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(1) Instituto de Ingeniera, Universidad Nacional Autnoma de
Mxico. Circuito Escolar s/n, Ciudad Universitaria. Delegacin
Coyoacn, Mxico, D. F. CP 04510, Mxico. 2)
Investigador, Instituto Mexicano de Tecnologa del Agua, Paseo
Cuauhnahuac 8532, Jiutepec 62550, Morelos, Mxico. E-mail:
[email protected]
DYNAMIC TORSION IN STRUCTURES COMPUTED FOR SEVERAL CONTROL
POINTS
Martha Suarez(1) and Javier Avils(2)
SUMMARY
In the dynamic design of structures the criteria used to account
for the dynamic torsional coupling effects generally suggest that
the design eccentricity most be obtained from computing the dynamic
and accidental eccentricities obtained from applying the
coefficients indicated in design codes. These coefficients were
computed considering the structures founded on a rigid base,
assuming that maximum translational and torsional responses occur
simultaneously and are additive, and also it is supposed that the
maximum displacement occurs in the periphery of the structure. In
this paper the dynamic coefficients for the design eccentricity are
calculated considering the effects of soil structure interaction,
the peak coupled lateral-torsional response and several observation
points located between the stiffness center and the periphery of
the structure. The purpose is to compare the computed coefficients
with those proposed in design codes. Key words: design
eccentricity; and coefficients; couple response; soil-structure
interaction
INTRODUCTION
On the static analysis for buildings, the design codes stablish
that the torque effects most be
considered by applying equivalent lateral forces to a distance
edis from the stiffness center. In particular, the Mexican codes
(NTCD DF, 2004) stipulate that the design eccentricity must be the
most unfavorable from the following equations:
1.5 0.1 0.1
(1)
Where e is the static eccentricity given by the distance between
the centers of mass and stiffness, and B is the length of the
building slab perpendicular to the seismic excitation. The
coefficient that multiplies the structural eccentricity is called
the dynamic eccentricity coefficient (named here as ) and has the
purpose to take into account the lateral and torsional movements of
the structure; the second addend considers the effects due to the
accidental torque, for example, among others, the rotation
generated from the wave passage and the current discrepancies
between the actual and computed eccentricities. In this second
addend, B is multiplying by a coefficient =0.1. This coefficients
values are based on the results obtained from the analysis for
structures founded on rigid base and on the engineering
judgment.
When the flexibility of the foundation is not taken into
account, their influence in the torsional
response of the system is neglected (Li Y y Jiang X , 2013). The
wave passage (Veletsos AS y Prasad AM, 1989) and the incoherence of
the soil movement (Veletsos AS y Tang Y, 1990) tend to reduce the
translational movement by filtering the high frequencies in the
base, and to generate rocking and torsion in the whole building
(Avils J y Suarez M, 2006). The depth of the foundation contributes
also to reduce the torque response in the structure (Iguchi M,
1982; Clough RW y Penzien J, 1975) if the structure is not so
stiff.
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In the definition of and it has been considered that the maximum
responses of translation and rotation take place simultaneously and
consequently are summed. This is a very conservative criterion
because it is assumed that the peak values of the natural and
accidental torque take place at the same time and in the same
direction, given results that overestimate the design moment to
torsion. A less conservative criterion implies to consider the peak
dynamic displacement as the simultaneous action of all the
components of the excitation at the base. This can be seen like an
extension of the approximation developed by Dempsey and Tso (1982)
to estimate the effects of the torsion due to the rotation of the
foundation. Another point to consider is the location on the
structure slab where the computations of the displacements are
performed. Generally the displacements are evaluated considering a
point located on the periphery of the floor diaphragm supposing
that the response variation to the rotational center is linear.
The system is considered linear. However, when we deal with
non-linear structures the global
effects to torsion are quantitatively similar to the elastic
ones because the differences in the responses are more pronounced
in translational movements than in torsional ones.
