dynamic interaction of surface machine foundations under vertical harmonic excitation

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DYNAJ\.f iC fNTER CTION OF SURF CE M CHlNE

F OUNDA TIONS UNDER VERTIC L U RMONJC EXCIT TION

Abdelmtmem Moussa', Turek Mack} , Mflhamed Murs/

a J a •ed E/-Sayt·d '

SUMMARY

In this paper. t h ~ illlcraction of mnchlne foundm•on< is studied by C Q d e r ~ n g n three

dimen,ional model of two square f001ings wilh d "ffriable spacu1g. IQ.Ided by vcn•cal

harmonic forces The finite elemeut method (PIZM) is used in this study wilh the soil

modeling as visco-elastiC half space. To calculate the displacement. the o m p l c responsemethod is adopte-d and Ute analysis is implemented in the frequctt<.:y domam A

paramct ir. udy is camed on to evaluate tlte effect of the spacing of he f o u n d a t i o n ~ . the

frequencies. the phase difference and the amplitude ot the rorclng futMlu•••· One resuhof the study shows tl1at the mteracuon of atljacent foundations ru y double the rcspo=of single footmg at low spacing/width ratios which may affect t11e scrviceAbtlity

rcquilcll\ent of ma<jhine foundations. Dunensionless design curves have been de-'Ciopcd

for mass ratio of 3.0 to take the intcl11Ction intoconsideration in the ruachine f o u n d ~ t i o udesigns, the CUIVCS incorporate the previously men tioned factors

INTRODUCTION

The magnitude of vibrauon of a footing In response to an acting dynamic loading •s

essentin1 for their serviceability S.1Lisf:l(tion We can say bat v ry sltghJ (of the order of

a hundredth of a centimeter) vibration ' lllgnitude may cause harm to function Qf themaclline and may even ternfy peopleor threaten the snfery of surroonding Structuresso

the codes and manufacturers of the machines limit the amplitude of the vibrat tons of

machine foundation to a certain l1m11 depcrdin ou tltcir frequencies and pe< AS'

Ftgurc (1) The response of footings subject to louds of dynamic na Ure has been Stttdied

excessive ly (Barkan 1962. Ricban and Hall 1970. Ayra et al 1979. Rocsset et al. 191 0

Prdkash 1981 . Ga1.etas 1984. Wolf 198S. Ga7ctas and Dobry t985and 1986).

Miller and Purscy (1954) showed 67% of total input energy of the venically

oscillated circular energy source. is transmitted by the R wave while 26% and 7% arc

transmitted by the S and the P wa\'es respect \ ely The fuct tllllt two-thirds of the total

r o ~ cf(ieo(echnlc..t Euaul«f'lnS. AinSboutll Uruva'liry· F k u h y o { i = . f l ~ l Cauu. Eg vpl

1IA t\rt« o[Geolcc;twal E n c . ~ ~ S b u n s l l n i \ ~ 1 ~ f*-'U hy ofF..t tgrnot('ftn&, Ca•ru

• w o f ~ ~ m Shams Un1 <nd) · f ~ c u t t y o En a iu«:ting. Duro. E ~ p

as



input energy is transmitted by the R wave. which decay much more slowly with distance

than the body waves, indicates that the R wave is of main concern for foundation on or

near the surface of the ground. so nearby footing may generate waves that creates what

we can can dynamic footing interaction.

Kumar. Bandyopadhyay and Lavania (1986) had shown the effect of spacing on

interaction of dynamically loaded footing using an experimenta1 model of two footings

rested on clean dJy sand. Warburton (1971) studied the problem of two periodically

excited masses with circular bases attached to the surface of an elastic half space.

Another studies performed by Kobori (1973) and Luco (1973) but these studies

concentrate on the geometric spacing and the damping ratio for a loaded footing near an

unloaded footing. In this paper the interaction of loaded surface footings are

investigated.

