Soil-structure interaction parameters from finite element analysis

Nuclear Engineering and Design 38 (1976) 289-302 © North-Holland Publishing Company

S O I L - S T R U C T U R E INTERACTION PARAMETERS FROM FINITE ELEMENT ANALYSIS * C.J. COSTANTINO, C.A. MILLER Department o f Civil Engineering, The City College of the City University of New York, New York, N. Y. 10031, USA and L.A. L U F R A N O Paul Weidlinger Associates, New York, N. Y., USA Received:30 January 1976

A series of two-dimensional finite element computer runs were made to compute the frequency dependent soil-structure interaction coefficients. Variations in the element size, mesh dimensions, boundary conditions, and soil hysteretic damping ratio to determine their influence on the computed interaction coefficients were made. From the calculations, it has been determined that the primary requirement of the mesh is a transmitting boundary formulation. For low damping conditions, roller support boundary conditions must be placed exceedingly far from the structure to ensure convergence of the ~esults to the analytic solution. In addition, with such boundary conditions, the addition of artificial hysteretic soil damping cannot be used to simulate radiation damping behavior of the continuum. A frequency dependent criteria is also presented to determine minimum size elements that must be used in any calculation.

1. Introduction

(2) the s o i l - r o c k free field can be adequately represented as a linear elastic system with all dissipation restricted to a hysteretic damping modulus in each soil layer; and (3) the seismic input basement motions are such that without the embedded structure at the surface, a given criteria motion history (or response spectra) will develop at the surface. These restrictions are typical of the assumptions usually made in such analyses. In treating this problem, two methods are generally available to the analyst. The first, as shown in fig. 2, involves a one-dimensional (shear wave) convolution of the surface criteria motion to obtain the corresponding basement motion, which in turn is used as input to the two-dimensional s o i l - s t r u c t u r e interaction problem, In this phase, the free field is represented by a finite element mesh which may or may not be continued through the structure. The entire s o i l - s t r u c ture problem is treated as one, i.e. the interaction is implicit to the analysis and the structural response is determined directly. The error inherent in this ap-

In treating the response analysis o f structures subjected to seismic loadings, the finite element method has been and will continue to be used to a significant extent. Unfortunately, for soil-structure interaction analysis, insufficient criteria on mesh construction are currently available to the analyst to assure adequate accuracy in the computations. During the course of this study, a significant number of two-dimensional finite element computations were made with the view o f generating such criteria. At the outset, we make the following assumptions to restrict this discussion to the more usual problems encountered: (1) the problem of interest is a two-dimensional one, plane or axisymmetric, as indicated in fig. 1 ;

* Paper U2/3 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguards and Containment Structures (ELCALAP), Berlin, Germany, 8-11 September 1975. 289

290

C.J. Costantino et al. / Soil-structure interaction parameters

q.

q.

I

EMBEDDED LINEAR FLEXIBLE STRUCTURE

FLEXIBLE LINEAR STRUCTURE

CRITERIA INPUT MOTION

I k

i

+1

11

/

LAYERED LINEAR HALFSPACE

,/./ /

/

+

?

i

/

/

!+

FINITE ELEMENT FREE FIELD MESH +,

Z t / j j /

CORIW.IPONOINO

"7"---" ED DYNAMIC INPUT IO

Fig. 1.2D plane or axisymmetric problem.

proach stems from the computational fact that at the bottom boundary, energy reflecting off the structure cannot radiate out of the system. The argument often used for locating lateral boundaries, namely that soil damping will eliminate the radiation effects, cannot be used for the bottom boundary, since the boundary must be driven at a high enough amplitude (over the entire frequency spectrum of interest) to regenerate the criteria motion. The higher the soil damping, the higher the driving amplitude. This fact stems directly from the specification of the criteria motion at the surface. A second method that can be used to solve the problem exactly (in the sense of numerical exactness) is indicated by fig. 3. The same free-field finite element mesh as used previously is uncoupled from the structure, unit displacements and rotations applied to the interaction surfaces, and (frequency dependent) interactive parameters determined. These parameters are in turn used directly in the structural response analysis, with the criteria motions applied to the in-

