Note: Descriptions are shown in the official language in which they were submitted.
CA 02415987 2003-O1-02
SLIDING CONCAVE FOUNDATION SYSTEM
FIELD OF THE INVENTION
The present invention relates to foundation systems and in particular
foundation
systems adapted to withstand earthquakes.
BACKGROUND OF THE INVENTION
The occurrence of earthquake is a worldwide disaster often causing the loss of
many lives and financial ruins. Studies have shown that an earthquake with a
magnitude of six to seven at a distance of &km from the causative fault can
produce
forces that are about 10 to 20 times the minimum forces given in existing
seismic
design codes. The elastic design of the structures for these forces, results
in irrational
costly construction. The inelastic design concept and ductility provision for
structures
can prevent the collapse of the buildings and reduce the cost of construction.
However,
the building and its contents could still be severely damaged. Therefore, the
seismic
isolation concept is considered a suitable solution in designing structures
that can
protect the buildings and its contents from serious damage.
Seismic isolation is the separation of the building (or any other type of
structure)
from the harmful motions of the ground by providing flexibility and energy
dissipation
capability through the insertion of the so-called isolators between the
foundation and
the superstructure.
The first application of the isolators was relatively recent. The first base
isolated
CA 02415987 2003-O1-02
building in United States of America was built in 1985. In terms of behavior,
isolators
are classified in two major groups: elastomeric and frictional isolators.
Kelly, Su et. al.
and Skinner provided comprehensive reviews on isolation devices and
techniques.
The use of base isolation systems has two major advantages. First, the
vertical
and horizontal loads are resisted in different ways. This results in more
stable structural
system and eliminates the need for a mechanism to dissipate energy while
preventing
structural collapse. Second, in the presence of a small restoring force, the
sliding
systems are (practically) insensitive to frequency content of the base
excitation and
always limit the transmitted shear force to the building. This feature
(insensitivity to
frequency content of the base excitation) is the most important benefit of a
sliding
system.
A large number of theoretical and experimental studies have shown that these
systems could decrease the damaging effects of the earthquakes. The oldest
base
isolation technique that had previously been successfully implemented in the
building
construction was the sliding support system that is also referred to as the
Pure Friction
(P-F) system. Numerous studies have been carried out concerning the response
analyses of the structures on sliding support. Westermo and Udwadia and
Mostaghel
et. al. studied the responses of the sliding structures under harmonic base
excitations.
Mostaghel and Tanbakuchi investigated the ability of the sliding supports to
isolate the
superstructure from the vibrations of the base during strong earthquakes
excitation. In
most of these studies, a 2-degree of freedom (DOF) model was employed, one for
the
single degree of freedom (SDOF) structure and one for the sliding raft. Yang
et al, Fan
et al and Vafai et al studied the effects of sliding supports on the responses
of the
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multistory structures subjected to different base excitations. In these
studies, it was
assumed that the friction was of the Coulomb type and it was independent of
the
contact pressure and the relative velocity of the sliding surface and also the
dynamic
and static coefficients of friction were considered equal.
Zayas in 1986 introduced one of the most effective isolation systems, namely
the Friction Pendulum System (FPS), which utilizes friction to dissipate the
transmitted
energy to the structure. Thereafter, numerous theoretical and experimental
studies
were conducted to investigate the FPS characteristics and its application in
bridges
and buildings. Principally, a building supported on FPS isolators behaves as a
simple
pendulum. The basic concept of the FPS system is shown in Figure 1. It is
shown that
the fundamental period of the FPS is determined by Equation (1 ):
T=2~~ (1)
Where, R is the curvature of the sliding surface and g is the gravity.
Equation (1 ) gives
the fundamental period of a rigid body supported on a FPS isolator. However,
if the
structure on the FPS is relatively short and rigid, Equation (1 ) can be used
as the
period of the combined system of the FPS and superstructure.
Accordingly it would be advantageous to provide a foundation system
that reduces the lateral seismic forces transmitted to the structure.