The purpose of this paper is to evaluate the dynamic and
accidental coefficients for torsion ( and
, respectively) considering the kinematic (wave passage effect)
and inertial (base flexibility) interaction between the soil and
the structure. The analysis is made considering control points
located between rotational axis and the edge of the slab. The
combined effects of structural asymmetry and rotation of the
foundation are interpreted by means of dynamic and accidental
eccentricities in a similar way that has been performed in previous
works where these effects are considered independent one from the
other. The eccentricities are computed taken into account the
maximum couple displacements due to translation and torsion. In the
analysis there were considered more than 400 earthquake records of
events with magnitudes equal or bigger than 6 registered in
stations located on the soft soils in Mexico City.
Even when the results presented here are directly applicable to
simple structures, it is considered
that the reported conclusions can be also applied to the
fundamental mode of more complex structures that satisfied certain
conditions (Kan CL y Chopra AK ,1977) and that remain in the
elastic range during moderated earthquakes..
COEFFICIENTS AND In figure 1 is presented the model used to
compute the coefficients and . It consists of an
oscillator with five degrees of freedom, two of them concern to
the structure (lateral (e) and torsional (e) displacement related
to the center of the slab) and the other three are the foundation
displacements (torsion (f), translation (f) and rocking (f))
related to the input motion. The excitation of the system is given
by shear waves with a motion parallel to the y axis, with
impingement on the foundation at an angle measured on the z axis.
With this antiplane shear waves, are computed the horizontal
displacements o, rocking o and torsion o input foundation motions
at its center. The distance between its mass center CM and its
stiffness center CT that defines the structural eccentricity is
called e. The side of the structure square rigid slab measures 2a,
and is at a height He helded by four inextensible columns conected
to the foundation with an embedment D. The building and foundation
masses, Me and Mf , are not uniformly distributed over their
identical square areas. The related parameters are the area moments
Ie and If to the horizontal centroidal axis and polar moments of
inertia Je y Jf to the vertical centroidal axis.
The structure is characterized by the uncouple periods to
translation and torsion when it is
considered on a rigid base, and their corresponding damping
ratios ( h 5%). The foundation soil is
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idealized like a viscoelastic homogeneous halfspace with a
velocity of shear wave propagation Vs, mass density s, Poisson
ratio s=0.4 and damping ratio s=5%.
The equations that govern the movement in the frequency domain
are: ooohgssss QQQi JIMMCK 22 (2) Where is the circular frequency
of the excitation and 1i the imaginary unit. Tfffees ,,,, is the
amplitude for the displacement vectors of the system, and ee d
and
ff d are the torsional displacements in the building and the
foundation, respectively, computed in a point x located on the
symmetry axis to a distance d from the shear center, and fef DH )(
is the displacement due to the rocking measured on the top of the
building. The ratios gohQ ,
goe DHQ )( and godQ represent the transfer functions of the
input motions, where g is the horizontal free field movement. The
input motions at a depth D are the horizontal displacement o , the
rocking o and the torsion o related to the x axis and the z axis,
respectively (see figure 1).
The equation (2) represents a system of five algebraic linear
equations with complex coefficients. In
Avils J y Suarez M (2006) are defined the load vectors oM , oI
and oJ , as well as the mass sM , damping sC and rigidity sK
matrices of the system.
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Figure 1. Model used in computations
By the computations of the input motions was used the Iguchis
method (1982), and for the impedance functions those reported in
Mita and Luco (1989) were employed. The calculus was performed in
the frequency domain using the Fast Fourier Transform and, for
simplicity, it was considered
efefef MMJJII .
The torque for the structural design was obtained as
follows:
andis TT=T (3) where:
;V~e=T odn by natural torsin (4)
;V~e=T oaa by accidental torsin (5)
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de and ae are the dynamic and accidental eccentricities to be
computed and oV~
is the uncoupled design shear due to the effective excitation
for translation and rocking, obtained as:
oeho KV ~~ (6)
is the lateral stiffness for the structure and |)0(|~ emax eo is
the maximum dynamic displacement in a symmetric structure.
de and ae are determined to satisfied:
dK
eeVKV
eado
eh
op
)(~~~ (7)
where ||~ eep max is the maximum dynamic displacement at dx . It
is considered that the force oV
~ is statically applied at a distance ad ee from the stiffness
center to produce a displacement in
the flexible side of the structure equal to p~ (figure 2).