SOIL BEHAVIOUR UNDER DYNAMIC LOADING

Under small strains soils will have a nearly constant modulii and a small amount of

damping but under large strains, stiffness degradation and large amount of damping will

be observed. It is the case in this research to consider the soil behaviour under only

small strain, which characteri1..e the Serviceability State of machine foundations. Even

under very small displacements soil exhibit hysteric behaviour as n Figure (2). ldriss et

al. (1978) proposed linear model with an equivalent shear modulus Goq at strain level Yand a damping ratio c which introduces the hysteric damping as viscous damping as

shown in Figure (3), where

1 . W (1)4?r. w

The soU is assumed to be homogeneous, isotropic and elastic. and described by the

shear modulus 0 and Poisson s ratio v. Theoretically, damping is zero in a perfect

elastic body however damping is introduced into the solution by radiation ofenergy

away from the footing through the half space of the soil (radiation damping).

The Finite Element Formulation

The 8-node tsoparamteric (Zienkiewicz 1977. Irons, and Ahmed 1980) is used in this

research. This element has linear pattern ofdisplacement along i ts edges. the degrees of

freedom (nodal displacement) vector a•-={a }• for element e is related to the genericdisplacement vector u through the shape function matrix N which are functions of the

spatial position only N = N(x, y, z) so we have

2)

86



Also the relationship between strain and displacement are given in a matrix fonn as

following

t =Au = AN =Ba

where the A matrix is the strain-displacement matrix and B is the strain-nodal

displacement matrix. The stress-strain relationship for linear elastic solid can be written

(4)

Using the virtual work principle we can obtain the following relation

Pe + N ~ d V = TDB dV ale 5)

v v

that may be written in the well-known fonn

6)

where q• is the equivalent nodal force element vector and K• is stiffness element matrix

In the undamped case, the development of inertial body force given by d Aiembert

principal as following :

d i = - i i p dV = p N e dV 7)

so the only change in equation (5) to get the element equations in this case; is to replace

the tenn b by term - p N a so we can get

(8)

where M is called the consistent mass matrix and is defined by

M = NTNdV 9)

v

In which K• and M• are evaluated using Gauss Quadrature with Gaussian points as

shown in Figure (5)

In the damped case, in addition to the inertial forces , damping forces will develop

thus the equations of an element will be in the form

( 10)

87



where is the damping matrix . Using the correspondence principle the damping can

be introduced into solution by replacing the elastic constants with the corresponding

complex ones so

K• =K l+2 ; i) 11)

where l is the damping ratio. For a forced vibration of frequency of frequency ro the

acceleration i e and the-displacement ac amplitudes are related by the relation

• = w2a• 12)

So a dynamic stiffness matrix se is introduced for each frequency ro

13)

It may be note that the dynamic stiffness matrix is dependent on the loading frequency. _

THE COMPLEX RESPONSE METHOD

The application of Fourier Transform in the numerical analysis is accomplished by

defining the forcing function R t) as a set ofN discrete points defined at q ~ interval so

that the time. By applying the discrete Fourier transform {OFT) to the loads at discrete

points R ~ ) . the amplitude of the loads in the frequency domain P ro 1) calculated at all

discrete frequencies 1 are determined. The equation of motion of the system in the

frequency <lomain is equal to

S cvt )u wt) = P wt) 14)

where S ro1) is the complex stiffness matrix defines as before. After solving for the

displacement. the inverse transform is used to obtain the time history of the displacement

can be obtained and scanned for the peak values.

The olkl

The model consists of two identical square surface footings rested on an isotropic

visco-elastic half space the spacing and the loading functions are studied as parameters.

The soil and foundation parameters are given in Table l ). The two footings F1 and F2

are loaded by vertical harmonic forces given by P1cos ro 1t and P7CQS ro2t+e) respectively.

