Elastic Plastic Foundation

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  • 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 Construction1 To t eho ro , Oha-machi. Tsukuba-gun,

    IBARAlO PREFECTURE, J APAN

  • 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 small buildings having an elasto-plastic behavior. It's foundedthat a relationship between the ratio building-stiffness/foundation-stiffness and the maximum shear force acting on the building exists.

    Some devices for obtaining the elasto-plastic behaviorare proposed and it suggests the possibility of designing theappropriate device for the kind of earthquake expected.

    1.- INTRODUCTIONSince 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 resistthe earthquake shock and the second one, how to avoid or decrease the earthquake imput. Nowdays, the first field got a hugedevelopment, but the second is still growing up. This study isinvolved in the second area, where one of the solutions is anelasto-plastic foundation.

    In 1960ths. Dr Matsushita and Dr. Izumi presented some devices for decreasing the earthquake imput, all of them shown a non-linear relationship between a restoring force and thedisplacement (f). In 1974 they presented to the Fifth World Canference on Earthquake Engineering some devices for tall buildings

    Private Consultant, calle Alemania 0381 Quito - ECUADOR** Tokyo University Professor and Research Engineer of Inter

    national Institute of Seismology and Earth. Eng. (Japan/74)

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  • ructed with foundation devices able to decrease the imput acc~lerations caused by earthquakes, and a few years ago a nuclearpower plant is projected by Koeberg S.A. in Africa with an elas to-plastic foundation system composed by rubber and sleek s

    ;,teel (3).

    2.- MODEL USEDIn order to pointed out the possible relationship bet

    ween the variables involved in the problem, a two degree offreedom system model was used (figure 1).

    In the model; the mass number one, ml, represents thefirst story and the foundation device; the stiffness number 0ne, kl, is the stiffness of the device which transmit the ground movement to the structure and has an elasto-plastic behavior; the mass number two, m2, is the building's mass and thestiffness number two, k2' is the stiffness of the building.

    The ground motion exitation was imput to the model bythe accelerograms of three earthquakes: San Fernando earthquake (recorded in Pacoima dam, S - 74W component), El Centro (N-S component) and Miyagi-oki earthquake (recorded in the basement 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 fora non-linear systems.

    3.- DAMPING FOUNDATION DEVICESSome systems were analyzed before one was chosen as th

    e most appropriate, but they could be divided in the followinggroups:3.1.- Hanging type:

    Fundamentally, this one consists of hanging the struc_ture so that it could work as a pendulum (figure 2), from thephysical pOint of view, the stiffness (kl) of the device is given by the formula:

    Research Engineer of I.I.S.E.E. (1980)

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  • rk1 = _-=-P_...:.x'--_E=---__

    sLn , r x Lt

    Where:k = stiffness value of the device; other words, restoring force1

    divided by deformation.P = weight of the structure~= angle between the cable and the horizontal planeE = Young's modulus~ = allowable stress in the cablet

    L = length of the cable

    where,In the practical field if steel is used, for example,

    2 r.- 2E= 2100000 kg/ cm, U = 1400 kg/ cm , 0< = 900 (in the mot

    st favorable case) and L= 100 cm.; kl becomes 15 x P, this is ahigh value for kl because instead of decrease the imput it willwork as an amplifier.3.2.- Sledding type:

    Mainly this type consists of defining one level of imput force by the friction angle between the building and the earth (figure 3). Nevertheless the behavior of this model is interesting, the analysis by computer is difficult.3.3.- Floating type:

    The idea in this type is floating the building in a medium where some frequencies could be missed and a medium withbig values of damping ratio will decrease the imput values hig~ly. From the practical point of view, the floating medium to beused could be soft soil and some researchers had pointed out th

    ,',is fact (4). Therefore the problem to be analized in this typebelongs to the soil-structure interaction.3.4.- Rolling type:

    Basically this type involves rolls, balls or lenses 10cated between the earth and the foundation (figure 4), with thecharacteristic that the shape and the size of the lenses or balIs gave different behavior to the whole structure by providingthe wished foundation's stiffness.