Accepting that:
dK
eVKV
edo
eh
ohp
~~)~( (8)
dK
eVeao
hpp
~)~(~ (9)
where is the torsional stiffness for the structure and
|)0()0(|)~( QQmax eehp is the peak dynamic displacement due to the
base excitation for translation and rocking. Substituting the
equation (7) on (8) and (9), the dynamic and accidental
coefficients ( and , respectively) are obtained by:
da
ar
eee
ro
hpd22
21~
)~( (10)
da
ar
be
o
hppa2
2
2
~)~(~ (11)
with aeer , 21)( ee MJr is the polar gyration radius and is the
ratio for torsional to translational uncoupled frequencies:
bh
b
h KK
r
1
(13)
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6
where 21)( eJK and 21)( ehh MK are the circular frequencies to
torsion and translation, respectively .
Figure 2 Equivalent static force oV
~ applied at that produces a peak dynamic displacement p~ in
the flexible side of the building.
The ratios ohp ~)~( and ohpp ~])~(~[ give a measure of the
modified structural response
due to the torsional coupled and to the rotational excitation on
the base, respectively. This is a natural and convenient way to
separate the torsional effects due to the rotation of the
foundation, from those generated by the structural asymmetry. When
and have negative values this means that the lateral displacement
is reduced due to torsional effects.
NUMERICAL RESULTS
A parametric study was performed using a database with more than
400 accelerograms (Sociedad
Mexicana de Ingeniera Ssmica, 2000) of earthquakes with
magnitudes bigger or equal to 6. The seismic stations that
registered them are settled over the soft soils in Mexico City. The
seismic records were normalized in order to have a one second
period approximately in the Fourier spectral peak. This guarantees
that only the parameters related to the soil-structure system
influence the torsional response and to avoid the site effects
where the stations are located.
The results presented here correspond to the mean values of the
system response subjected to the
accelerograms from the database with specific soil
characteristics and the structure defined by the parameters shown
in table 1. The accelerograms chosen were supposed to generate an
antiplane movement (on the direction of axis y, figure 1) impinged
on the foundation with an angle . The incident angle is related to
a relevant parameter in the seismic characterization (the apparent
velocity = c) through the following equation:
cVssin (14)
The relation between the stiffness of the structure and the
soil, shew VH2 , is related with the
transient time of shear waves generally used to determine the
foundation flexibility as hhwsVa 2 (15)
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If h is inversely proportional to eH , then w measures only the
soil flexibility. The results showed consider fixed values by cVs
and sVa because their interpretation by these terms is more useful
than by terms of and w . The values considered in the parametric
study were cVs 0.025, 0.05 and 0.1, and for
sVa 0.15 and 0.3 s, that correspond to foundations with 30 to
60m of width embedded in soft soils with shear velocity ranges sV
50 to 200 m/s and phase velocities c bigger than 500 m/s, similar
to those observed in seismic records obtained from dense arrays
(Bolt et al., 1982; ORourke et al, 1982). The wave passage effects
are controlled by the effective phase velocity that is equivalent
to the order of rupture velocity or the propagation velocity in
rocks (Luco y Sotiropoulos, 1980; Bouchon y Aki, 1982). Then the
huge values for c , in comparison with those for sV , seem to be
reasonable for engineering applications.
