ML20102A682

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Forwards Corrected Suppl to 850201 Response to NRC Design Audit Action Item 7.Discussion of Response at 841130 Meeting Re Draft SER Open Item on Soil Structure Interaction Addressed
ML20102A682
Person / Time
Site: Beaver Valley
Issue date: 02/01/1985
From: Woolever E
DUQUESNE LIGHT CO.
To: Knighton G
Office of Nuclear Reactor Regulation
References
2NRC-5-016, 2NRC-5-16, NUDOCS 8502080510
Download: ML20102A682 (32)


Text

,, _~ ~. .

,t THIS 2NRC LETTER BEING REISSUED' }

TO CORRECT DUPLICATION OF 2NRC t 015 DATED 02/01/8N s TO NOW READ 2NRC-5-016 1 78 5 41

.412i 92WO Nuclear Construction Division Robinson Plaza, Buildir.g 2. Suite 210 Wecm N2s 78tm Pittsburgh, PA 15205 February 1, 1485 United States Nuclear Regulatory Commission Washington, DC 20555 ATTENTION: Mr. George W.-Knighton, Chief Licensing Branch 3 Office ot Nuclear Reactor Regulation

SUBJECT:

Beaver Vallay Power Station - Unit No. 2 Docket No. 50-412 Response to NRC Structural and Geotechnical Engineering Bran 6's Dreft SER Open Item on Scil-Structure Interaction Centlemeni 7 This letter provides our response to the NRC Structural and Geotech-nical Enginearing Branch's (FGEB) Draft SER open item on Soil-Structure Interaction (Item SRP 3.7.3 [ Audit Action Items 4, 7, and 23]) . This submit-tal suppleme nt s our response to NRC Structural Design Audit Action Item 7, which~ was provided in Reference (a), and.. addresses the discussion of that response at our November 30, 1984, meeting with the SCEB (Reference (b]).

In Action Item 7, the SGES reviewers requested that additional soil -

structure interaction analyses be performed for the containment and intake structures in order to demonstrate that SVPS-2 meets the intent of SRP 3 . 7 . 2 .11.4 ~. No further analyses . were performed for the intake structure '

because, as stated in References (a) and (b), the adequacy of this structure was addressed under the BVPS-1 docket.

To demonstrate that BVPS-2 meets the intent of?SRP 3.7.2.11.4, DLC's '

. response to Action Item 7' provided an alternate soil 7structure interaction-analysis for the coit' inment a structure'. As .discused fin FSAR Section 3.7.2, ,

the original soil-structure interaci. ion analysis for the1 conti% ment i; sed the

./

. finite element methed (PLAXLY computer code), in which the 66il was nodeled'.

as - finite ' elements and the structure as a lumped mass elastic beam. The 4,'

r alternate' soil-structure interaction analysis, provided in, the Action Item 7 4 3

response, was- based og cthe three-step solut ion devebned by Kausel and ,j -/ "

Whitman. This! analysis used .' the same lumped. ~mdss elastic beam model' to ,

- represent ~ the Containment structure; the soil - was modeled .as a half-space using th'e frequency-dependent compliance function met'aod of anlaysis. The

' design'e'arthquake input'mo' tion was. defined to occur at the ground surface in the , - free field. '. Kinematic 4 interaction was ualh to, transform the ; purely

' translational motion at the ground surface into. combineditranslational and-rotational motion'at the foundation level. O',

At - our meeting with the SCEB on' November 30, 1984' , our response to

~

4' ,

~

' Action . Item - 7 was : discussed. The. SGEB reviewers requested that a further -

soil-structure interaction analysis be 'perforir.ad' by using either a simplified l 1

8502080510 850201's'h.'

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Uni",e'd 3itates Nuclect Ragulatory Commission

. Mr.Jeor'ge W. Kni'ghton,- Chief

'Page 2 ,,

, 9' .