THE PARAMETRIC STUDY

The PtUameten

A computer program was prepared using the previous procedures Smith 1982) to

calculate the dynamic and static response of the model, the parameters studied are :

1. The spacing of the footings s) siB= 0.5.1,1.5.2.3).

2. The active passive dynamically loaded-unloaded) footing interaction.

3. The forcing frequency ratio FFR= ro2/c.o1) FFR=1.2.3).

88



4. The forcing amplitude ratio FAR=(Pi P 1) (FAR=O.O.5,1 1.5,2).

5 The forcing phase difference FPD=(e) (fPD =0 . 45°. 90°. 180 . 270u 315 ).

The results of the dimensionless magnification f ~ t o r M z= ~ J y n l \ m • c I 7 . Alc are plotted vs.

the dimensionless. frequency factor A.= m

lin Figures (7) to (25)

THE RESULTS

ffect o Spacing and The Forcing Frequency Rutio (FFR)

The results are shown in Figures (7) to (I 1). Figure (12) shows that the most

pronounced effect of the of the interaction appears when FFR =1 and dlis effect decreases

by increasing the spacing.

ffed of he Forcing Amplitude Ratio (FAR)

The results are shown in Figures (13) to (19). Figure (13) shows the effect of FAR vs.M / M ~ The results show that the response is linearly proponional to fAR if the FFR=1and non-linearly proportional to FAR i f FFR has other values

ffect Qf Forcing Phase Dlfference (FPD)

The results are shown in Figures (21) to (24) It must be observed l t any value ofeless than 180 has a corresponding angle 8+180° where the footings Fl and F2 exchange

their response due to the symmetry of the problem. Figure( 18) shows that the out of

ph.1se response may be greater the in phase response 10 for s/B=l.S and FFR=l.O. for

sib ~ O Sand FFR=l.O. out of phase response may be m ~ t e r the in phase response 3 % .

The effect of the phase difference is much reduced in higher FFR.

CONCLUSION

1. The closer the two f{)()t/ng, the greater the magnification factor is not always a true

statement because at higher FFR (FFIQJ.O). the footings are not suffering from the

effect of interaction, thus different machine foundation can be located as close to each

other as siB =O S i f heir FFR is higher than 3.0 .

2. The most pronounced interaction effect appears at the least distance s/B=O.S and at

FF R • 1.0. At s/8=3.0 the interaction practically vanishes .

3. The greatest interaction effect appears when the FFR=l but decreases as this ratio

increases and almost vanishes at FFR=3.0 .4. The interaction increases as the FAR increases in a linear fonn for FFR= I but for the

other values ofFFR is not linearly.

S. The greatest interaction happens when the two footings have same phase angle this

effect is small n case of small siB ratios (s/B=O.S) but of greater values at higher siB

ratios (siB= l.S)

89



REFERENCE

1. Ayra S. , O'Neil M. and Pincus G., Design of Structures and Foundations for

Vibrating Machines ,GulfPublishing Co . Houston. TX. 1979

2. Barkan, D.D., Dynamics ofBases and Foundations , McGraw-Hill Book Company.New York, 1962

3.Dobry, R and Gazetas, Dynamic Stiffness and Damping Using Simple Methods ,Proc. Symposium: Vibrations Problems in Geotechnical Engineering ASCE pp.75-

107, l98S

4. Dobry, R and Gazetas, Dynamic Response of Arbitrary Shaped Foundations .Journal ofGeotechnical Engineering. SCE Vol.ll2, pp. 109-135, 1986

S. Gazetas, 0 . et al, Vertical Response ofArbitrary Shaped Embedded Foundation .Journal o Geotechnical Engineering ASCE Vol.llO, pp.20-40, 1984

6. Idriss, I.M et al., Nonlinear Behaviour of Soft Clay During Cyclic Loading ,Journal ofGeotechnlcal Engineering ASCE Vol.l04, pp.l427-1447, 1978

