ML19242D187
| ML19242D187 | |
| Person / Time | |
|---|---|
| Site: | South Texas  |
| Issue date: | 08/08/1979 |
| From: | Eric Turner HOUSTON LIGHTING & POWER CO. |
| To: | Vassallo D Office of Nuclear Reactor Regulation |
| References | |
| ST-HL-AE-367, NUDOCS 7908140668 | |
| Download: ML19242D187 (75) | |
Text
.y a Houston j Lighting 8TPower company Electric Toner August 8, 1979 RQ Box 1700 ST-HL-AE-367 HoustcnTexas77001 SFN: V-0100 Mr. Dominic B. Vassallo Assistant Director Division of Project Management United States Nuclear Regulatory Congnission 1717 H. Street Washington, D.C. 20555
Dear Mr. Vassallo:
South Texas Project Units 1 & 2 Docket Nos. STN 50-498, 50-499 Transmitta] of Technical Publiutions Related to Soil Structure Interaction The attached technical publications discussing analytical methods related to soil-structure interaction were requested by Mr. E. Licitra of your office in a telephone conversation between Houston ighting and Power, Mr. Raja Gupta and Mr. Licitra of the Nuclear Re 71atory Commission. The articles provide a detailed discussion of the analytical methods proposed to be used in performing soil-structure interaction studies as required by NRC question SEB 130.12.
Mr. Gupta was provided a copy of the attachments by Federal Express courier on August 7,1979. Should additional information be required, please contact Mr. L. R. Jacobi at (713) 676-7953.
Very truly yours ,
4 %
E. A. Turner Vice President Power Plant Construction
& Technical Services LRJ:bf Attachments (4)
CC: Without Attachme +
Director, NRC Office ut Inspection & Enforcement
[
M. D. Schwarz (Baker & Botts)
R. Gordon Gooch (Baker & Bottsi v@p]\ \
J. R. Newman (Lowenstein, Newman, Reis, Axelrad & Toll) l D. G. Barker A. J. Granger R. A. Frazar g ,, q
/
'> ', ') 7 9 0 814 9 L'h'
_m
'4
/
g3, -
[
a e
~
a
. - .. ~ . - - ~ ~ - Ata d
() em i a +
9412 DECEMBER 1972 SM 12 n Cos=siox F^cumS JOURNAL OF THE
, tw d the ASCE Board M Direcion, which SOIL MECHANICS AND 5.xuty stand list all measurements in both customary
_ , _ s . th, is ,, beio conmns con,m,on famon FOUNDATIONS DIVISION 51 vn : ,alves of measurements. A complete guide to
.se from the Arnerm Sooety for Testmg & Materials.
e .1910) at a pnce of $1.25 Mr copy (trurumum smgfe il fa cisJ ar4:ncenr4 s ia,a.lable f rom ASCE headquarters.
vt twmg askcJ to prepare their paMrs in ttus dual.urut
- i t,*w majunty of papers pubhshed oc ad! conunue to yert*i cal iribration of $p"'b' edded rooting yurt;psy By Milos Novak' and Youpele O. Beredugo 2 To by raineten (min) 25A0 INTRODUCTION cenarneters (cm) 2
- " (*) '
Most of the existing theoretical solutions treat the footing as a rigid body aters tm) 0.3c5 attached to the surface of an clastic half-space. However, real footings are k4ometen M) usually partially embedded, and experiments indicate that embedment can
" *" (* ) NI g considerably affect the dynamic response of footings (7.14).
.quare centuneten (cmq 6A5 No rigorous analytical solution of embedded footings is available because squie meren (m3 083 0.836 of the obvious mathematicaldifficulties. The most promisirig way of approaching Muare meten (mg this problem seems to be the finite element analyses as used by Lysmer and
( 3 2 59 Kuh!cmeyer (13), and by Kaldjian (9) for static stiffness. Nevertheless, there is a need for alternative approximate solutions that would be able to precict cubic centimeten um') 16.4 the motion in more degrees-of-freedom and to yield the stiffness and damping cub.c am (m9
- "'*"(* g$ '
characteristics of embedded footings reably applicable in dyna. tic analyses of various structures.