Wh itman -type s soil spring approach or a frequency-de pendent impedance 4'

approach. Yn't either approach, the SGEB reviewers specified that the free-field ground surf ace earthquake input motion was to be applied at the founda-tion level of the structure. DLC agreed to perform an analysis for the

, M cont aimment structure that uses the frequency-de pendent impedance approach

}i)' with ' the free-field ground surf ace motion applied at the foundation level. l 3

j , 4pon further consideration of the ECEB reviewe rs ' request, we con-cludenthat such an analys,is would yield results which are neither physically 9

'" represents.tive of the ' actual site conditions nor technically sppropriate. l The SCEB reviewers' suggested analysis neglects two physical phenomena which are well-recognized by prof essionals in the field of seismic analysis (Refer-ence ,(c}) and are.very impo rt ant to specifying the appropriate vibratory

. groand motion to be applied at the foundation level of the structure, cc,nsis- j I tent with the: requirements of 10CFR100, Appendix A . These two phenomena t are:

(1) the soil laye r between' the ground surface and foundat ion level modifies the foundation level vibratory motion compared with the

, . greund surface vibratory motion; and

. i

'(2) the geometric ef fects of the structure also modify the vibratory 1

motion at the foundation level relative to the ground surface vibratory motion.

The Kausel-Whitman three-step anal) tical method, used in our Action Item 7 response, . has c - sound engineering basis and accounts for both ' the

'ef fects of ' the soil layer and the geometric ef fects of the structure on the

. vibratory motion at the foundation level compared with the ground surface -

+ - vibratory motion. We believe that the results of the ' analysis presented in 3

Lour Action Item 7 response are physically. consistent with these well-recog-17 . nised principles of soil-structure interaction and are therefore technically appropri ate. Attachment A provides - a detailed description of the Kausel-Whitman three-step analytical method.'

< :J-hr

. Attachment - B presents ' a comparison of' the one percent damping curves

~

j of. b'othj the: BVPS-2 design - response spectra and those resulting from the J Kausel-Whitman three-step method' for several typical locations . Both spectra .

",' '~

compare favorably with only ' minor exceedances which are- ins ignificant consideving the conservative value -(one percent) used for equipment damping.

This ' demonstrates that BVPS-2 meets the intent of SRP 3.7.2.11.4.

n In DLC's a plication for the..BVPS-2 Construction Permit, the soil-s structure . interaction was analyzed as directed by your staff. (See PSAR

= Question 3.19, . Amendment - 7, July 9, 1973.) In the course of the present Operating License Application' review, the docket has been augmented with a responsive, . technically' appropriate analysis which indicates that BVPS-2 meets the intent of . SRP 3.7.2.11.4 of NUREG-0800, the most recent formal NRC

-guidance concerning this ?isJue. Therefore, we believe that the BVPS-2 PSAR,

'FSAR, the supplemental informationL provided in the response to NRC Structural s Design l Audit Action' Item 7, and' this' letter provide a complete record for the.

satisf actory closure of this issue. DLC is willing to again meet with the

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y b-qmm-- --w ,,,-C.eN**-d'-*w-ed F- e m a w- e +=ww= --w+csas--%y syg..g-w.- .,,m-vewin ==e-Mw---tw*'-eww,w--@-+,-w-- y-e+w+-*=~ee-- 1- -e

  • m United States -Nuclect Rtgulatory Connaission Mr. Gaorge W. Knighton, Chief

.Page 3' SGES staf f to clarify any points in this or previous submittals. If such a

. meeting is desired, DLC requests that the Assistant Director for Licensing participates to ensure involvement of appropriate NRC management personnel.

However, DLC believes that further requests for analyses utilizing al t e rne-tive methodologies .will of fer no, meaningful additions to the existing record.

Therefore, DLC is requesting that further requests fo r information on this issue be addressed by.the NRC staff as a backfit in accordance with the

- provisions of 10CFR50.109, CNLR 84-08, and NRC Manual Chapter 0514.

DUQUESNE LIGHT COMPANY By .