7. 1rons, B.M. and Aluned, S., Techniques ofFinite Elements , Ellis HorwoodLtd.,

Chichester, 1980

8. Kobori, T. and Minai R Dynamic Interaction of Multiple Structural Systems ,

Proc. Of the yA World Conference on Earthquake Engineering pp.206l-2069, New

Delhi, 1973

9. Kumar. V., Bandyopad.hyay, S. and Lavania, B.V.K., Dynamic Cross Interaction

Between Two Foundations Under Horizontal Vibrations . 8' Symposium on

Earthquake Engineering Roorkee, Vol.l , pp.22l-228, 1986

lO .Luco, J.E., Contesse, L • Dynamic Structure-Soil-Structure lnteractionn, Bulletin of

The Seismological Society ofAmerica Vol.63, pp.1289-1303, 1973

ll.Lysrner, J. and Kuhlemeyer, R. L., Finite Dynamic Model for Infinite Media .

Journal o Engineering Mechanics Division ACSE Voi.9S, 1969

lJ .Miller, G.F. and Pursey, H. , The Field and Radiation ImpedanceofMechanical

Radiators on The Free Surface of a Semi-Infinite Isotropic Solid. Proc. o Royal

Society London, Vol.223, pp.S2l-SS4, 1954

14.Prakash, S., ..Soil Dynamics , McGraw-Hill Book Company New York, 1981

15.Richart, E.E, Woods, RD., and Hall, J.R., Vibrations ofSoils and Foundations··.

Prentice-Ha/1/nc. Englewood Cliffs, N.J.. 1970

16.Roesset. J.M., Stiffness and Damping Coefficients in Foundations . Dynamic

Response o Pile Foundations ASCE pp. l-30, 1980

17.Smith, I.M., ..Programming the Finite Element Method with Application to

Geomechanics , John Wiley Sons London. 1982

90



18 .Warburton, G.B. Forced Vibrations of Two Masses on an Elastic Half-Space ..

Journal of ppliedMechanics ASA1E Vol.38, pp.148-156, 197 I

19.Wolf, J.P., Dynamic Soil-Structure Interaction Primice-Ha/1 Inc. Englewood

Cliffs, N J 1985

20 .Woods..

Screening of Surface Waves in Soils .Journal f . ~ o i l Mechanics andFoundation Division SCE Vol.94, No .SM4, pp.95 J-979. 1968

2l.Zienkiewics, O.C., The Finite Element Method , r ed. McGrow-llt/1 ook

Company London, 1977

91



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IJ

.001

1

/

..

1

Frequency z

92

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loop for Jollr111der

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Fig. J) q t ~ i V G i m lJhfOr 1J1odwlu1 tJifd

motnial tloiJiplttg

Fig. 4)

D l s c ~ t l z a t l o n o f

3D solid

C{Hit/nullm Into

j j n l l ~ ~ l e m e n t susing tire fl-notle

brick l e m e n t



ncS

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If

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0

Node

Gauu poonl

4 - 1 - 1- + - - - ·

A - L- - ·- -

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110 o : ~ o s u50 ? 05u·. 10 10~ - - - 1 - - - - t - t t ~ · ~ ~ · - • - - - ~ 1 - t - 1 ~ ~ · ~ • - - ~

f l V .

y

-- ·

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PlAN

Item F u u n d ~ t t iModulu orEla•ticih (kN/m1

) 2 1 1 ~ 1 1 1 ~PoiSion•s ralio (u) 0.17

Damplnt ratio (C) I) II

llnit ·eistht (y) (kN/m1

) IR-1 7

. I - v) mThis val iC is ~ n to mal.c the \ Crtlcal m u s . ~ rollo 8

1=--- · -, u

4 pr_

93

h j (51 - i l l t l \ ''

' or t · H ,,.,J,

' I . . .,,

Soli1 \IU'

O.:l

0

Ill II



.._ _ . - _ . -- 21 '

__ I4

•1'• 0

• A.0

A. •

....,.. II•_,, --- 2 _,,-- a I

•. .t

•• A. •• A.t

Figure 8) s/B=l .O P1- P2 , 9 0

94



I~ lllngM .o_ - I 1-· --- ..1 :-··-?Jii lI

_jj·•• A.