IIIlOBLEM

|AIEnb*s'li'lr ED

MOTION PROEKEM

Fig. 2. Convolution method of analysis. teraction elements. It should be noted, of course, that if the finite element analysis is performed correctly, the interaction elements (for lateral, vertical and rocking motions) can be represented with both spring and dashpot elements, the coefficients of these elements being frequency dependent. The use of 'virtual' mass additions to the structure are of course unnecessary. In what follows we will restrict our discussion to this second method of analysis, and concentrate on the criteria that must be used to generate an adequate finite element mesh. It must be realized of course that the correct solution cannot be determined by the finite element analysis. Rather, the point of view should be taken of developing criteria which will yield results that are within acceptable error. 2. Technical formulation

In determining the response of the finite element mesh, specified displacements are applied to the inter-

CJ. Costantino et al. /Soil-structure interaction parameters

291

N {F} = ~ ) {Fn} e x p ( i w n t ) , n--1 ,~

FI.EXIBLE LINEAR STRUCTURE

~T~~_ s

FREQUENCY DEPENDENT INTERACTION PANAMETEN8

ORITImlA 140TIOIll

where {Sn ) and {F n} are complex vectors associated with each radial frequency ~on . Typically, for seismic problems, the transient seismic pulse can be suitably represented by frequencies between 0 and 30 cycles/ sec spaced at 0.05 cycles/sec intervals, so that N will be 600. Substituting eqs. (2) into (1), and considering a single solution n, the equations of motion become

UNIT DIIPL8. AND NOTATION

IlK - 6o2M] + ico n [C] ]{Sn} -- { F n ) .

(3)

11]

I J' 11

)

I Iqg. 3. Exact method of analysis. action surface and the corresponding forces necessary to generate these displacements are determined. The equations of motion of element nodes can be written in matrix notation as [M]{S) + [C](;~} + [K]{S} = { F ) ,

(1)

where [M] is the mass matrix which is diagonal if lumped nodal masses are used; [C] is the damping matrix generated from hysteretic damping moduli of the soil, if any; and [K] is the corresponding stiffness matrix. The vector {S } are the nodal displacements; {F} are the applied.loadings; and the dot represents the time derivative. For the problem of interest here we consider the total transient response as being composed of a series of steady state solutions, so that the load and displacement vectors can be considered as

{S}

=

N Z ) {S *,, } exp(i,,.,,,t) n=l

(2)

For the problem of concern here, we will consider only surface structures and neglect any depth of burial considerations. Various finite element meshes were investigated and these are shown in figs. 4 and 5. Both the width and depth of the meshes were varied as seen in fig. 4 as well as element sizes as shown in fig. 5. For each mesh considered, four different hysteretic damping ratios were used (0, 5, 10 and 20%) which span the range usually encountered and solutions obtained for each one. Of primary interest for this study has been the formulation of the boundary conditions required at both the lateral and bottom boundaries. These conditions fall into two categories, which can be.labeled as the conventional and the transmitting or quiet boundaries, as shown in fig. 6. For the conventional boundary condition, roller supports are applied at the sides of the mesh. When treating the lateral or rotational imposed interface displacement problem, horizontal rollers are used while vertical rollers are used for the case of vertical motion of the interaction surface. The transmitting boundary consists of applying both horizontal and vertical transmitting elements at each node of the boundary. Each transmitting element consists in turn of a spring-dashpot model which transmits radiated energy from the mesh, with little reflection back into the mesch. The exact solution of course allows no reflection back into the mesh. The parameters of the transmitting element [!,2] are obtained from the one-dimensional wave propagation solution and would yield exact results for that problem. The approximation stems from its application to the two-dimensional problem.

292

C.J. Costantino et al. / Soil-structure interaction parameters

q_ I

q_ I

FOOTING

FOOTING

FT]IIII

LI.'.LL I I •

i

III

i i I I I

III I q_

I SHRLLOH

q_ q_ I

FOOTING

CL

FOOTING

MESHES

q_

I

FOOLING

FOOT NG

FR I I l

I

tI I I I I

DEEP MESHES

q_ Fig.