E. J. Woolever Vice President JD0/wjs A t t achment s -

cc: -Mr. B. K. Singh, Project' Manager (w/a)

~

Mr. G. Walton, NRC Resident Inspector (w/a)

Mr. G. ' E. Lear,- NRC SGER Chief (w/a)

References:

(a) Letter 2NRC-4-080, dated June 15,L1984 (b) Letter. 2NRC-4-207, d sted December 17,- 1984

...c)

( NUREG/CR-1780, " Soil-Structure Interaction: The Status of Current Analysis Methods and Research" prepared for U.S. .

Nuclear Regulatory Commission ' by Lawrence Livermore Labora-tory, October 1980 COMMONWEALTH OF PENNSYLVANIA -)

) SS:

COUNTY OF ALLEGHENY -)-

' On this /44 day of _e m_> > f , /976 , before me, a Notary Public in and - for , said Commonwealth and County, personally. appeared E . _J . Woola.er, who being duly sworn, . deposed and said that (1) he is Vice President ~o f Duquesne Light, (2) he - is . duly mathorized to execute and file the foregoing Submittal ; on behalf of said Company, and (3) the statements set forth in; the Submittal are true and correct to the best -of his knowledge.

0 Notary Public ANITA ELAINE RI!rtn. TiOTARY FUGLIC ROBINSON TOWNSHIP, ALLEGHEN'/ CCUNTY -

' MY COMMISSION EXPIRES OCTOBER 20,1986 i

gr e w--wen---p%- -w hy9-e w ut-ger ye*TN^8' T'-

i

v ATTACHMENT A

^

I. DESCRIPTION OF THE THREE-STEP ANALYSIS

The solution of soil-structure interaction problems can be reduced to the following three steps:
1. Calculations of frequency-dependent soil stiffnesses
2. Modification of the specified surface motion to account for structure
embedment

]

l :3. Interaction Analysis ,

These steps are illustrated in Figure I-1 (see Reference 2).

I.l.1 Frequency-Dependent Soil Stiffnese f -

The frequency-dependent stiffnesses of a rectangular footing. founded at the i
surface of a layered medium are computed with the program REFUND, discussed in Section II. The program solves the problem of forced vibration of a i rigid plate on a viscoelastic, layered stratum using numerical solutions to i the generalized . problems . of Cerruti and Boussinesq (see Figure I-2) . The effects of unit harmonic horizontal and vertical point loads are combined by ,
superposition to produce the behavior of a rectangular plate.

Solutions to the problem of a point load on the surface of continuum require an assumption about the behavior of ypium directly under the load; for example, see Ti.aoshenko and Goodier. In REFUND, a solution directly under the ' load is achieved by employing a columen of elements for which a .

linear displacement function is assumed. Away from this-centr 31 column, in the "far-field,"'the solution for a viscoeleastic layered medium is obtained (see Figure I-3). I g, If the _ central column under the point load is removed and replaced by equi-valent distributed forces corresponding to the internal stresses, the dyna-mic equilibrium of the far field -is preserved. Since no other prescribed i- forces -act on the far _ field, the' displacements at the ' boundary (and any

, other point in the far field) are uniquely defined in terms ' of these bound-

- a ry fo rces . . The problem is thus to find the relations between these bound-ary forces _and the corresponding boundary displacements.

LB4-12241-7772 1

In" REFUND's cylindrical coordinates, loads and displacements are expanded in Fourier Series around the axis:

oo oo Ur- 0Eu n cos nO Pr " E' 0 Pr cos n s Uy

  • u" cos n O Py n py cos n O n n Ue ' h-u, sin ne P, = g - p, sin nO For the problem at hand, only the first two components of the series are needed. The (unit) vertical force case corresponds to the Fourier component of order zero (n = 0), and the horizontal unit force case corresponds to the Fourier component of order one (n = 1). The cartesian displacement (flexi-bility) matrix (F) at a point then follows from the cylindrical displacement components.

(ur+ )+ (Up U )COS 26 Uf CoS6 (u p-u g) sin. 20 u y coS e .u y u'y sin e -

(ur-uh Sin 29 UfSinO 1 (yI+yg).1 p (yl.u r g) cos 29 .