Table 1. Parameters used in the analysis
DESCRIPTION NOMENCLATURE VALUES Incident angle 1.5, 3 and 6
Slendernees aH eh 1 and 3
Foundation depth aDd 0, 0.5, 1.0 and 1.5 Normalized eccentricity
aeer 0.10 and 0.20
Transient time hhws TVa 4 0.15 and 0.30
Structural period to translation
hT 0.25s, 0.5s, 1s and 2s
Uncoupled ratio frequencies
between 0 and 3
The analysis was performed for short structures (h = 1), with
translation periods Th = 0.25 and 0.5 s,
and for medium structures (h = 3), with Th = 1 and 2 s. In spite
of the structures hardly will reach the uncoupled ratio frequencies
() bigger than 2, the computations were done for 0 3 in order to
know the results tendency. There were considered shallow structures
(d = 0) and with foundations depths (d = 0.5, 1.0 y 1.5), with
eccentricities er = 0.1 and 0.2. The results were computed
considering different point positions located on the flexible side
of the structure at distances 0.1 d/a 1.0 from the stiffness
center.
In order to avoid strong variations in the structural response
for an specific earthquake, and to
identify the trends of the results, there were obtained the mean
values for the factors and .
and variations due to the structural period and the incident
angle of the excitation
Figures 3 and 4 show some of the results obtained for the
coefficients and , respectively. There it can be appreciated how
the angle of incidence of SH waves, the depth of the foundation,
the structural eccentricity and the period for translation
influence in the values of these coefficients.
The advantage in normalizing results with respect to the
uncoupled shear generated in the structure
on a flexible foundation, is to obtain small variations in the
values of when the parameters, but Th and er, vary, that is, the
graphics tend to gather depending on Th and er values (figure 3).
This implies that the other parameters like the incident angle of
the excitation and the foundation depth have a very little
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influence on the value of . For , the values are small but,
considering the proportion among them, the appreciated variations
are very important and it is not so clear the gather of the curves
related to the structural period (figure 4). Due to the small
values obtained for when the point of observation is located at the
periphery of the slab, the design eccentricity is almost the same
as the dynamic one because the wave passage effect, that is one of
the elements that has an important contribution to the accidental
eccentricity when it is considered founded on a rigid base, here is
taken into account when there are considered the soil-structure
interaction effects for the computation of both coefficients ( y ).
In spite of the fact that the curves have a specific pattern for a
given period Th, the amplitude differences imply a dependency on
incident angle of the excitation and the foundation depth. The
angle of incidence for SH waves has a negligible influence on . For
, the highest values are for the biggest incident angles, keeping
approximately the same curve form.
Figure 3 Variation of related to for x/a=1, considering the
values of d and h reported on table 1and
and er=0.1, for Th = 0.25s, 0.5s, 1s and 2s
0 1 2 3-1
-0.5
0
0.5
1
1.5
2
2.5
3=1.5
0 1 2 3
-1
-0.5
0
0.5
1
1.5
2
2.5
3=6
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Figure 4 Variation of related to for x/a=1, considering the
values of d and h reported on table 1and
and er=0.1, for Th = 0.25s, 0.5s, 1s and 2s
The negative values for and can be deduced from the analysis of
equations 10 and 11, respectively, and they are obtained mainly for
torsionally flexible ( < 1) and high (h = 3) structures. For
these negative values mean that the shear basal force is reduced
with respect to the uncoupled one which represent a favorable
effect. For , the negative values indicate that the displacements
due to the horizontal movement and the base rotation, are opposed,
causing a reduction on the coupled basal displacement. This
indicates that the torsional coupled effects and the rotational
excitation could be advantageous.
The system behaves like a torsionally uncoupled one ( = 0) or
like a static one ( = er /22 ) when
the values for are very small or very large (figure 3). On this
last case, the torsion for the foundation can be computed with the
product between the uncoupled shear and the static eccentricity.
The variation of related to do not follows a clear and consistent
pattern like the one showed by coefficient . approaches to cero for
very small values of . When is very large, there are no theoretical
indications that consider this factors limits (see figure 4). The
computed values can be negative as a consequence of the base shear
reduction caused by the torsional coupled effects. This is evident
for some values, mainly when the points of observation are located
near the stiffness center of the structure (figure 6). For is also
appreciated this tendency in points that are not located near the
structural periphery (figure 5).