4. Mesh size variations with

The coefficients of the element can be specified as follows. The force transmitted by the element is F* = - b l

( r? c02a ) S* Vs + i a w v s ,

(4)

where S* is the complex nodal displacement; Vs is the shear wave velocity; co is the radial frequency of the motion; 77 is the hysteretic damping ratio; and b is the spacing of the nodes. The parameter a is determined by

cx={II- i ( ~ ) ] / I 1

q_

+ (~-~)2 ] ) 1/2 ,

(5)

where G is the shear modulus of the soil. Using this notation, the transmitting element will allow shear motions to transmit through the boundary with little (hopefully) reflections generated. Thus, eq. (4) will apply to the vertical elements along the side boundaries and the horizontal elements along the bottom

0.5 X 0.5

square elements.

boundary. For the horizontal elements along the side boundary, compressional waves are to be transmitted. Thus, eqs. (4) and (5) can be utilized with Vs being replaced by the compressional wave velocity and G by the modified modulus G* =

E(1 - v) (1 + v)(1 - 2 v ) '

(6)

where E is the elastic modulus and v is Poisson's ratio. The effectiveness of this boundary will be indicated in the following paragraphs. It should be noted of course that if conventional boundaries are used and no soil damping considered, the solution to eq. (3) cannot duplicate the known exact solution [3] since no damping exists in the system. However, if the transmitting boundaries are utilized, damping is available at the boundaries even though no soil damping is available. Thus, radiation damping effects can be determined.

293

C.J. Costantino et al. / Soil-structure interaction parameters CONVENTIONAL IOUNDMY

¢ONDITIONB

i9

FOOTI NG

VERTICAL. ~.~OLLI[RS

HORIZONTAl.

~r

CORRSE ELEMENT 1.0~i.0

~7 1

llllII M

I

BObL[R8

FIXED

VIrATICAL. FOUNDATION MOTION

r

M E [ OII FIXRD HORIZONTAL FOUNDATION MOTION

FOOTING TNAHSMITTING (QUIET) BOUNDARY COND|TIONI

REGULAR ELEMENT 0.5*0.5

I I

FOOTING

~~ALL

OOUNDARII[ll

QUIIT

FINE ELEMENT 0.25.0.25

I q_

Fig. 6. Boundary condition types.

Fig. 5. Variation of square element size used with small mesh.

3. Numerical results As mentioned previously, various two-dimensional plane problems were run as indicated in figs. 4 and 5, in a variation-of-parameter study to determine the characteristics of the computed responses. Comparisons between computed and available analytic solutions then can be used to generate the desired mesh criteria. The parameters varied in this study are as follows: (a) Six different mesh sizes were used as shown in fig. 4 with the location of the lateral and b o t t o m boundaries varied.

(b) For each mesh three different element sizes (fig. 5) were utilized, although all the elements in each problem were maintained as uniform square elements. (c) For each problem, two different boundary types were utilized, conventional and transmitting. (d) For each problem, four different values of hysteretic damping ratio were used, from 0 to 20%. Thus, a matrix of 144 different finite element problems were run. In each case solutions for stiffness and damping parameters were determined over a range of frequencies, with the results presented in dimensionless form as indicated by table 1.

294

CJ. Costantino et al. / Soil-structure interaction parameters EXRCI SOLUIION, OX DRMPING A OX SOIL ORMPING + 5X 501L DRMPING x I0~ 5OIL O R M P I N G m----e 20~ B O I L ORMPING

Table 1. Definition of dimensionless parameters.

. + x

Parameter

Dimensioned variable

Dimensionless variable

Frequency

to ( 1 / s e c )

~. = ato/V s 0

Stiffness parameters: horizontal vertical rocking Damping parameters: horizontal vertical rocking

3d

kHH (psf) kvv(psf) kTT (lb)

KHH = kHH/nG K V V = kVV/nG K T T = kTT/*rGa2

eHH ( p s f / s e c ) cVV (psf/sec) CTT (lb/sec)

CHH = toCHH/~rG CVV = wcvv/~rG CTT = COCTT/nGa2

U_ Lid E3 {J ' u3 Ll_ . Ll_ o

Oq

N 02

3.1. Conventional boundary conditions The first set of problems were run using the small mesh with regular elements and conventional boundary conditions, and these results are shown in fig. 7. As

E X R C T SOLLI[ION, OX A-----. OX S O I L I]QMPIN~ +----+ SX SOIL O ~ M P I N C x x I0% SOIL OAMPINO m • 20X $01L ORMPINO

~0.0

0 .5

1.0

OIMENSIONLESS

~0.0

0.5

1.0

1.5

2.0

DIMENSIONLESS FREQUENCY

Fig. 8. Horizontal stiffness, large mesh, roller sides, effect of soil damping.