% J and the displacement vector for arbitrary loading is U = FP where m e m (UX EX U-- uy v P-i p  ?

y Sz; sL P B4-12241-7772 2

~ . . . - .- . _ _ _ .

4 U - is the displacement vector at a point (x,0,z), while P is the load vector at (0,0,0). The' coordinate system is illustrated in Figure I-4.

For p,oints along the free surface,.the reciprocity theorem requires tL at US Uy . Hence, F is chessboard symmetric /antisynssetric. REFUND then ecs-putes the cylindrical displacement components for the two loading cases, and determines the cartesian flexibility matrix F under the load (axis) at the boundary and at selected points beyond the boundary. ,

To compute the subgrade stiffness functions for a rigid, rectangular plate, the program discretizes the foundation into a number of points and computes the global flexibility metrix F from the nodal subnatrices F using the technique just described. Imposing then the conditions of unit rigid body-displacements and rotations, it is possible to solve for the global load vector from the equation ,

FP = U where U is the global displacement vector satisfying the rigid body condi-tion. It follows that U is of the form U = TV

.I where V is a (6 x 1) vetor containing the rigid body translations or rota-tions of the plate and T is the linear transformation matrix assembled with the coordinates of the nodal points. The stiffness functions are then obtained from e

Z=TP l

l

(=

l whi-h corresponds formally to l

Z = T P' TV l.

B4-12241-7772 3

j. A comparison of REFUND results with another method is shown in Section II.1.

.I.1.2 Embedment Correction The effects of foundation embedment on the impedancesgre included by employing correction factors described by Kausel et. al. These correc-tion factors are determined from parametric studies of embedded fcundations and are of the form

-Cg =(1+C h )(1+Cd)(1+C 3

3 k) in which C * *******i** ****#

R.

R = foundation radius E = embedmont depth H = depth to bedrock Ct = constanta, different values for each degree of freedom.

. The frequency-dependent stiffnesses, K, determined by REFUND, are modified -

to become K1 =KxC R ,

f I.1.3 Kinematic Interaction l In the second step of the analysis shown in ' Figure I-1, " kinematic inter-action" modifies the purely translational input specified at the surface of

[ the stratum to both a translational :and rotational motion at- the base of the l rigid, massless. foundation. The existence of the additional input can be 1 inferred from Figure I-5. In a stratum undergoing . translational motion

! .only, the boundary. conditions ' at the " excavation" require the foundation to rotate. Ignoring the rotational component would result in an unconservative solution. Note that 'the modified motion at the base of the foundation is not equivalent to a deconvolution.

The solution to the kinematic interaction portion of the analysis is based on .Kausel's adaptation of Iguchi's (1982) generalized weighted averaging technique. In essence, the method requires solving the 6 x 1 equation U.H(A f T UT *d A + K [A TTg* dA 1

84-12241-7772 4 b

w f

l l

l l

l

- where:

r.

~

T'= '

1 o o o (z - z o) -(Y - Yo)  ;

. o 1 o -(z - zo) o (X - Xo)  ;

o o 1 (Y - Yo) -(X - Xo) o (Xo,Yo,zo ) = coordinates of the centroid of the foundations .

I contact area (X,Y,Z) = coordinates of foundation / soil interface A = surface area of foundation U* = U*(X,Y,Z) = the free field displacement vector along the foundation / soil interface (before excavation)

S = 'S*(X, Y, Z) -

the free field tractions vector at the foundation / soil interface -

1 T

n. = T TdA

.A K = Foundation impedance matrix Uf = matrix of transfer functions for motion of the massless foundation To obtain the actual motion to be*used as support motion in the- three-step method, the transfer functions must be convolved with the-Fourier transforms of .the ' accelerations' of the surface earthquakes, resulting in the following

. solution:

i (t) = IFT (F(O)uf S (t) = IFT (F(O)*)

t F(O) = Fourier. Transform of surface motion

. IFT = ~ Inverse Fourier trans form sin (P E)