0 1 2 3-0.005
0
0.005
0.01
0.015
0.02
0.025=1.5
0 1 2 3
-0.02
0
0.02
0.04
0.06
0.08
0.1=6
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Figure 5 Variation of related to for x/a=0.4, considering the
values of d, and h reported on table 1
and er =0.1 for Th = 0.25s, 0.5s, 1s and 2s
Figure 5 Variation of related to for x/a=0.4, considering the
values of d, and h reported on table 1
and er =0.1 for Th = 0.25s, 0.5s, 1s and 2s
0 1 2 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2=1.5
0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2=6
0 1 2 3-5
0
5
10
15
20x 10
-3 =1.5
0 1 2 3
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07=6
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Foundation embedment effects
The influence of the foundation embedment on is negligible. Only
for shallow foundations there
are differences in the peak value with amplitudes that are
smaller for deep foundations. In structures with short periods (Th
= 0.25 s) this is reversed because there is no reduction in the
basal shear streess due to the foundation depth (increase in
stiffness). For structures with larger periods, the influence of
the foundation depth is almost absent (figure 7).
The accidental eccentricity decrease when the depth of the
foundation increases because it generates
wave diffraction, reducing the effects of excitation that
directly affect the value of (figure 8).
Figure 7 Variation of related to for x/a=1, er = 0.1, = 1.5 and
d = 0.0, 0.5, 1 and 1.5. Foundation flexibility influence
The results computed for the transient times sVa 0.15 y 0.3 s
that generally characterize the
foundation flexibility, are shown in figures 9 and 10. For the
structures with short periods (Th 0.5 s), the absolute value for
tends to be a little bit higher for systems with sVa 0.15 s than
for those with
sVa 0.3 s. This behavior is reverse for medium structures ( h 3)
(see figure 9). The coefficient values are smaller when sVa 0.15 s
than those when sVa 0.3 s (figure 10).
0 1 2 30
0.5
1
1.5
2Th=0.25s
0 1 2 3
0
1
2
3Th=0.5s
0 1 2 3-1
0
1
2
3Th=1s
0 1 2 3
-1
0
1
2Th=2s
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12
Figure 8 Values for coefficient when x/a = 1, er = 0.1, = 1.5
and d = 0.0, 0.5, 1 and 1.5.
Figure 9 Values for coefficient when x/a = 1, er = 0.2, = 1.5
and 0.0, 0.5, 1 and 1.5. In solid line
sVa 0.3 s, and in dash line sVa 0.15 s.
0 1 2 3-5
0
5
10x 10
-3 Th=0.25s
0 1 2 3
-5
0
5
10x 10
-3 Th=0.5s
0 1 2 3-5
0
5
10x 10
-3 Th=1s
0 1 2 3
-0.01
0
0.01
0.02
0.03Th=2s
0 1 2 30
1
2
3
4Th=0.25s
0 1 2 3
0
2
4
6Th=0.5s
0 1 2 3-2
0
2
4Th=1s
0 1 2 3
-2
0
2
4Th=2s
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13
Figure 10 Values for coefficient when x/a = 1, er = 0.2, = 1.5
and 0.0, 0.5, 1 and 1.5. In solid line
sVa 0.3 s, and in dash line sVa 0.15 s.
Response to wave passage
Some of the results for the dynamic torsional coefficients when
cVs 0.025, 0.05 and 0.1 are
shown in figures 11 to 13. There it can be appreciated that is
insensitive to the variation of cVs (figure 11), contrary to that
is strongly dependent, as is illustrated in figures 12 and 13. is a
function that increases with cVs because the torsional excitation
is proportional to the incident angle, , which also increases with
cVs . These increments are small when the control points are nearer
to CT (figure 13). Due to the increase of the stiffness for depth
foundations, values are bigger for shallow foundations, and even
the peak value occurs for small values of .