O£MPINO

t.5

o

2.0

FREQUENCY

Fig. 7. Horizontal stiffness, small mesh, roller sides, effect of soil damping.

may be noted, the computed solutions contain severe oscillations (corresponding to the natural frequencies of the mesh). The addition of hysteretic soil damping does not help in making the solution converge to the exact solution. Thus, it is clear at the outset that hysteretic soil damping cannot be used to simulate radiation damping effects. This result is of course enforced in the data from further studies. Similar results using the large mesh are shown in fig. 8 with the same conclusions. The oscillations generated by the natural frequencies of the mesh are sometimes called the 'box effect'; that is, energy is trapped within the mesh by the conventional boundary roller conditions. As may be noted, at low frequencies all the computed solutions approach a finite stiffness value, where as the exact solution for the two-dimensional plane problem has a zero stiffness at zero frequency. By comparing figs. 7 and 8, the deeper the mesh the lower the stiffness to be expected. This result points out another characteristic of the finite element solution, namely that static finite element results cannot

CJ. Costantino et aL

/ Soil-structure interaction parameters

EXQCT SOLUTION, 07. DQMPING

~

• NARROW

EXRCT SOLUTION. 0% DQMPING • NQRROW MESH + INTERMEDIRTE MESH x WIDE MESH

a

MESH

X

f

AIL WITH

SQIL ORMPING

5Z

295

QLL WITH 20Z SQIL DQMPING

~L LIJ CO

u 6 '

LL_ L F-0"]

m OJ L'~ I

o [ 'O.O

o

'O.O

O.S

l.O

2.0

1.5

i

I

~ I

i

I

I

I

L ]

I

I

I.O

I I 1.5

i

i

I

; 2.0

DIMENSIONLESS FREQUENCY

Fig. 9. Horizontal stiffness, 5% soil damping, shallow meshes, roller sides, effect of lateral extent• •

]

0.5

DIMENSIONLESS FREQUENCY

Fig. 11. Horizontal stiffness, 20% soil damping, shallow meshes, roller sides, effect of lateral extent.

EXRCT SOLUTION. 0% DRMPING NQBBOW MESH

A l . O ,l.O SQURBE ELEMENI + 0 . 5 w O . 5 SQURBE ELEMENT x 0 . 2 5 w 0 . 2 5 SQURBE ELEMENT

* INTEBMEOIRTE MESH x WIDE MESH RLL WITH I0% SOIL OQMPING

o

B ~co

W~

S6 LL CO CJ 0

co

m

"r-

\

?

c~

oO.O L

I

I

I

l

0.5

I

I

I

t

I

l.O

I

I

[

1

I

i

I

I

2,5

DIMENSIONLESS FREQUENCY

Fig. 10. Horizontal stiffness, 10% soil damping, shallow meshes, roller sides, effect of lateral extent•

I

I

[

2.0

~O.O

[

I

I

i

0.5

I

I

L

]

L

l.O

DIMENSIONLESS

L

I

I

I

I

I

1.5

I

I

I

]

2.0

FREQUENCY

Fig. 12. Horizontal stiffness, small mesh, 10% soil damping, roller sides, effect of element size.

296

C.J. Costantino et al. / Soil-structure interaction parameters

reasonably be used to determine stiffness coefficients, as these are strictly a function of the boundary distances of the mesh. To determine the influence of lateral boundary location on computed results, figs. 9 - 1 1 indicate that as the soil damping increases the location of the lateral boundary becomes less critical and the solutions coalesce. Unfortunately they still do not converge to the exact solution, again due to the 'box' effect caused by the bottom boundary. Based on the restricted data from this study, the relation between the minimum acceptable frequency, the lateral boundary extent, and the hysteretic soil damping is 1 -(0.0175 +0.247?)(L) ~'min + (0.03875 - 0.06 rT),

~ > 0.05,

(7)

where Xmin is the minimum acceptable dimensionless frequency; ~7is the hysteretic soil damping ratio; L is the distance from the foundation centerline to the lateral roller boundary; and a is the half-width of the foundation. For frequencies less than Xmin, the results

A +

• +

will definitely be in error due to lateral boundary interference. Fig. 12 presents a comparison of results for comparable problems using different element sizes. These results agree with data previously presented [4] for studies on one-dimensional wave propagation. The maximum frequency that can be adequately transmitted through an element is approximately Xmax