SUMMARY OF THE INVENTION
The present invention, a new base isolation system, namely the Sliding
Concave Foundation (SCF) is introduced. The new system utilizes friction to
dissipate
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the earthquake energy. However, it has other features that make it an
attractive base
isolation system.
A foundation system for a building or other load includes a lower part and an
upper sliding raft. The lower part has a generally concave surface at the top
thereof.
The upper sliding raft has a convex surface at the bottom thereof adapted to
rest on
the concave surface of the lower part and allow for sliding rotational
movement
therebetween. The building or other load is attached to the upper sliding
raft.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will now be described by way of example only, with reference to
the accompanying drawings, in which:
Fig.1 is a schematic showing the basic principles of prior art friction
pendulum
system;
Fig. 2 is a side view of the sliding concave foundation system of the present
invention;
Fig. 3 is a schematic showing the basic principles of the sliding concave
foundation system of the present invention;
Fig. 4 is a model of a single degree of freedom structure on the sliding
concave
foundation system of the present invention and showing the operating forces at
the
contact points thereof;
Fig. 5 is a model of a single degree of freedom structure on the sliding
concave
foundation system of the present invention for the determination of the
fundamental
period of the superstructure;
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Fig. 6 is graphs of the fundamental period for the superstructure and the
foundation;
Fig. 7 is comparative models comparing a fixed base, a sliding support , a
sliding concave foundation and a friction pendulum system;
Figs. 8 and 9 are comparative structural responses using different isolation
systems and without them subjected to Tabas earthquake;
Figs. 10 and 11 are comparative structural responses using different isolation
systems and without them subjected to long period harmonic excitation; and
Fig. 12 is a graph showing a comparison of the structural damage to a
structure
on a fixed base versus the sliding concave foundation of the present
invention.
DETAILED DESCRIPTION OF THE INVENTION
The main components of the new system (SCF) are shown in Figure 2 at 20.
The building foundation 22 in this system consists of two parts. The lower
part 24 has
a cylindrical concave surface at the top and hereafter is referred to as the
fixed
foundation or, briefly, the foundation. This part moves with the ground and
has no
sliding displacement relative to the ground. The top part 26 has a cylindrical
convexity
and is called the sliding raft. The sliding raft 26 is attached to the
building and can be
considered as its floor so that the columns are connected rigidly to this
raft. In practice,
the sliding raft 26 can be constructed in different ways. One way involves a
concrete
platform resting on a number of short and rigid stands that follow the
curvature of the
fixed base foundation 24 at their ends as shown in Figure 2. The sliding raft
has a
sliding-rotational movement on the sliding surface of the fixed foundation 24.
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The interfaces of these two parts 24, 26 are made of very low friction
materials
such as Steel-Teflon or Teflon-Teflon interfaces. Sliding of the raft 26
provides
excessive flexibility at the foundation level that causes the separation of
the
superstructure from the damaging vibration of the base 24. In addition, a
great amount
of the earthquake energy is dissipated due to friction at the sliding surface.
Therefore,
base shear drifts and consequently the total damage to the building
significantly
decreases.
Since the sliding raft 26 always moves with the building, hereafter the
combination of the building and the sliding raft is called superstructure 28.
It should be
mentioned that Figure 2 is a two dimensional model. In practice, however, the
problem
is three-dimensional and there are two cylindrical sliding surfaces that are
perpendicular to each other. Since the sliding on each surface is independent
of the
other surface, the characteristics of the SGF can be investigated using a two
dimensional model.
The basic concept of SCF is shown in Figure 3. The motion of the building on
this isolation system is similar to a compound pendulum shown at 30 and 32.
