dynamic interaction of surface machine foundations under vertical harmonic excitation
TRANSCRIPT
8/12/2019 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
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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)
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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)
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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).
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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)
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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
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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
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6
IJ
.001
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Frequency z
92
w
<lg /) Typical
peifonHOIIce
r e q t ~ l , . . , e t f t formachine
fourrdatiotrl. Nfer
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Flg. 1 ) 1/yttmc
loop for Jollr111der
cycl1c lotJdlng
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
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ncS
n0
If
cS
0
Node
Gauu poonl
4 - 1 - 1- + - - - ·
A - L- - ·- -
+10
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
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93
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.._ _ . - _ . -- 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
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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
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-...... ...... - •
, . 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
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-· ...... .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
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-- -- ·- ··- - · -- - · 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
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-.. ~ : : : ~ - · - = - ~ : : ~ ~ = - ---. 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
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-
· · ~ - - - - ~ ~ - - - - ~ - - - - - - - - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - - - - - - - - ~• 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
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•••
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
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·· ·· · · 1
ll
u r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - -
~
· ~ - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - ~• • • • . •
Figure 24) he effect o FPD
102