E cos(PE)' g sin (PE) ) ,,, 77 PR (f.{}

~

u =

R PE 4 t 2E

,, g (Go, +5pXlt 0.6i PR )

R 84-12241-7772 5

  • 74- - -p w- r-i +%v y,.m.,v g- -- g ep.-4-y*p.-we-g y. *g-- 99e.,---g.,%,pp.-%.*y -,Wewp,. .ywyp mww 99 q.- is,ye,ay,. ,-p,ew- g.-e.wneg,pp.-- e_wy 9-ye.d'F

I sin E f

\+sv(g hos(PC-ph sm(PET

, (PE)1-( g) p(sd(

, ,p '

e 4}(f cos (PO)+ ( .

8 2 g (PR)*

MPR)

I+ j 4({ } 4 -

R = foundation radius P =

h(MS s

E = foundation embedment depth V = Poisson's ratio Gg = shear modulus of soil adjacent to foundation

, G 2 = shear modul'us of soil below foundation l h = height of the foundation's area center of gravity above the base of the foundation 4

c, = shear wave velocity I.1.4 Interaction Analysis The third step of the procedure- illustrated schematically in Figure I-1 is the analysis of the structural model supported on the frequency-dependent springs from Step 1 for the modified seismic input from Step 2. The solu- _

, tion is achieved using the progrse TRIDAY.

FRIDAY evaluates the ' dynamic response of as assembly of caatilever struc-tures supported by a common est and subjected to a seismic excitation. The support of the met can be rigid, or it can consist of frequency-dependent /

independent springs and dashpets faubgrade .stiffnesses). The equations of motion _are selved is the frequency densia, dece d =4=- response time histor--

ies by coevolution of the transfer functions and the Fourser transfone of the imput escitation. The dynamic equilibrium equations can be written in estriz notation as:

i- . . . .

M U t CY +KY=0 (1) where M,- C, and K are the assa, damping, and stiffness matrices, respec-

, tively. . and U Y are the absolute and relative (to the moving support) dis-placement vectors.

These two vectors are related by:

U = Y~+ EU (2) l'

.84-1224'.-7772 6

-where U is the base excitation vector (three translations and three rota-tions), and E is the matrix:

e m I T1 O I I T2>

4 O I I Tn O I

% J (3) where I is the (3 x 3) identity matrix, o is the null matrix, and

- 3 O Zj Zo 4Yj - Yo)

T; = ' d -(Zj-Zo) o X;- Xo P Y;-Yo MX;-Xo) o u s with x g , y , z g being the coordinates of the corresponding mass point; x,,

y,, z, are he coordinates of the common support.

In the frequency response method, the transfer functions are determined by setting, one at a time, ge ground motion components equal to a unit har-monic of the form u =e . It follows then that U, Y are also harmonic:

84-12241-7772 7

~

iwt = iwt U = Hj e (Hj - Ej) e C*{lwH je iwt 9 gl(Hj- Ej)e iwt .

~ iwt y ,- (H;-Ej)e iwt U- rHje I

(4) gre H k groun(d = H g m) is j . inpu motion, and E. the vector containggs column of the E intransfer functions Equation 3. Sub- for the.

g - is the j stitution of Equation 4 into Equation 1 yields:

(-w aN + imC + K)M = (5) 3 (imC + K)E)

Ifthedampingastriz.isoftheformCsh.D,whichcorrespondstoalinear hysteretic damping situation, the equation reduces to:

(-wsM +-K + iD)Ey = (K + iD)E (6)

In view of the correspondence principle, it is possible to generalize the equation of motion allowing at this stage elements in the stiffness matrix K.

~with aa arbitrary variation with frequency. This enables the use of frequec.cy-dependent stiffness functions or impedance (the inverse of flexi-bility functions.or compliances). l

l. Defining the. dynamic stiffness matrix: l Ed = E + iD - way I

(7) -

The solution for the transfer functions follows formally from:

B4-12241-7772 8 1 ._

sy r+- yv v sq-

i Mg a - g",' (K + 10) Ej a

= -(I+w gpgyg)

(8)

~ Note that the dynamic ' stiffness matrix K does not depend on the loading condition E g. Also, for m = 0, H g(0) = E g.

