This document analyzes the response of buildings with elastic-plastic foundations to earthquake ground motions. It describes a two-degree-of-freedom model used to represent the building-foundation system. A range of system parameters are varied in the model, including the stiffness of the foundation, its mass, and its yield point. The response of the system is computed for records from three earthquakes with different intensities. Graphs of the maximum shear coefficient, displacement, and acceleration are presented for different parameter values. The results show relationships between the response and parameters like the foundation to building stiffness ratio. The study evaluates different types of damping devices that could provide elastic-plastic behavior to the foundation.

1. INDIVIDUAL STUDIES
BY PARTICIPANTS AT THE
. INTERNATIONAL INSTITUTE OF SEISMOLOGY
AND EARTHQUAKE ENGINEERING
Vol. 16
October 1980
Interna tional Institute
of Seismology and Earthquake Engineering
Building Research Institute, Ministry of Construction
1 To t eho ro , Oha-machi. Tsukuba-gun,
IBARAlO PREFECTURE, J APAN

2. ELASTO - PLASTIC FOUNDATION
by
Fabian Ed. CEVALLOS larco
(For the Earthquake Engineering Course, 1979-1980)
ABSTRACT
This paper is an effort in analyzing the response of s
mall buildings having an elasto-plastic behavior. It's founded
that a relationship between the ratio building-stiffness/founda
tion-stiffness and the maximum shear force acting on the build
ing exists.
Some devices for obtaining the elasto-plastic behavior
are proposed and it suggests the possibility of designing the
appropriate device for the kind of earthquake expected.
1.- INTRODUCTION
Since the begining of the earthquake engineering subje
ct, the attitude of the researchers was pointed out in two ways;
the first one, how to design a building strong enough to resist
the earthquake shock and the second one, how to avoid or decre
ase the earthquake imput. Nowdays, the first field got a huge
development, but the second is still growing up. This study is
involved in the second area, where one of the solutions is an
elasto-plastic foundation.
In 1960ths. Dr Matsushita and Dr. Izumi presented so
me devices for decreasing the earthquake imput, all of them sho
wn a non-linear relationship between a restoring force and the
displacement (f). In 1974 they presented to the Fifth World Can
ference on Earthquake Engineering some devices for tall buildi
ngs
Private Consultant, calle Alemania 0381 Quito - ECUADOR
** Tokyo University Professor and Research Engineer of Inter
national Institute of Seismology and Earth. Eng. (Japan/74)
-205-

3. ructed with foundation devices able to decrease the imput acc~
lerations caused by earthquakes, and a few years ago a nuclear
power plant is projected by Koeberg S.A. in Africa with an el
as to-plastic foundation system composed by rubber and sleek s
;,
teel (3).
2.- MODEL USED
In order to pointed out the possible relationship bet
ween the variables involved in the problem, a two degree of
freedom system model was used (figure 1).
In the model; the mass number one, m
l
, represents the
first story and the foundation device; the stiffness number 0
ne, k
l
, is the stiffness of the device which transmit the gro
und movement to the structure and has an elasto-plastic behav
ior; the mass number two, m
2
, is the building's mass and the
stiffness number two, k2' is the stiffness of the building.
The ground motion exitation was imput to the model by
the accelerograms of three earthquakes: San Fernando earthqua
ke (recorded in Pacoima dam, S - 74W component), El Centro (N-
S component) and Miyagi-oki earthquake (recorded in the basem
ent number two of the Sumitomo Bank Building, E - W component).
The analysis was carried out in a digital computer, b~
sically using a program already performed by Dr. Yamazaki for
a non-linear systems.
3.- DAMPING FOUNDATION DEVICES
Some systems were analyzed before one was chosen as th
e most appropriate, but they could be divided in the following
groups:
3.1.- Hanging type:
Fundamentally, this one consists of hanging the struc_
ture so that it could work as a pendulum (figure 2), from the
physical pOint of view, the stiffness (k
l
) of the device is gi
ven by the formula:
• Research Engineer of I.I.S.E.E. (1980)
-206-

4. r
k1 = _-=-P_...:.x'--_E=---__
s Ln «, r x L
t
Where:
k = stiffness value of the device; other words, restoring force
1
divided by deformation.
P = weight of the structure
~= angle between the cable and the horizontal plane
E = Young's modulus
~ = allowable stress in the cable
t
L = length of the cable
where,
In the practical field if steel is used, for example,
2 r.- 2
E= 2100000 kg/ cm, U = 1400 kg/ cm , 0< = 900
(in the mo
t
st favorable case) and L= 100 cm.; kl becomes 15 x P, this is a
high value for kl because instead of decrease the imput it will
work as an amplifier.
3.2.- Sledding type:
Mainly this type consists of defining one level of imp
ut force by the friction angle between the building and the ear
th (figure 3). Nevertheless the behavior of this model is inte
resting, the analysis by computer is difficult.
3.3.- Floating type:
The idea in this type is floating the building in a me
dium where some frequencies could be missed and a medium with
big values of damping ratio will decrease the imput values hig~
ly. From the practical point of view, the floating medium to be
used could be soft soil and some researchers had pointed out th
,',
is fact (4). Therefore the problem to be analized in this type
belongs to the soil-structure interaction.
3.4.- Rolling type:
Basically this type involves rolls, balls or lenses 10
cated between the earth and the foundation (figure 4), with the
characteristic that the shape and the size of the lenses or ba
lIs gave different behavior to the whole structure by providing
the wished foundation's stiffness.
-207-