Therefore, both the linear inertia (mass) and the rotational inertia of the
superstructure
28 contribute to the dynamic natural period of the system. The contribution of
the
rotational inertia is unique to the SCF system and is not available in other
existing
isolation systems. An interesting feature of the SCF system, is that the
center of gravity
of the superstructure falls under the center of curvature of the foundation as
can be
noted at 34 and 36 in Figure 3. Therefore, for any rotation 9 of the
superstructure
around the center of curvature, the weight of the superstructure produces a
restoring
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force that always opposes the overturning moment. This means that the SCF
system
improves the stability of the structure.
EQUILIBRIUM EQUATIONS
A simplified model of a SDOF structure supported by a SCF, which is used to
develop the governing equations of motion, is shown in Figure 4 generally at
38. The
frictional forces at the contact points i ( F~; ), shown generally at 40 in
Figure 4b, can be
determined from the equations of equilibrium (Equations 2 to 4). These forces
develop
due to the very small rotation B of the superstructure around the center of
the
foundation curvature. The equilibrium of the system is considered along the
tangential
direction of the sliding surface and along the radial direction, i.e.
Tangential equation of equilibrium:
F;cos(a;+e)-W;sin(a;+e)-F~ =0 (2)
F; = F~' +W; tan(a; +e) (3)
cos(a; +e)
Radial equation of equilibrium:
N; = F, sin(a; +e)- W; cos(a; +e) (4)
where W;, F, and N; are the weight, friction and the normal forces at the
contact point i,
respectively, and angles a and A are defined in Figure 4. In Equations 2
through 4, F; is
the inertial force that acts at the ith contact point on the sliding surface
(i.e. the shear
force that acts on the sliding part at the ith contact point) and can be
obtained by
solving the dynamic equations of motion of the system.
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For Coulomb type friction F,; _ ~ N; , where w is the coefficient of friction,
and
Equation (4) can be rewritten as:
Fj; _ -i [ F; sin(a; +e)+ W; cos(a; +e)]sgn(e) (5)
The negative sign of sgn(e) implies that the friction force is always in the
opposite
direction of the velocity.
Another interesting feature of the new system is that the value of lateral
(inertial)
force at the ith contact point (F;) is proportional to the part of the weight
that is
supported at the same point (W;), as suggested by Equations (3 to 5). This
means that
the center of lateral stiffness always coincides with the center of gravity of
the system.
This feature of the SCF system, which is similar to FPS, principally
eliminates the
undesired torsional vibrations and the consequent damage to the unsymmetrical
buildings during an earthquake.
When the structure goes through a small rotation 8 from its initial position,
a
restoring force develops in the center of gravity of the superstructure due to
the weight
of the superstructure. This restoring force and the frictional forces at the
sliding surface
would produce a restoring moment around the center of curvature of the
foundation,
which can be calculated from:
M = WR~.~e -F~~ R (6)
where R is the radius of the sliding surface of the foundation, R~.~ is the
distance
between the center of gravity of the superstructure and the center of
curvature of the
foundation and W is the total weight of the superstructure. In Equation (6),
the first
term in the right hand side is the share of the gravity in the restoring
moment (M9ra~ic~)
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and the second term is the share of the frictional force, which is opposite to
the first
term.
The rotational stiffness of the system around the center of the foundation can
be
determined by dividing the restoring moment due to gravity (M9rav~ty) by the
angle of
rotation 8, i.e.
K = Mgrnvity = WR
8 a c.g
The rotational inertia around the center of curvature of the foundation is
given as:
W z
Io = Ic.g +~ g ) Rc.,y
Thus the oscillation period of the superstructure around the center of
curvature of the
foundation becomes:
T = 2c~ ( I-°° ) = 2~ ( Iw + Rc.a ) (9)
KA WR~.~ g
It can be noted from Equation (9) that the period of the system is the same as
the
period of a compound pendulum consisting of two parts. In comparison with the
period
of a FPS isolator (Equation 1 ), the presence of the first part (which is
absent in the FPS
case) results in a large increase in the period of the system. The form of the
second
part is the same as that of a FPS isolator, however, the value of R~,g for the
SCF is
much larger than that of the FPS and this increases the period furthermore.
