccin x, modelos asimtricos en
rigideces123456780 1 2
3EntrepisoFADyM8-M10EUROCOD8REALESUBCCOVENINRDF123456780 0.5 1 1.5
2 2.5EntrepisoFADyM8-M20EUROCOD8REALESUBCRDFFigure 22: Factores de
amplificacin dinmica, direccin y, modelos asimtricos en masasIn "Y"
direction in both models 4 and 8 levels, amplification factors
obtained with the UBC have taken an opposite trend to the one
presented in direction "X", and obtained very conserva-tive values
for large asymmetries. The COVENIN remains very close to actual
values, but is ap-plicable only for very small asymmetries flutes.
For its part, the Eurocode, yields valuestoo conservative in both
cases of asymmetry.The behavior of models with asymmetric rigidity,
as expected, was very different from the asymmetric mass models,
this because even when these models are associated with a similar
level of asymmetry, these structures are completely different. 15A.
G. Ayala and O. Garca 12340.5 1 1.5 2 2.5 3 3.5
4EntrepisoFADxM4-R10EUROCOD8REALESUBCRDF12340.5 1 1.5 2 2.5
3EntrepisoFADxM4-R20EUROCOD8REALESUBCRDFFigure 23: Factores de
amplificacin dinmica, direccin y, modelos asimtricos en
rigideces123456780 0.5 1 1.5 2
2.5EntrepisoFADxM8-R10EUROCOD8REALESUBCRDF123456780 1 2 3 4 5 6 7 8
91011121314EntrepisoFADxM8-R20EUROCOD8REALESUBCRDFFigure 24:
Factores de amplificacin dinmica, direccin y, modelos asimtricos en
rigidecesFor "X" direction, which corresponds to a width
perpendicular to the minor earthquake di-mension, the real FADs are
lower than those obtained using the expressions of the UBC and EC8
and generally are considered less than the value of amplification
factor in RCDF.12340.5 1 1.5 2 2.5 3 3.5
4EntrepisoFADyM4-R10EUROCOD8REALESUBCRDF12340.5 1 1.5 2 2.5
3EntrepisoFADyM4-R20EUROCOD8REALESUBCRDFFigure 25: Factores de
amplificacin dinmica, direccin y, modelos asimtricos en
rigideces123456780 1 2 3
4EntrepisoFADyM8-R10EUROCOD8REALESUBCRDF123456780 1 2 3
4EntrepisoFADyM8-R20EUROCOD8REALESUBCRDFFigure 26: Factores de
amplificacin dinmica, direccin x, modelos asimtricos en
rigideces16A. G. Ayala and O. GarcaAs shown in Figs. 25 and 26, the
real dynamic amplification factor obtained for the address "Y",
associated with increased plant size in the direction perpendicular
to the earthquake, is al-ways greater than the specified value of
1.5 the RCDF. In regard to COVENIN, because the
ex-pressionsusedtocalculatethe FADs,were obtained by adjusting
model-based extreme dis-placements whose asymmetry is given by
masses yields values out of context to asymmetries in stiffness, so
not included in the charts.In general, we see that in models with
asymmetric rigidity, the FAD is not constant in height, which is
associated with the level of asymmetry of each mezzanine. Analyzing
the results of the models, one can observe that the values of the
FADs are mainly associated to and directly to the plant dimension
perpendicular to the direction in which it operates the earthquake
and the turn-ing radius of inertia. Thus, one factor was proposed
amending the square of the inertial turning radius content Eurocode
expression for calculating, resulting the following equations: 22 2
2 2 2 2 2 2 2100 1 0 11 4 442 5 5ss s s s sse. ( L B) . ( L B)Le
mnB Bl e r ( l e r ) e re L L+ +' 1 + + + 1 ] (10)where B and L,
correspond to the plane dimensions perpendicular and parallel to
the direction of the earthquake, respectively. By including this
factor was a reasonable approximation can ad-just
theamplificationfactorvaluestoactual
valuesobtainedfortheanalyzedmodels.The results are shown in Fig. 27
to 30:12340.5 1 1.5 2 2.5 3 3.5
4EntrepisoFADxM4-R10EUROCOD8REALESUBCRDF12340.5 1 1.5 2 2.5
3EntrepisoFADxM4-R20EUROCOD8REALESUBCRDFFigure 27: FADs obtenidos
con e2, direccin x, modelos asimtricos en rigideces 123456780 0.5 1
1.5 2EntrepisoFADxM8-R10EUROCOD8REALESUBCRDF123456780 1 2 3 4 5 6 7
8 91011121314EntrepisoFADxM8-R20EUROCOD8REALESUBCRDFFigure 28: FADs
obtenidos con e2, direccin x, modelos asimtricos en rigideces 17A.