5. Some devices and the kl value formula are listed below:
W
k
l
=-----
Z(R-r)
Where:
W= weight of the building
R,r,e,a showed in graphs above.
3.5.- Mixed type:
The combination of two of the types mentioned before
generates the mixed type and this one seems to be the most sui
table to solve the problem. The nuclear power plant projected
in Africa includes a floating-sledding type (figure 5). And t~
is study is related to this type, where the foundation behavi
or is elasto-plastic.
4.- VARIABLES CHOSEN
A floating-sledding type was selected because it is po
ssible for construction. For that purpose a slab foundation mu
st. be done, and over that the lenses will be located and over
the lenses a steell plate with lubricant, then the foundation
of the building will be placed (figure 6).
Since the variables involved in the problem are eight:
stiffness of the device (k
l
), stiffness of the building (k
2
),
mass of the foundation (m
l
), mass of the building (m
Z
)' damping
ratio of the foundation (hI)' damping ratio of the building
(h
2
), sledding level of the foundation or yeilding point of th
e foundation (Q ) and imput accelerations (x); some of them m u
y
st be fixed for the analysis. Therefore a three stories build
ing was considered, in order to avoid the influence of the roc
,I~
king problem (5), with a hundred square meter per floor. The
foundation weight was included in the first floor. In this ma
nner the upper stories weight was rounded in 300 tons.
By keeping the period of the building between 0.2 to
-208-

6. 0.3 seconds the stiffness of the building is in the range betwe
en 150 to 300 tn/ern. The damping ratio of the building was fix
ed in 0.02, considered as the worse case, and fa r the foundati
on it was fixed in 0.001 to assure tha t the only way where the
energy will dissipate in the foundation is throught the pla~
tic deformation beyond the sleeding level. Due to this cons ide
rations, the variables
m 2 ' k2' hI and h2 were fixed. The varia
bles k
l
, m
l
, Q
y
and x were the values used as parameters.
The stiffness of the foundation, k
l
, w a s varying from
the 10% to 80% of the building's stiffness in the following ran
ge:
Name k2 kl ratio Name k2 kl ratio
ROLING 0 150 20 0.13 ROLLING7 2S0 40 0.16
ROLLING 150 40 0.26 ROLLINGS 250 SO 0.32
ROLLING2 150 SO 0.53 ROLLING9 250 120 0.4S
ROLING2A 150 100 0.66 ROLING9A 250 160 0.64
ROLLING3 150 120 0.80 ROLING9B 250 200 0.80
ROLING3A 200 20 0.10 ROLLINGO 300 40 0.13
ROLLING4 200 40 0.20 ROLLINGA 300 80 0.26
ROLLINGS 200 80 0.40 ROLLINGB 300 120 0.40
ROLLING6 200 120 0.60 ROLINGBI 300 180 0.60
ROLING6A 200 160 0.80 ROLINGB2 300 240 0.80
The stiffness of the foundation, k ,
1
are varying mean
while the stiffness of the building is fixed due to the fact
there is more uncertainties about the behavior of the device than
the behavior of the building.
The weight of the foundation was defined by three valu
e s : 50, 100 and 150 tons, in other words, one sixth, one third
and one half of the building weight.
The yeilding point of the foundation (where the plastic
range start) was defined by three values: 0.1, 0.2 and 0.3 of
the whole weight over the damping devices.
The maximum imput acceleration given by the three eart~
quakes mentioned before, was adjusted to reach three levels 350
-209-