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  • Some devices and the kl value formula are listed below:

    Wkl=-----Z(R-r)

    Where:W= weight of the buildingR,r,e,a showed in graphs above.3.5.- Mixed type:

    The combination of two of the types mentioned beforegenerates the mixed type and this one seems to be the most suitable to solve the problem. The nuclear power plant projectedin Africa includes a floating-sledding type (figure 5). And t~is study is related to this type, where the foundation behavior is elasto-plastic.

    4.- VARIABLES CHOSENA floating-sledding type was selected because it is po

    ssible for construction. For that purpose a slab foundation must. be done, and over that the lenses will be located and overthe lenses a steell plate with lubricant, then the foundationof the building will be placed (figure 6).

    Since the variables involved in the problem are eight:stiffness of the device (kl), stiffness of the building (k2),mass of the foundation (ml), mass of the building (mZ)' dampingratio of the foundation (hI)' damping ratio of the building(h2), sledding level of the foundation or yeilding point of the foundation (Q ) and imput accelerations (x); some of them m u

    yst be fixed for the analysis. Therefore a three stories building was considered, in order to avoid the influence of the roc,I~king problem (5), with a hundred square meter per floor. Thefoundation weight was included in the first floor. In this manner the upper stories weight was rounded in 300 tons.

    By keeping the period of the building between 0.2 to

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  • 0.3 seconds the stiffness of the building is in the range between 150 to 300 tn/ern. The damping ratio of the building was fixed in 0.02, considered as the worse case, and fa r the foundation it was fixed in 0.001 to assure tha t the only way where theenergy will dissipate in the foundation is throught the pla~tic deformation beyond the sleeding level. Due to this cons iderations, the variables m 2 ' k2' hI and h2 were fixed. The variables kl, ml, Qy and x were the values used as parameters.

    The stiffness of the foundation, kl, w a s varying fromthe 10% to 80% of the building's stiffness in the following range:

    Name k2 kl ratio Name k2 kl ratio

    ROLING 0 150 20 0.13 ROLLING7 2S0 40 0.16ROLLING 150 40 0.26 ROLLINGS 250 SO 0.32ROLLING2 150 SO 0.53 ROLLING9 250 120 0.4SROLING2A 150 100 0.66 ROLING9A 250 160 0.64ROLLING3 150 120 0.80 ROLING9B 250 200 0.80ROLING3A 200 20 0.10 ROLLINGO 300 40 0.13ROLLING4 200 40 0.20 ROLLINGA 300 80 0.26ROLLINGS 200 80 0.40 ROLLINGB 300 120 0.40ROLLING6 200 120 0.60 ROLINGBI 300 180 0.60ROLING6A 200 160 0.80 ROLINGB2 300 240 0.80

    The stiffness of the foundation, k ,1

    are varying meanwhile the stiffness of the building is fixed due to the factthere is more uncertainties about the behavior of the device thanthe behavior of the building.

    The weight of the foundation was defined by three value s : 50, 100 and 150 tons, in other words, one sixth, one thirdand 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 ofthe 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

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  • 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 foundation'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 followingvalues: maximum shear coefficient for the building (SCMAX), ma~imum absolute displacement of the foundation (with respect tothe earth, ABMAX), frequency for the first mode (OME) and themaximum acceleration imput to the first story (ACMAX).

    5.- RESULTSThe results of the computation are shown in the Table 1

    and they are listed in the annex number 1, 2, 3, 4 and 5. Wherethe first column is the value of ACHAX, the second column isSCMAX, the third column is ABMAX, the fourth is the earthquakename, the fifth is the maximum acceleration of the earthquake,the sixth is the type of the building structure and the seventhis OME's value.

    To visualize the relationship between the variables, acomputer program was performed to draw the following graphs:

    Ordinate (Y) vs. Abscissa (X)- maximum shear coefficient vs. stiffness ratio- maximum .absolute displacement vs. st