Equation
(1 ) suggests that the period of the FPS isolator is independent of the mass
of the
superstructure. Similarly, the period of the SCF system is independent of the
total
mass but it depends on the distribution of the mass ( y.g ~ w )
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CA 02415987 2003-O1-02
The fundamental period of a SDOF structure supported by SCF system is given
in Equation (9)), which simply gives the fundamental period of a compound
pendulum.
In this equation, it is assumed that the period of the main structure is very
small
compared to the fundamental period of the total system, thus the period of the
superstructure is not affected considerably by the stiffness of the main
structure. To
verify this assumption, the fundamental period of a SDOF structure supported
by a
SCF (shown in Figure 5) is determined through the solution of the governing
equations
of motion. This was achieved by computing the time required for the system to
complete one cycle under free vibration conditions.
To investigate the effects of the SCF on the behavior of structures, a
computer
program was developed that is capable of analyzing the response of a SDOF
structure
supported by a SCF system (the program is also capable to analyze the
responses of
the SDOF structure supported by FPS, sliding base and fixed base foundation).
The structure whose stiffness, damping, mass and geometrical properties are
listed in Figure 5 was used as an example to verify the performance of
Equation (9).
Specifically the particulars are as follows: rn~ = 350.2 kg; I~ = 4109 kg-m2;
mb = 350.2
kg; Ib = 4109 kg-m2; K=8.64 x 104 N/m; C = 550.1 N.sec/m; T=0.4 sec; R = 15.0
m; h~ _
6.0 m; hb = 1.0m; ~, = 0.08; and ~ = 0.05. The fundamental period of the
structure was
calculated using Equation (9) and its value was 7.20 sec.
The system shown in Figure 5 was subjected to an harmonic base excitation for
5 sec., then the amplitude of the excitation was set to zero to allow the
system to go
through a free vibration with no external excitation. The response of the
superstructure
was calculated for a period of time long enough for the system to complete at
least one
CA 02415987 2003-O1-02
full cycle of oscillation after the base excitation had stopped. To minimize
the strong
dissipative effects of the friction (so that the free vibration of the system
could last long
enough), the Coulomb coefficient of friction for this example was assigned a
small
value (p,=0.003). The results of the analysis are presented in Figure 6 in
terms of the
absolute acceleration time histories of the story at 44 and the sliding raft
at 46. The
fundamental period of the structure obtained from the absolute acceleration
time
histories of the story and the sliding raft is ~ .21 and 7.23 sec.,
respectively, as shown in
Figure 6. As can be noted, the fundamental period value calculated using
Equation (9)
agreed well with those obtained from the free vibration response analysis.
A computer program developed was used to calculate the response of a SDOF
structure for different support conditions: fixed base 48, sliding support 50,
SCF 52 and
friction pendulum system (FPS) 54. The necessary information used in this
example is
given in Figure 7. Specifically for the fixed base has the following
particulars: m~ _
350.2 kg; K=8.64 x 104 N/m; C=550.1 N.sec/m; T = 0.4; and sec ~ = 0.05. The
particulars for the sliding support are as follows: m~ = 350.2 kg; mb = 350.2
kg; K=8.64
x 104 N/m; C=550.1 N.sec/m; T = 0.4; ~, = 0.08; and sec ~ = 0.05. The
particulars for
the SCF are as follows: m~ = 350.2 kg; I~ = 4109 kg-m2; mb = 350.2 kg; Ib =
4109 kg-m2;
K=8.64 x 104 Nlm; C = 550.1 N.sec/m; T=0.4 sec; R = 15.0 m; h~ = 6.0 m; hb =
1.0m; ~,
= 0.08; and ~ = 0.05. The particulars for the FPS are as follows: m~ = 350.2
kg; mb =
350.2 kg; K=8.64 x 104 N/m; C = 550.1 N.sec/m; T=0.4 sec; R = 15.0 m; h~ = 6.0
m; hb
= 1.0m; p, = 0.08; and i; = 0.05. The friction is assumed to be of the Coulomb
type and
the radius of the curvature of the foundation for the SCF is 31 m. The radius
of the FPS
used in the analysis is assumed to be 1.0m, a value that is larger than those
normally
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CA 02415987 2003-O1-02
used in practice.