Structural eccentricity The natural torque can produce dynamic
amplification or attenuation on the design eccentricity. In
figures 14 and 15 are reported the effects that re has over . In
general, the absolute value for is increased when re increases,
contrary to that is reported by Chandler y Hutchinson (1987) who
considered that the mximum responses to torque and to translation
occur at the same time. This reveals that the effect of
re can be important for very stiff structures where the biggest
differences occur. For torsionally flexible structures with hT 1 s,
0 for all the analyzed cases when the control point is located at
the edge of
0 1 2 3-5
0
5
10x 10
-3 Th=0.25s
0 1 2 3
-5
0
5
10x 10
-3 Th=0.5s
0 1 2 3-5
0
5
10x 10
-3 Th=1s
0 1 2 3
-0.01
0
0.01
0.02Th=2s
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14
the slab. When it is located near the stiffness center, also 0
for structures with hT 0.5 s. Besides, shows negative values for a
wide range of values for (figure 15).
Figure 11 values for x/a=1, er = 0.1, sVa 0.3 s and 0 (solid
line) and 1 (dash line). In
graphics, cVs 0.025, 0.05 and 0.1.
0 1 2 30
0.5
1
1.5
2Th=0.25s
0 1 2 3
0
1
2
3Th=0.5s
0 1 2 3-1
0
1
2
3Th=1s
0 1 2 3
-1
0
1
2Th=2s
0 1 2 30
0.01
0.02
0.03
0.04Th=0.25s
0 1 2 3
-0.02
0
0.02
0.04Th=0.5s
0 1 2 3-0.02
0
0.02
0.04Th=1s
0 1 2 3
-0.05
0
0.05
0.1
0.15Th=2s
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15
Figura 12 values for x/a=1, er = 0.1, sVa 0.3 s and 0 (solid
line) and 1 (dash line). In graphics, cVs 0.025, 0.05 y 0.1.
Figura 13 values for x/a=0.25, er = 0.1, sVa 0.3 s and 0 (solid
line) and 1 (dash line). In
graphics, cVs 0.025, 0.05 and 0.1.
In figures 16 and 17 it is observed that increases when re
decreases but only when 1. This implies that the effects for the
accidental torque, when they are meaningful, are bigger for
symmetric systems compared with the asymmetric ones as have been
observed by De la Llera and Chopra (1994). In all these cases, the
effects of re on are small. From these results it can be seen that
is more sensitive to the values of re and to those of d .
Torsional response considering the location of x related to
CT
The location of the point of observation related to the
stiffness center in the computed values for
and has important implications on the structural design, because
the support elements in the buildings subjected to the biggest
torsional moments not always are located on its periphery, neither
on their flexible side. To know the torsional response, several
points of observation were located on the flexible side of the
structure at different distances from the CT, covering the interval
x/a= [0.05, 1] with a span distance 0.05 between them. In figures
18 to 23 are shown some of these results. It is observed that the
negative values obtained in torsionally flexible structures are
bigger for points located near CT than for those placed on the
periphery of the slab. This is favorable because it implies that
the lateral displacement is reduced because
0 1 2 3-0.02
0
0.02
0.04Th=0.25s
0 1 2 3
-0.01
0
0.01
0.02Th=0.5s
0 1 2 3-0.02
-0.01
0
0.01
0.02Th=1s
0 1 2 3
-0.02
0
0.02
0.04
0.06Th=2s
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16
of the torsional effects. Even these negative values were
present in structures with hT 0.5 s, behavior that was not observed
for the points located at the periphery of the slab.
Figure 14 values for x/a=1, cVs 0.025, sVa 0.15 s and e= 0.1 and
0.2 for 0 (solid
line) and 1 (dash line).
0 1 2 30
1
2
3
4Th=0.25s
0 1 2 3
0
2
4
6Th=0.5s
0 1 2 3-1
0
1
2
3Th=1s
0 1 2 3
-1
0
1
2
3Th=2s
0 1 2 30
1
2
3
4Th=0.25s
0 1 2 3
-2
0
2
4
6Th=0.5s
0 1 2 3-10
-5
0
5Th=1s
0 1 2 3
-2
-1
0
1
2Th=2s
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17
Figure 15 values for x/a=0.25, cVs 0.025, sVa 0.15 s and e= 0.1
and 0.2 for 0 (solid line) and 1 (dash line).