=

(8)

where d is the smallest dimension of an element. By noting the results of fig. 13, it may be seen that doubling the depth of the mesh does not materially influence the accuracy of the solution, with strong oscillations still in evidence. Thus, to eliminate the box effect using conventional boundary conditions would require extremely deep meshes. 3.2. Transmitting boundary conditions

Similar results using transmitting boundaries along all sides are shown in fig. 14. By comparing these re-

EXRCT SOLUTION. 0% OAMPING OEEP MESH SHRLLON

1/[(2)l/2(d/a)] ,

EXRCT

BOTH NITH 5% SOIL OAMPING

SOLUTION

A

•

NARRON

x

+ x

INTEBMEDIRTE NIOE

MESH

MESH MESH MESH

-r22 5:2

b_

u~

•

0_

to .

co

to

N

Gz ~D

OZt~ _ o

o,i

,

°'0.o

o.s

I

I.o

DIMENSIONLESS

I

I

L

I

I

1.s

2.o

FREQUENCY

Fig. 13. Horizontal stiffness, 5% soil damping, widemeshes,

roller sides, effect of mesh depth.

°o.o

o.s

1,o

l.S

~,o

OIMENSI~3NLESS FREQUENCY

Fig. 14. Horizontal stiffness, 0% soil damping, shallow meshes, transmitting boundaries, effect of lateral extent.

297

C.J. Costantino et al. / Soil-structure interaction parameters

sults with those of fig. 9, it may be noted that a marked improvement in accuracy results. It should also be noted that the results with the transmitting boundaries do not include any soil hysteretic damping. Thus, radiation damping effects are accounted for by the damping introduced along the boundaries of the mesh. Fig. 15 shows similar data using the deep meshes which improve the results, although some small oscillations remain, but do not appear to have a significant effect on structural response for some typical structures encountered in reactor facilities. Fig. 16 indicates the results obtained when soil hysteretic damping is included together with the radiation damping of the boundaries. As may be noted, the smaller oscillations of fig. 15 are dissipated by the soil damping. The relation comparable to eq. (7) for the case of transmitting boundaries is 1/)kmin = - 0 . 1 6 7 + 0.168 ( L / a )

(9)

for the case of no hysteretic soil damping. This result indicates that for the given minimum frequency of in-

• 4 x

terest, a nmch smaller mesh is required with transmitting boundaries than with conventional boundaries, with the corresponding savings that accrue. Fig. 17 indicates a comparison between the results using the conventional and transmitting boundaries and the improvement in results is obvious. Figs. 18 and 19 present comparable data for cases including hysteretic soil damping. As may be noted, the conventional boundaries are always in error at the low frequency end of the spectrum and .become reasonably accurate at the higher frequency end only when significant soil damping is available. 3. 3. Other interaction coefficients

The results for the other stiffness coefficients tend to indicate similar behavior patterns as those just discussed with respect to the horizontal stiffness coefficient. The results with respect to the damping coefficients are even more striking since they represent the effects of radiation damping which cannot be accounted for by the conventional boundaries.

EXRCT SOLUTION ~ NAflflOW MESH + |NTERMEOIQTE MESH x WIOE MESH

A A NRRROW MESH ÷-----+ INTERMEOIRTE MESH x x WIOE MESH

"l-t-

II

L~ o t_)

LL

• p.(/3

N

v.4

Off

%

"1-

.0

0.s 1.0 z.s 0IMENSIONLESS FREOUENCY

e.0

Fig. 15. Horizontal stiffness, 0% soil damping, deep meshes, transmitting boundaries, effect of lateral extent.

°O 0

0,5

l.O

1.5

2.0

DIMENSIONLESS FREQUENCY

Fig. 16. Horizontal stiffness, 10% soil damping, deep meshes, transmitting boundaries, effect of lateral extent.

298

CJ. Costantino et al. / Soil-structure interaction parameters

EXACT S O L U T I O N • ROLLER SLOES, F I X E O 6RSE + T R A N S M I T T I N G BOUNDARIES

• +

R O L L E R S I D E S , FIXED 86SE + T R R N S M I T T I N G BOUNDRRiES

.----A +

2C

JbJ

IJ_

LJ "Ln 0

LL (:3 I-03

~J g

~-

:]z

o,

~.o

0.5

1.5

1,0

DIMENSIONLESS

2,0

• R O L L E R SLOES * TRQNSMITTING

I

I

i

I

I

~

I

0.5

FREQUENCY

t

I

I

I

I

I

~

1

I

~

I__J

1.5

].0

DIMENSIONLESS

Fig. 17. Horizontal stiffness, 0% soil damping, large mesh, effect of boundary type.