The fundamental period of the structure supported by the SCF is 10.4 sec.
(from
Equation 9), which is much larger than that of the structure supported by the
FPS (2.0
sec. from Equation 1 ). The fundamental period of the structure supported by
the SCF
is far from the predominant frequency associated with destructive earthquakes.
Therefore, the SCF renders the structure insensitive to the frequency content
of the
base excitation.
The results of the analysis of the system response under the Tabas earthquake
excitation are presented in Figures 8 and 9. Figure 8(a) 56 shows the
displacement of
the story relative to the base and Figure 8(b) 58 shows the absolute
acceleration of the
story, for different support conditions. Figures 8 shows that the ability of
the SCF to
reduce the responses of the structure is comparable to those of the sliding
support
system and FPS. The three systems significantly reduce the displacement and
acceleration of the story compared to the case of the fixed base. Figure 9(a)
60 shows
that the SCF system, similar to the FPS and sliding support system, is very
efficient in
reducing the base shear forces transmitted to the structure. Figure 9(b) 62
presents a
comparison between the sliding displacements of the three base isolation
systems (i.e.
SCF, FPS and sliding support). Figure 9(b) 62 shows that the maximum sliding
displacement of the structure supported by the SCF system is significantly
reduced
compared with that of the sliding support system, but is higher than that of
the FPS.
However, the SCF system may perform even better than the FPS as shown below.
The same structures shown in Figure 7 are subjected to a harmonic excitation
(representative of earthquake excitations in soft soils) with an amplitude of
0.5g and a
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CA 02415987 2003-O1-02
circular frequency of 4 rad/sec (period of 1.57 sec). The results are shown in
Figures
and 11, for different support conditions. Figure 10 shows the relative
displacement
64 and absolute acceleration 66 of the story and Figure 11 presents the base
shear 68
of the SDOF structure and the sliding displacement 7U of the raft. It can be
noted from
5 Figure 10 that the SCF reduced the structural response compared to the other
support
conditions. Furthermore, Figure 11 shows that the base shear transmitted to
the
structure and the sliding displacement of the foundation are significantly
reduced for
the SCF system relative to the other systems, including the FPS. It should be
noted
from Figure 11 (b) that the residual displacement for the SCF system is much
less than
10 that of the FPS.
The results presented in Figures 8 through 11 show that the relative
advantages
of the SCF and FPS depend on the amplitude and frequency contents of the
excitation.
Studies show that for stronger earthquakes the SCF is more efficient than FPS.
A very
important feature of SCF is that because of its long period, it never
experiences
resonance during an earthquake. On the contrary, most of the isolating systems
that
have been introduced so far can undergo resonance under especial conditions.
As
shown in Figure 10 and 11, both the sliding support system and FPS fall in
resonance
under a long-period harmonic excitation. Similar results were obtained through
the
study of numerous examples under different types of base excitement.
The inelastic energy absorption demand is a suitable criterion for predicting
the
level of damage to a structure. Figure 12 shows a comparison between the
inelastic
energy absorption demands of two similar structures; one supported by a SCF
and the
other with a fixed base. The structures were loaded according to the seismic
code of
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Iran and designed according to AISC allowable stress method. The inelastic
energy
absorption demand of the structure was determined utilizing the method of
Nurtug and
Sucuoglu. The yielding energy (defined as the reserved energy in the structure
at the
commencement of the yielding) is used to normalize the absorbed energy. Figure
12
confirms the positive effects of the SCF in reducing the total damage to a
structure. As
can be seen from the figure, for weak earthquakes with a magnitude of four or
five the
amount of absorbed energy is nearly equal for both foundation systems. This is
so
because weak earthquakes do not impose large sliding displacement to the SCF
system (i.e. the system has not been fully Exploited), and thus the two
systems perform
nearly the same. But for strong earthquakes (i.e. magnitude > 6), the amount
of energy
absorbed by the structure on SCF and, consequently, the total damage to the
structure
decrease dramatically compared with that of the structure on the fixed base
foundation.