Figure 16 values for x/a=1, cVs 0.025, sVa 0.15 s and e= 0.1 and
0.2 for 0 (solid
line) and 1 (dash line).
0 1 2 3-1
0
1
2
3x 10
-3 Th=0.25s
0 1 2 3
-2
0
2
4x 10
-3 Th=0.5s
0 1 2 3-1
0
1
2
3x 10
-3 Th=1s
0 1 2 3
-5
0
5
10
15x 10
-3 Th=2s
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18
Figure 17 values for x/a=0.25, cVs 0.025, sVa 0.15 s and e= 0.1
and 0.2 for 0 (solid
line) and 1 (dash line).
The values for , gradually diminish or increase when the control
point where the computations are
performed moves farther or closer to the CT. This gradual
tendency is nonlinear as can be appreciated more vividly when x is
less or equal to a 0,3a.
In torsionally flexible short structures the values for are
higher when points of observation are at
the periphery of the slab, and this tendency reverse for
torsionally rigid structures when sVa 0.15 s (figure 18). If the
ratio between the structure and the soil stiffness is higher ( sVa
0.30 s) this reverse does not happen (figure 19).
0 1 2 3-4
-2
0
2x 10
-3 Th=0.25s
0 1 2 3
-6
-4
-2
0
2x 10
-3 Th=0.5s
0 1 2 3-4
-2
0
2x 10
-3 Th=1s
0 1 2 3
-5
0
5
10x 10
-3 Th=2s
-
19
Figure 18 values for cVs 0.025, sVa 0.15 s, er=0.2, 0, x/a=1
(solid black line), x/a =
0.05 (dash line) and x/a = (0.05, 1) (orange lines) with x/a =
0.05.
The values computed for the coefficient in points located at x/a
0,3a. In torsionally flexible structures when x is close to CT some
values for are negative, changing to positive when the structures
increase their stiffness, but contrary to the behavior shown by ,
this phenomenon occurs for tall structures having values very high
when they are compared to the peak values computed for a point
located at the periphery of the slab (figure 22). For most of the
analyzed cases, when x/a=
-
20
Figure 19 values for cVs 0.025, sVa 0.30 s, er=0.2, 0, x/a=1
(solid black line), x/a =
0.05 (dash line) and x/a = (0.05, 1) (orange lines) with x/a =
0.05. .
Figure 20 for x/a = [0.05, 1], cVs 0.025, sVa 0.15 s, 0, and
er=0.1 and er=0.2.
Dash lines show the values for x/a=0.05.
0 1 2 3-2
0
2
4Th=0.25s
0 1 2 3
-4
-2
0
2
4Th=0.5s
0 1 2 3-15
-10
-5
0
5Th=1s
0 1 2 3
-10
-5
0
5Th=2s
0 1 2 3-2
0
2
4Th=0.25s
0 1 2 3
-5
0
5
10Th=0.5s
0 1 2 3-60
-40
-20
0
20Th=1s
0 1 2 3
-15
-10
-5
0
5Th=2s
-
21
Figure 21 for x/a = [0.05, 1], cVs 0.025, 0, er=0.2, sVa 0.15 s,
and sVa 0.30 s.
Dash lines show the values for x/a=0.05.
0 1 2 3-2
0
2
4Th=0.25s
0 1 2 3
-5
0
5
10Th=0.5s
0 1 2 3-60
-40
-20
0
20Th=1s
0 1 2 3
-15
-10
-5
0
5Th=2s
-
22
Figure 22 values for cVs 0.025, sVa 0.15 s, er=0.2, 1.5, x/a=1
(solid black line), x/a
= 0.05 (dash line) and x/a = (0.05, 1) (orange lines) with x/a =
0.05.