-+

I

°0.0

2.0

FREQUENCY

Fig. 19. Horizontal stiffness, 20% soil damping, large mesh, effect of boundary type. Ex~c~ SOLUTION oz O~.~N~

;

F I X E D BQSE B O U N O R R ES

-

; sx SoIL oAMPI~C -

K .....

I0% SOIL ORMPING x 2O% SOIL ORMPING

]E L.P

Eo

LL LU

g

C~

i O3

r',-J

2E

I

'0.0

l

I

O.S

I

I

I

I

I

1,0

O [MENSIONLESS

I

l

I

l

I

I

1.5

I

I

l

I

2.0

FREQUENCY

Fig. 18. Horizontal stiffness, 5% soil damping, large mesh, effect of boundary type.

O;

C~O.O

0.5 OINENSi@NLESS

1.0

1.5 F6EOUCNCY

Fig. 20. Horizontal damping, shallow narrow mesh, roller sides, effect of soil damping.

2.0

C.J. Costantino et al.

/Soil-structure

~" °I

EXACT SOLUTION NABBOW MESH + INTEAMEDIATE MESH x WIDE MESH

"

•

+ x

interaction parameters

i

299

EXACT SOLUTION NAABO~ MES~I MESH

•

+ INTEAMEOIATE x WIDE

MESt'l

z> Lo ~> c~

"7-

°

b.. LU ED 0

2,

•

o

f,)

b_

.

LI- 0 l'-(d9

f~

l--n-c:,

uJS >

-r

°

E).__

I

I

I

l

I

~0'~0

I

0.5

I

I

I

I

1.0

DIMENSIONLESS

I

I

I

I

I

I

1.5

I

I

L

10.0

2.0

1.0

1.5

2.0

Fig. 23. Vertical stiffness, 0% soil damping, deep mesh, transmitting boundary, effect of lateral extent.

EXACT SOLUTION, 0% DAMPING SOIL DAMPING 5% SOIL DAMPING x 10% SOIL DAMPING 0 2 0 % SOIL DAMPING •

0,5

DIMENSIONLESS FREQUENCY

FREQUENCY

Fig. 21. Horizontal damping, 0% soil damping, shallow meshes, transmitting boundaries, effect of lateral extent.

• + x •

ol

I

0%

EXACT SOLUTION • 5% SOIL DAMPING * 10% SOIL DAMPING x 20% SOIL DAMPING

• + x

÷

c) 0

L..) ° u_";

LLJ E3 LJ

o_ :E_ CE

o p-El2 LLJ >-

-

'0.0

o.s i.0 1.s DIMENSIONLESS FREQUENCY

e.0

Fig. 22. Vertical stiffness, large mesh, roller sides, effect of soil damping.

~

~.o

]

]

1

I

]

I

]

I

1

I

I

[

I

I

]

L

0.5 1.o t.s 2.0 OIMENSIONLESS FREQUENCY Fig. 24• Vertical damping, large mesh, roUer sides,effect of

soil damping•

CJ. Costantino et al. / S o i l - s t r u c t u r e interaction parameters

300

o

-

I

EXACT SOLIJ[ION • FINITE ELEMENT SOLUTION

EXACT

SOLUTION

NARROW INTERMEDIATE

•----•

÷-----÷

MESH MESH

o

F-

(J

it_ ILl

LL_ LIJ L~ ea u3

~o Lf3

O

7 (J cI2

O

• l

I

~

I

I

I

'::b.

J

0.s

I

I

J

l

J

I

t.0

OIMENSIONLESS

I

I

J

I

1.5

I

I

I

%,

J

I

EXACT SOLUTION, 0% OX SOIL ORMPING

•

•

• x •

x 10% SOIL DBMPING • 20% SOIL ORMPINQ

J

J

r,

I

0,5

FREQUENCY

J

J

J

I

i I.O

DIMENSIONLESS

Fig. 25. Vertical damping, 0% soil damping, small mesh, transmitting boundaries.