kJosrains (ig) 0 '53 An approximate analytical approach was formulated by Baranov (I), who Wsrams (kg) 9C.2 assumed that the soil underlying the footing base is an clastic half-space and nr. tons (N) 4 45 that the overlying soil is an independent clastic layer composed of a series erwtons (N) 9 81 of infinitesima!!y thin independent clastic layers. The compatibility condition y, ,qu,,, between the clastic half-space and the overlying clastic layer was satisfied only meter (N/m3 47.9 et the body and very far from it. Nevertheless, the solution seems to yield wne. tons per square reasonable results in closed forms, and is very versatile and easily applicable meter (kN/m3 6.9 to any vibration mode (3,16). For these reasons, this solution is further extended cut =c meten (m3 0m38 in this paper and compared with finite element solutions and with experiments hter (dm3 3.8 in order to verify its applicability. Embedment into a stratum is also investigated.
#'C "" (S D I Note.-Discussion open unta May 1,1973. To extend the closing date one month, cubic exten/mmute (m'/cun) 0238 a wntten request must be filed with the Editor of Technical Publications ASCE. This paper is part of the copyrighted Journal of the Soil Mechanics and Foundations Division.
Proceedings of the American Society of Civil Engineers, Vol. 98, No. SM12. Decernber, N*N 1972. Manuscnpt was submitted for review for possible publication on Aprd 21,1971.
' Prof., Faculty of Eng g. Sci.. Univ. of Western Ontario, London, Ontano, Canada.
8Shell BP Nigeria Ltd., Lagos, Nigeria.
1231 CWF Ww" vtnmspf.9p*'WW" *WW PTW~'W.\cw y y gr ww -
- . . , , , n.,
a Y W =4 I' f '_ e M *4 8 I, s.N ~ , . . .v
-L g
s y
A
/ .4s%
9\ h
h:..-
o! o
. e
~
~
N /[
- ' .g
- . g% f /*
. , ,'a % M~ 9# Y w
p
',NY i.s. m :[-[,
.'t% %.O i .,
( :* z. [ W w . E ..
a!MkE%q!k0:E d k n ~ ... p _ m m - . -
SM 12 EMBEDDED FOOTINGS DECEMBER 1972 LA 12 12 2 Special attention is devoted to the vertical motion because this is the fundamen-tal vibration mode for checking the theory. A brief consideration of all vibration pg,)
modes is t iven in R f.16. Coupled horizontal and rocking motion is analyzed in detail in Ref. 3.
I EQUATION OF MOTION i
- Fig. I shows a model of the embedded footing-soil system, and the forces ! ,
acting on the footing. The basic differential equation of motion is f __ _
. .. .. (1) m,
- mw(t) = P(t) - R,(t) - N,(t) y, it ), jg (,
in which: m = mass of footing; w = vertical displacement of footing; P(t) i
= time dependent sertical excitation force; R,(t) = dynamic vertical reaction j at base of footing; and N,(t) = dyramic vertical reaction along the side surface ,! . " tt) of footing. H , ,
In order :o solve Eq.1, the following assumptions are made: '
Z I
W.$:ixlp; '
, 'aA,(tl
- 1. The footing is a rigid cylindrical body with radius r . g[ ,'
'm R'f'fj'Q, '
- 2. Linear elasticity is assumed. { - -.
- " R:
The dynamic reaction, R.(r), is independent of the depth of embedment. [ , Y'y '"[-7' 5' f' ' ,
( There is a perfect bond between the sides of the footing and the soil.
- 5. The excitation force. P(t), is harmonic and acts along the vertical axis.
Fig.1.-Embedded Footing Excrted Under these assumptions, the dynamic reaction at the base can be expressed usins clastic half. space solutions.The relation between footing base displacement '
12 = n :c c , g }
w(t) and clastic ha:f-space reaction R,(t) can be written as i ! t
- - - - (2) ,
l l l l f l
..gp'I'T -Q R,(t) = Gr,(C, + iC2) *(8) - -
[
10 ' I i1 I !
I I t
- f, f, ' '
in which: C, = 4 + f2 , C2 " f2 + f2 -
- -(3) ,
I t I el I l l l i
l 1cc ' { l ul 7 2 l .sa'-!
o ' " I ,I' d Herein, f,'s. = functions of dimensionless frequency a, = wr Vp/ G,5*Poisson' . c. stressL .,
I distnbution A,J U ~ 77 ratio and , ' , i and r, = radius of footing. (Functions fi', were introduced by Reissner and I y5 '
can be taken from later solutions, e.g., Refs. 4,12,18.) l "4 .