0 \ A.I

~ II:I

I' - - - ' 1 -nl

•-L

I ;r •I A. •• A.

Figure {9) s.B • / .5, P1• P:. 9-0

_-

II

IL......, --·1I I __ . I It ......_. . - - - - J

a,

a I• •• A • A

I _ .l·

:l _._ i • II ___ .

•21a.

a:

lj•• A • •• A

Figure 10} slB=J.O P1=P . e-o

95



-...... ...... - •

, . I ' ~ - - f 1 -•-Fll

l :i

& 0

0.,

A. •• • •• Ao '.- · --·

•r·-

0

- -F I - f'2 I - - ~__ :z

•:£ ~ ~

• 0

•• A. •• 0 ..A. ••

Figure 1 I s/8 =3.0 P1=P1 , 0=0

£ ~ ~ a c t ~ Tl R and Spacing

I I

u

•••tS

1.4

~ I

u

l .f

•••...

• ... • I 2 1

FFR-co2/Q I

1 a/8=0.5 wB=1 a/8=1.5 al8=2

Figure 12) Effect o FFR and siB ratios

96



-· ...... .I

_ __E

-- '• i

0 •• A. • A

...I

10

... . '

••E --nl •

... _ .aj

• •• ., A.

• A.f •

Figure J.I) s =0 5 e»z•JOJ1 , e-o

9



-- -- ·- ··- - · -- - · r

----,••r---------------------------------,

.

••U ~ : _ _ s ; : ; ; ; t ; ; ; ; ; ; a = ~ _ J I I

_ _ , __ _ ·-. - - ·= _ = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = ~ r . - ---· -- - - · ·- .- • ·•·• '· I q • •

2·

"'

F;gure 15} siB= 1 5, ev = lVI 0 =0

·" ••. ...

, _ . __ 2 , _,--nl

•• :i2

•0 .. ••

.,A.

,,

• P •t.S • tOf • f

••

, _nl

a ·

, _ ___ l

:j•

•.. 0 01 ..

Figure 16) SJB=l.5, m,=3 1)1 8=0

98



-.. ~ : : : ~ - · - = - ~ : : ~ ~ = - ---. c : : ~ : ~ ~

t a \~

.L ~ s : t o : = = : : _ j '_·L_=_ = = · ~ • A . ~ : : = : : I t : = : _ _ _ _ J , ....., ..

0• • t .S ,.,

••

1 ~ 12 ~ ~ :1.

...__• •• t A. •

Figure 17) s B := J.O. DJ;=DJ,. 8=0

... -o .... .•• , _ . ___ . _

i.

0 •0 t A

,, • •• A.

.t.l ' '-* ·1--

__,,• ,....

2

i ·

• A.•

•• • A

Figure (18) s/8=3 0, W] =3m1 , fJ=>O

99



-

· · ~ - - - - ~ ~ - - - - ~ - - - - - - - - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - - - - - - - - ~• t .2t Ol en 1 lS •• us

• 1118 0.5 •w2• x = .5 . w2=3w2

• 1/9.::3.0 .w2•3w2

Figure /9) The ej}t CI o Fl·U

••• ..••• FF1 F21

••

• p .. F2]

•I

• •• ..A.

I 1,1 • A. ••

•••~ I-nl

••

A.I

···•

1 F1 F1

••

- • A.

Figure 10) si/J.,.OJ, P1=-P O rt»r

100



•••

a ,

L-

..

c:-fl

•·of

. LFt ::?J.aJ

I

-

I

A.I

:1 A . 2

•I

A.I

·· · · - ·-.

a ,

Ll o

-···

lf

..••

r F 1 . . . F2 ] I

A.

•

I

• I

I

•II

A.II

Figure {12 s/8=1.51 P1 P1 OJz=/»1

1 1



·· ·· · · 1

ll

u r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - -

~

· ~ - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - ~• • • • . •

Figure 24) he effect o FPD

102

dynamic interaction of surface machine foundations under vertical harmonic excitation

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