0 1 2 3-0.01
0
0.01
0.02Th=0.25s
0 1 2 3
-0.01
0
0.01
0.02
0.03Th=0.5s
0 1 2 3-5
0
5
10
15x 10
-3 Th=1s
0 1 2 3
-15
-10
-5
0
5x 10
-3 Th=2s
0 1 2 3-10
-5
0
5x 10
-3 Th=0.25s
0 1 2 3
-10
-5
0
5x 10
-3 Th=0.5s
0 1 2 3-0.03
-0.02
-0.01
0
0.01Th=1s
0 1 2 3
-0.04
-0.02
0
0.02Th=2s
-
23
Figure 23 values for cVs 0.025, sVa 0.15 s, er=0.1, 0, x/a=1
(solid black line), x/a = 0.05 (dash line) and x/a = (0.05, 1)
(orange lines) with x/a = 0.05.
.
CONCLUSIONS
It has been studied the combined torsional effects of the
structural asymmetry and the foundation rotation for structures on
a flexible base. The computations were performed for different
points of observation located between the periphery and the
stiffness center of the structure. The objective was to obtain the
values for the coefficients and taken into account the coupled
lateral-torsional effects in asymmetric buildings, and to determine
if there is a tendency on the results associated to the studied
parameters. To achieve this, the structures were idealized as a
simple oscillator with several degrees of freedom subjected to the
seismic excitation of accelerogrames registered on the soft soils
in Mexico City. The mean values of these factors were computed for
several configurations of the system considering the soil structure
interaction and the embeddement of the foundation.
It has been shown that and depend on the selection of the design
basal shear stress to be used in
the computations and on the way that the torsional effects are
separated from the foundation rotation and the structural
asymmetry. From the parametric study it was concluded that::
a) The shapes of the and curves widely vary for , with values
that go from small to big in
comparison with those proposed in design codes. The values for
torsionally flexible structures when hT 1 s and x/a=1 could be
negative. When x is close to the stiffness center these values can
be also
negative for hT 0.5 s, and to a wide range of when hT 1 s. For ,
the negative values are also for torsionally flexible structures
for any specific hT , nevertheles it is more frequent to find them
when 0.5 s
hT 1 s depending of the combination of the different parameters.
In these cases, the displacements to translation act in the
opposite direction to those of torsion.
b) The dynamic excentricity is sensible to the changes in re and
sVa , and the accidental excentricity to d and cVs . The effects of
sVa and d are reflected mainly for short stiff structures in an
uncoupled shear reduction, oV
~, used in the computations, for this reason it is wrong to
select constant
values for y in order to cover all the combine parameters of the
system.
c) The effect of sVa on the response of the system, related to
the response of the structure on a rigid base, is to increase ( 1)
or diminish ( 1) the values for . In most of the cases analyzed in
this work, the peak values for exceed the 1.5 value proposed in
codes. For parameters considered in this study, the highest values
for are presented when the point of observation is located at the
periphery of the slab, but the highest negative values correspond
for those points located near to CT.
d) In general, the values for increase when d decreces and cVs
raise. never exceeds the value considered in codes.
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24
e) and have the highest values for the observation points
located at the edge of the structure and, in general, the highest
negative values for and are obtained for the points located near
the stiffness center for torsionally flexible structures. These
results are favorable because they imply that the lateral
displacement is reduced for the torsional effects.
f) The location of the point where the displacements are
computed related to the stiffnes center, has a significant
influence on the values computed for and . This influence can be
more important than the influence excerted by the parameters of the
system itself, mainly for the points located near to the CT. This
has significant implications on the design of the structures,
because their columns subjected to a big torsion not always are
located at the periphery on their flexible side.
g) The values for coefficients y increase or diminish gradually
when the point of observation is
approaching or rifting to the CT. This gradual tendency is not
linear as can be appreciated more clearer for the points located at
x 0,3a.
ACKNOWLEDGEMENTS
This work has been supported by the Institute of Enginnering and
the School of Engineering of the National Autonomus University of
Mexico under the Project 3561.
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