7. 700 and 1400 gals.
Therefore, the total number, N, of cases analyzed is gi
ven by:
N= earthquakes x imput levels x building stiffness x foun
dation stiffness x building's damping ratio x foundati
on's damping ratio x yeilding point values x building
's masses x foundation's masses.
N= 3 x 3 x 4 x 5 x I x 1 x 3 x 1 x 3 1620 cases
The output requested to the computer was the following
values: maximum shear coefficient for the building (SCMAX), ma~
imum absolute displacement of the foundation (with respect to
the earth, ABMAX), frequency for the first mode (OME) and the
maximum acceleration imput to the first story (ACMAX).
5.- RESULTS
The results of the computation are shown in the Table 1
and they are listed in the annex number 1, 2, 3, 4 and 5. Where
the first column is the value of ACHAX, the second column is
SCMAX, the third column is ABMAX, the fourth is the earthquake
name, the fifth is the maximum acceleration of the earthquake,
the sixth is the type of the building structure and the seventh
is OME's value.
To visualize the relationship between the variables, a
computer program was performed to draw the following graphs:
Ordinate (Y) vs. Abscissa (X)
- maximum shear coefficient vs. stiffness ratio
- maximum .absolute displacement vs. stiffness ratio
- maximum shear coefficient vs. frequency
- maximum acceleration of the first story vs. frequency
The typical graphs obtained are shown in the figure 7
in the annex number 6, where each line represents the same bui!
ding (fixed the stiffness, k2' of the building) with different
kinds of foundation devices (varying the stiffness, k
l
, of the
foundation). Therefore, each cartesian coordinates contains 20
points and 324 graphs were plotted for the analysis.
-210-

8. 6.- CONCLUSIONS
The analysis of the graphs pointec out the following
fdctS:
- the shear coefficient to be applied to the building keeps a
relationship with foundation's stiffness/building's stiffness
ratio, in other form: if the ratio increases, the shear coeffi
cient increases.
- the maximum absolute displacement increases for stiffness ra
tio values below some level.
- for the same conditions, the shear coefficient increases when
the mass ratio ml/m
Z
(foundation's mass/ buildine's mass) also
increases.
- for big earthquakes, the maximum absolute displacement of the
foundation is almost directly proportional to the yeilding poi
nt Q of the elasto-plastic device; then, it is proportional to
y
the frictional angle between the structure and the earth.
- in the case of big earthquakes analyzed (1400 gals) the maxi
mum shear coefficient obtained is independent of the earthquake
imput, but depend on the yeilding point Q of the elasto-plas
y
tic foundation device.
- there is some value of the stiffness ratio where the absolute
displacement and the shear coefficient are in a reasonable valu
es for each kind of earthquake analyzed.
7.- ADVANTAGES
Special buildings, for example, for computer systems or
military purposes, could be constructed with a great safety fa~
tor using an elasto-plastic foundation; even near to the faults
where the accelerations got unexpected values.
After the structural design of the building, the stiff
ness value of the structure must be computed, with this value
the appropriate stiffness of the foundation could be defined
and designed using the formula (a) of the section three.
8.- LIMITATIONS
- the small earthquakes amplified by a factor do not show the
-211-

9. same characteristics as the big ones. For that analysis another
kind of simulated earthquakes must be applied, for instance, th
e synthetic earthquakes obtained by the multiple source method.
- after the earthquake jacks will be used to return the struc
ture to the original po~ition.
- water, sanitary and gas installations must be provided with a
flexible joint to out-side of the building.
9.- SUGGESTIONS
- a floating-sledding type using clay and sand could be develo
ped, for that purpose some tests will be carried out to analyze
the real behavior of this system.
- a computer program must be developed to analyze the pure sled
ding type.
- using the results showed in the table 1, a formula could be ob
tained to establish the relationship between the variables.
REF ERE N C E S
(£) MATSUSHITA and IZUMI et al. "Study about the process to red!:!.
ce the seismic force on the buildings"; Transaction of the
Architectural Institute of Japan, Vol. N° 122, April-1966
pp. 15-22
,',
(2) MATSUSHITA and IZUHI. "Application of input controlling me_.
chanism to the structural design of a tall building"; Proce
edings of the Fifth W.C.E.E. Rome-1974, Vol. N° 2 pp. 2948-
2955.
(~) KOEBERG S.A. Mr. Marc Richli, Spie-Batignolle Paris
(4) BYCROFT "Soil-structure interaction at higher frequency fac
tor"; Abstract journal in earthquake engineering, Vol. N° 7
Dec-1978. Section soil-structure interaction 0.8 - 9 pp.l69
1,
(5) OSAWA, KITAGAhlA and IRIE. "Evaluation of various parameters
on response analysis of earthquake motions, including soil-
building system"; Proceedings of the Sixth W.C.E.E., Vol. 2
New Delhi - 1977, pp. 1461 - 1466
-212-

10. MODEL USED
W/.@IW$W/~
Figure 1
SLEDDING TYPE
Figure 3
FLOATING SLEDDING
Figure 5
HANGING TYPE
slab
Figure 2
ROLLING TYPE
'~~<>
slab foundation
PB~I!@W~_$A_
Figure 4
ROLLING - SLEDDING
steel plate
wi th
lubricant
lenses
foundation slab
Figure 6
-213-

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16. Annex N° 6
EXAMPLE OF THE TYPICAL GRAPHS OBTAINED
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