---

c3

J

2.0

I

i

I

J

I

i

t,5

I

}

l--J

2.0

FREQUENCY

Fig. 27. Rocking stiffness, 0% soil damping, shallow mesh, transmitting boundaries, effect of lateral extent•

OAMPING

5Z SOIL ORMPING

• + ru

×

EXACT SOLUTION 1.0 *I.0 SQUARE ELEMENT 0.5 wO,5 SOUARE ELEMENT x 0,25w0,25 SQUARE ELEMENT

•

p-~-- c3

J

LI_

LU (J

ta_ O-

Lt_ • IJ_ o

O3

O'3 Z

~xE cu

fU 0 C]Z o

[J O ED

(:3

~0.0

0.5

1.0

1.s

2.0

BIMENSIONLESS FREQUENCY

Fig. 26. Rocking stiffness, large mesh, roller sides, effect of soil damping.

'0,0

0.5

1.0

DIMENSIONLESS

t,5

2•0

FREQUENCY

Fig. 28. Rocking stiffness, 0% soil damping, small mesh, transmitting boundaries, effect of element size.

301

CJ. Costantino et al. /Soil-structure interaction parameters

• ----•

+x

EXACT 50LUTION. D Z 5X SOIL DAMPING + I0% SOIL DAMPING × 20% SOIL DAMPING

OAMPING • + x

EXACT SOLUTION • NAAAOW MEStl ÷ INTEBMEDIATE MESH ..... x WIDE MESH

t--F--

ta_ CD co O_ O CE C3

~E3 O:

~o.o

o.s ~.o 1.s DIMENSIONLESS FREQUENCY

2.o

5

2.0

OIMENSIONLESS FREQUENCY

Fig. 29. Rocking damping, large mesh, roller sides, effect of soil damping.

Fig. 30. Rocking damping, 0% soil damping, deep mesh, transmitting boundaries, effect of lateral extent.

Fig. 20 indicates the results for horizontal damping coefficients• It should be noted that for the case of no soil damping, this coefficient is identically zero when using conventional boundaries since no damping exists anywhere in the finite element system. The comparable results for the case of transmitting boundaries is shown in fig. 21 and the improvement is obvious• Figs. 2 2 - 3 0 present comparable results between conventional boundaries and transmitting boundaries for the remaining stiffness and damping interaction coefficients. As may be expected, the transmitting boundaries improve the calculations significantly, Fig. 28 presents the influence of element size on the rocking stiffness coefficient, KTT. As may be noted, the element size criteria presented by eq. (8) is not adequate for this parameter at the lower frequency end of the spectrum. For frequencies greater than about 1.2, these results appear to be adequate but below this frequency, further improvement in the element size is required.

4. Conclusions The conclusions developed from this numerical study concern the requirements needed to develop adequate finite element meshes for interaction studies. The primary requirement, of course, is that a transmitting boundary formulation is a necessity when considering the adequacy of the computed results. Of particular interest in this respect is that the inclusion of this capability in the calculations has negligible effect on computer run time. The criteria specified for required lateral boundary extent and element size are approximate and therefore do not include the effects of element aspect ratio on the calculations. This is a subject for further studies. However, it is recommended that these be adhered to until further data is available to improve the calculations. One further comment concerns a typical procedure often used in finite element interaction analyses.

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That procedure uses coarse elements away from the structure to minimize the number of degrees of freedom in the mesh. It should be noted that these coarse elements are opaque to the higher frequencies transmitted through the finer elements, i.e., the higher frequencies will be transmitted back into the mesh. Thus, the coarse elements will act as conventional boundaries at the higher frequencies thereby eliminating any advantages thought to be gained by the coarse elements.

References [ 1] J.L. Lysmer and R.L. Kuhlmeyer, Finite dynamic model for infinite media, J. Eng. Mech. Div., ASCE 95 (EM 4), Aug. (1969) 859-877. [2] A.T. Matthews, Effects of transmitting boundaries in ground shock computation, Report S-71-8, for US Army Waterways Experiment Station, Sept. (1971). [3] R. Parmalee and J. Wronkiewicz, Seismic design of soilstructure interaction systems, J. Struct. Div. ASCE, Oct. (1971). [4] C.J. Costantino, C.A. Miller and L.A. Lufrano, Mesh size criteria for soil amplification studies, Proc. 3rd SMIRT Conference, Paper K2/3, London, Sept. (1975).