As discussed above the Sliding Concave Foundation (SCF) system of the
present invention can reduce the lateral seismic forces transmitted to the
structure by
introducing the flexibility and energy dissipation capability at the
foundation level of the
structure. Furthermore, the new system has a number of advantages. A building
supported on the new system behaves like a compound pendulum during seismic
excitation. The pendulum behavior accompanied by the large radius of
foundation
curvature shifts the fundamental period of the system to a high value (e.g.
more than 8
sec.), which falls in a frequency range in which none of the previously
recorded
earthquakes had considerable energy. This results in a large decrease in the
structural
responses. This represents an important advantage of the SCF system over most
of
the other systems including the FPS, as it renders the system insensitive to
the
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CA 02415987 2003-O1-02
frequency of the base excitation. Although the pure frictional sliding systems
have the
same efficiency as the SCF, in reducing the responses of the superstructure,
the main
advantage of the new system is a significant decrease in sliding displacement.
The
period of the system increases with the radius of the foundation curvature and
the
mass moment of inertia of the superstructure. The center of lateral stiffness
of the
superstructure coincides with the center of gravity of the system, so the
effects of
torsional vibration are minimized (similar to FPS). Because of friction, SCF
system,
like other frictional isolation systems, can resist the lateral forces of
winds and small
earthquakes. This resistant continues until the level of lateral forces exceed
the
frictional resistant of the sliding surface. However, the SCF system can
resist higher
level of wind forces than FPS far an equal frictional resistance. This is
mainly because,
in the SCF the resultant of the wind forces acts at a point that is closer to
the center of
curvature of the foundation than that of the frictional resistance. The SCF
can reduce
the base shear and the inelastic absorbed energy of the structure and
therefore total
damage to the structure. These reductions are completely comparable with those
of
sliding support system, however the sliding displacement in the case of the
new
system is significantly less than that of the sliding support system. The
structure on a
SCF system is more stable than a structure on most of the other isolating
systems, due
to the relative locations of the center of gravity of the superstructure and
the center of
curvature of the foundation.
It will be appreciated by those skilled in the art that a number of the
parameters
may be varied while staying within the invention described herein. As
discussed
above, the centre of curvature of the foundation should be above the center of
gravity
CA 02415987 2003-O1-02
of the building so that the restoring moment can develop. In addition, the
radius of the
curvature should be at least equal to half of the building height. However,
the value of
radius of curvature is a matter of design which is based on the level of
expected
isolation. Further, the restoring moment depends on the radius of curvature
and so the
value of this moment should be decided in the design process. As well it
should be
noted that the coefficient of friction is a matter of design and the value of
which
determines the level of isolation. Similar to any other isolation system that
uses friction
to dissipate energy, as long as a material can provide the required
coefficient of friction
and its properties are constant over time, it can be used in SCF.
As used herein, the terms "comprises" and "comprising" are to be construed as
being inclusive and opened rather than exclusive. Specifically, when used in
this
specification including the claims, the terms "comprises" and "comprising" and
variations thereof mean that the specified features, steps or components are
included.
The terms are not to be interpreted to exclude the presence of other features,
steps or
components.
It will be appreciated that the above description related to the invention by
way
of example only. Many variations on the invention will be obvious to those
skilled in the
art and such obvious variations are within the scope of the invention as
described
herein whether or not expressly described. Further, it will be appreciated
that the
equations herein are to further describe the invention and they should not be
seen to
limit or bind the present invention to any particular theory or hypothesis.
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CA 02415987 2003-O1-02
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