Having found the transfer functions, the acceleration time-histories follow then from the inverse Fourier transformation:

e U*hJ.

l 1Hgfgfe""de ia (9) where, f = fg (w) is the Fourier transform of the j'h input acceleration componend: ,

T f) a y e*IM dt 0

84-12241-7772 9 4

- 4~-- -, .,

(10)

The procedure consistr then of determining the dynamic stiffness matrix K '

' d

' solving Equation 6 - fo e the six loading conditions H = {H } , determining the six Fourier transforms of the input components F = f; , and perform-ing the inverse transformation (Equation 9), whichcorres[pbdsformallyto:

e i Us Hp eh de

-e The dynainc equations are solved in FRIDAY by Gaussian elimination, and the

-Fourier. transforms are computed by subroutines using the Cooley-Tuckey FFT (fast Fourier transform) algorithm. A comparison of the results of FRIDAY j with another solution is shown in Section II.3.

. I.2 REFERENCES

1. Timoshenko in Goodier, Theory of Elasticity, Third Edition, McGesw-Hill Book Co., pp.97-109.
2. Eausel, Whitman, Morray, la Elsabee, The Spring hethod for Embedded Foundations. Nuclear Engineering and Design 48(1978): 377-392.
3. Michio Iguchi, An Approximate Analysis of Input Motions for Rigid Embedded Foundations. Trans of A.I.J. No. 315 May 1982.

84-12241-7772 10

.. . . - .. ~ . .. - _ _ . - - - .-. - . --

d 4

s If. DESCRIPTION OF COMPUTER PROGRAMS II.1 REFUND AND EMBED i The computer program REFUND is used for computation of the dynamic stiffness functions (impedance funcitons) af a scisid , massless, rectangular place

welded to the surface of a viscoelastic, layered stratum. The subgrade stiffness matrix is evaluated for all six degrees of freedom for the range of frequencies specified by the user. Embedmont effects are applied subse-

.quent ly by the program EMBED.

i The program reads the topology and material properties, assembles the sub- l grade flexibility matriz, and determines the foundation impedances by inver- l l sion. The subgrade flexibility matrix is determined with discrete solutions ,

to the problems of Cerruti and Boussinesq. A cylindrical column of linear l elements is joined to a consistent transmitting boundary, and the flexibil- l ity coefficients found by applying unit horizontal and vertical loads at the axis. h rectangular plate is discretized into a number of nodal points, i and the global flexibility matrix found using the technique just described.

The foundation stiffnesses are then determined solving a set of linear equa-

. tions which result from imposing unit-rigid body translations and rotations to the plate.

f Since REFUND is restricted to surface-founded plates, the effects of embed-l meet are included by adjusting the REFUND results with the program EMBED. ,

' h theoretical bases of these programs and their application to the ' solu- t

, tion methodology are described in Section I.1.2.

l h results of REFUND compare very well with published results. The com-

parisons shown in Figures II.1-2 through II.1 ' are based upoa " Impedance
  • Functions for a Rigid Foundation on a I.ayered Medium," J. E. I.uco, Nuclear
Easiaeering and Design, Vol. 2, 1974. Of the various solutions presented by
. I.uco, the following was selected for comparison (see Figure II.1-1)

I.ayer 1 I.ayer 2

Shear wave velocity 1 1.25 Specific weight 1 1.1764 Poisson's ratio 0.25 0.25 l

The - comparisons shown are of the coefficients k and c f rom which the verti-cal, translational, and rocking impedances can be expressed: i E = Ee [k + iso el 4

in which so is a dimensionless measure of frequency and Ko is a zero-p frequency stiffness.

! The< minor differences shown between the RETIMD result and Luco's analysis I

can be attributed to the use of an " equivalent" rectanau t a r plate tn the REFUND analysis (Luco's is circular) and to differences in hnunila ry condi- .