G. Ayala and O. Garca 12340.5 1 1.5 2 2.5 3 3.5
4EntrepisoFADyM4-R10EUROCOD8REALESUBCRDF12340.5 1 1.5 2 2.5
3EntrepisoFADyM4-R20EUROCOD8REALESUBCRDFFigure 29: FADs obtenidos
con e2, direccin y, modelos asimtricos en rigideces 123456780 1 2 3
4EntrepisoFADyM8-R10EUROCOD8REALESUBCRDF123456780 1 2 3
4EntrepisoFADyM8-R20EUROCOD8REALESUBCRDFFigure 30: FADs obtenidos
con e2, direccin y, modelos asimtricos en rigideces It is observed
that by using expressions of Eurocode with the proposed
amendments,the FADs in most reasonably fit mezzanines values
amplification considered as real in both models 4 and 8 levels for
the two levels asymmetry in both mass and stiffness as in the two
directions of analysis. As an exception when the last mezzanines
for models with asymmetric rigidity, however, this amplification
factor is associated with a very small eccentricity, so the
torsional moment is not very high, so it is considered that its
effects are not critics on the mezzanine. On the other hand, if we
consider that the asymmetry in stiffness was provided by concrete
walls, in corners, strictly considered as irregular structure for
which the RCDF, specifies a height not less than 20 m (6 levels
approximately) to apply the static method, the range in which
amplification factors are adjusted properly.CONCLUSIONSAfter
analyzing the results in models 4 and 8 levels studied, it can be
concluded that, unlike the procedures based on mezzanines
rigidities, the eccentricity obtained by three-dimensional matrix
procedure (Damy, 1987) is not constant and tends to decrease with
height, and can even change the direction of rotation of the upper
mezzanine, which is consistent with previous stud-ies, Chipol
(2001), Ortega (2001). This seems to be a common feature in all the
buildings so far studied.Another important aspect is that with
increasing weight in the last level, there is a decrease in the
structural asymmetry in asymmetric building models studied, which
is consistent with stud-ies in the inelastic range in which to
consider a major force in the last level, decreases the duc-tility
demands in recent interstoreys, De la Colina (2003). Based on the
foregoing, it is impor-tant to consider that the current regulation
will incorporate additional concentrated force at the last level,
as in codes such as the UBC 1997, among others.The group of
buildings regulatory studies considering extreme values of live
load, it was ob-served that the variation which presents the
distribution of lateral forces are not significantly 18A. G. Ayala
and O. Garcamodition and therefore has little effect on the
calculation of the location of the CT inter- floor. This means that
although the matrix formulation presented in this paper, the CT is
also a func-tion of lateral load distribution, it is mainly
influenced by the structure of the building, though
certainlynon-permanentlocationwiththeconstantneedheightwhilemaintainingthesame
structural configuration in all mezzanines.In evaluating the
expressions for calculating the dynamic amplification factor twist
set in some codes, it was noted that no expression can adequately
represent the actual responses of the amplification factors
obtained from a spectral modal analysis.The FADs obtained with
COVENIN-2001, was adjusted to adequately re-ales values in the
range is limited to the application according to the specifications
of the standard, but only for asymmetric mass models. For
asymmetries rigidities given by the expression valuesreported out
of context even in the range of application.By comparing the value
of 1.5 which establishes the real RCDF with FADs can say in
gener-al that asymmetries in mass, this value is appropriate in any
direction because this value is ex-ceeded. However for asymmetries
in rigidity, the value of 1.5 is clearly exceeded.Meanwhile the
values obtained with the Eurocode8 and UBC fit only for certain
levels of asymmetry depending on the direction in which analysis
takes place. Although-comes clear that the manner in which the
amplification factor applied to the UBC (affecting only the
accidental eccentricity is not correct) and that asymmetries are
produced rigidities higher magnifications as shown in this work.By
altering the expression of the factor Eurocode 4 / 5 (B / L) showed
that the values ob-tained for the amplification of asymmetric
models, reasonably adjusted to the actual FAD ob-tained by
three-dimensional spectral modal analysis. On this basis, in a
subsequent study will examine the influence of different parameters
involved in seismic torsional response, in order to obtain a factor
applicable to this expression which is valid for any type of
asymmetry.ACKNOWLEDGEMENTSThe constructive comments and the
interest shown to the work presented here We thank Dr. Mauro Nio
for his and the sponsorship of the Government of the Federal
District of the project "Development of the conceptual, theoretical
models and simplified methods for evaluation and seismic design of
structures based on performance " and the scholarship of the second
author during his graduate studies. REFERENCES [1] J.Damy, S.
Alcocer, Obtencindel CentrodeTorsindeEdificios, MemoriasVII
CongresoNacional deIngenieraSsmica,
SociedadMexicanadeIngenieraSsmica, 1987. [2] E. Bazn, R. Meli,Diseo
Ssmico de Edificios, Ed. Limusa, Mexico, 2000.[3] A.Chipol, Estudio
de la Respuesta Ssmica de Modelos Tridimensionales de Edificios
Torsionalmente Acoplados, Tesis de Maestra, DEPFI, UNAM, 2001.[4]
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Asimetra en Planta y Elevacin, DEPFI, UNAM, 2001.19A. G. Ayala and
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enlaRespuestaSsmica Inelstica de EdificiosTorsionalmente Acoplados,
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