I tions at the footing (rough vs. smooth).

84-12241-7772- 11 1-

. _ , _ _ . _ _ _ . _ _ _ _ _ _ _ , - . __,_.._.a.______,__._________.._______.___.__._

i II.2 KINACT I

KINACT is a computer program used in the three-step solution of soil-structure interaction problems. Briefly, the program modifies the specified i translational time history at the, surface to tranlational and rotational time histories at the base of a rigid, massless foundation.

The theoretical basis for the program is derived from wave propagation theory as described in Section I.1.3.

II.3 FRIDAY The computer program FRIDAY is used for dynamic analysis of structures sub-jected - to seismic loads, accounting for soil-structure interaction by means of frequency-dependent complex soil springs.

The structure is idealized as a set of lumped masses connected by springs or linear members, and attached to a common suFPort, the mat. The latter is supported by soil springs or impedances, which may or may not be frequency-dependent. Alternatively, the est may rest on a rigid subgrade. The structure may be three-dimensional, but cannot be interconnected; each structure has to be simply connected. Fourier transform techniques are used 3

to determine time histories; cutoff frequency is prescribed internally to 15 Hz.

4 The theoretical basis and implementation of the program is described in Section I.l.4. A comparison of FRIDAY with a public domain program, STAR-DYNE, for the seismic response of a fiaed-base, multi-mass, cantilever model is chown in Figure II.3.-l. The model is shown in Figure II.3-2.

a.

a 4

4 84-12241-7772 12

!l

~

y (t ) (K],,, y(t)

O O u f(t) k - QF gh[--

,g, Q ,Is(t) j(t)

" * +

7(g) 55 1 anVM menev/

  • merve

,. ,ni V '

'[K] nun '

FREQUENCY KINEMATid INTERACTION

  • DEPENDENT INTERACTION ANALYSIS

~ STIFFNESS

. REFUND KINACT FRIDAY O

FIGUREI-1 THE THREE STEP SOLUTI0li

i, lz

,v.. a f s

t )

\

~ ~

\ /

BOUSSINESQ 3

/z

/r

' ~

,.. im .

r X

\ (

~

q .

CERRUTI FIGURE I 2 THE SOUSSINESQ AND CERRUTI PROBLEMS

1 i

"l H

\

\

CENTRAL cowaus%

R g

\

1 FIGURE I-3

. IDEALIZATION OF THE BASIC ' REFUND' SOLUTION FOR CONCENTRATED LOADS s

U Un Z

i 0 -

ex 1

Y l

i i -

l L

FIGURE I-4

'REFU.'10 COORDINATE SYSTEM i

)

, i >  :  ! , - '

! !i l  ! ll

~

a'

_ b('

Yf

)

l I

( t b

mF 1

1 O

b .

I

_ T C

A R

_ E T

_ N I

5

_ C II T

EA -

nN uE l Gl Ii FK l

as i

e

b. i y

A A

N a O A -

I a T A TA y AD N

p 1 I

Il n

1 1

/ -

NU OO I

T AS F

N O

I

) RS T

)

g t EE A g ( LL R

" h ES E u .. CS CA L AM E C

L 0. C A1 A N0 L O0 I 2 A T N AF O LO I S T NE A AS T RA O TB R a =

l t t i i l

, Is , (

~

!lt

I e

o I

i l

1  :

i l

l F

I UNIT RADIUS rsssssssssssssss\ . .. . . . ',

>a as sw a

,, 8 1 - -

^

t i

I' I

{

l q'

t i

r I I

FIGURE E.1-1 LUCO'S TWO-LAYER PROBLEM

1 s

e f-e i

1 l

e 1

  • I eumaus eneum M f

/ \

/ \

as = /

y

/

es - g / -

~

g .

e 1 a s' 4 s e 7 s g

. o FIGURE H.1-2 ROCKING STIFFNESS COMPARISON -

4 REAL PART

  • m~ ,---.*vw--ww-- - -- ,- ----.w-we-----.-v.--.-,-w--we---*- ,wm-. -

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