The dynamic reaction, N (t), acting en the vertical sides of the footing is
- 3 J I I%
d 15 a complex function of the embedment depth, I, the dimensionless frequency j l l l .
a , the shear modulus, G,, and density p,, of the adjacent soil layer and also , 2 , g of the quality of the contact between the soil and the footing. If s = the dynamic I
f g l l 1 ,
%.C ' .
reaction per unit depth of embedment, then:
0
.. . (4) { civE ON;.ESS FREcuE N,(t) = s(:,1)d: .. .... . . ..
- e c,ig. 2.-Steiness Parameters C,,5, and Damping Ps If the approximate assumption is accepted that s is independent of 2, then Stratum (Dashed Lines) and Side Layer s = s(t) and Baranov's solution (1), of which the basic assumptions are outlined in the Introduction, can be used. According to Ref.1, the unit reaction:
. . (5) f
( 6 = G,(S, + iS ) w(t) . . . .
2 uo 5 c s, , s ) vco -._,-, -..
-, .-n....
f* I
'r !
p ry!,.~}
,_.,..-._-2 -.
. -. --="~--a-
- .~=-<-=m--- - - ' ^ . - -
12$3
( CEMBER 1972 SM 12 SM 12 EMBEDDED FOOTINGS J to the sertical motion because thiG the fundamen-
- q ihe theory. A brief consideration of a!! vibration
- c. spied horizontal and rocking motion is analyzed P(r) the embedded footing-soil system. and the forces am .' fferential equation of motion is '!
3 =(f3 tre ....... (1)
.m _ f. _ _ _ -
atin2; w = vertical displacement of footing; P(t) :: Ng (t) gg ,Njt).
gg y maation force; R,(t) = dynamic vertical reaction ~~~:p' 'i: G.
- dynamic vertical reaction along the side surface :a L iUs'es :
+ =(t)
.c fot!oming assumptions are made: H ._
fa <p;'/ w ,. -' '~--~~l, "N , %
iImdrical body with radius r,. N f5 ;', ,3 p '[e i; nmed. f, ' ,jQ','(j'y,'f'd'yl'(f,',
, ,xy;;g:'<g ' j g j f,f R,tt). is independent of the depth of embedment r 7 'y~,,j,/
~ ,
~ ; ,, g , , 4,f
[4'y: n/
between the sides of the footing and the soil. r
--7 g '4 7 _ f -
t!). is harmonic and acts along the vertical axis. A the dynamic reaction at the base can be expressed
. The relation between footing base displacement c . R,(r) can be written as 12 c: EnT ,g, 9,A 2s -
j ; g
.............. .. ... .. .m "
l 7 7 t'i i i
. c) o lI I'l L i M f). ........ .... .
i l ; l ; ,.y q timensionless frequency a, = wr,VpiG. Poisson's *17 ----l--e. ,c l
2 l l 'c : ld e
m the base, G = shear modulus of half space. ! "
l a'-! 'u I,'k '
I I C2' f I (Functions f,, were introduced by Reissner and 5 ' C' e.'L-sons. e.g., Refs. 4,12,18.) S len ' _I' I I I l
,(f). acting on the vertical sides of the footing is E } } j gi l lN y4 ; l5 g N. 4 embotment depth, l. the dimensionless frequency sai density p,. of the adjacent soil layer and also j 3
2
.f i i ! 3y I I I l, xtseca t!.e soil and the footing. If s = the dynamic
/ \ l .3 I' 2 I o
// l l_ _ _ .d . -
.................. ...........{4) 0 05 f.0 t. S DIMENSh0NLESS FREQUENCY c.
en is accepted that s is independent of : then Fig. 2.- Stiffness Parameters C,. S, and Damping Parameters C3, S ifor Half. Space.
'on(l). of which the basic assumptions are outlined '
Stratum (Dashed Lines) and Side Laysr wd. According to Ref.1, the unit reaction: *
=
.- - .................. (5)
%M Jw+-.mmmmmyn ?.s;.yy vg;:_ . y.yyy . .
--r
. * $ 9feW%M h9
..,o 7 }y@wl%&,w:? :...
,~
~
o u.; # w . S . .
() 4(1 p .& c Q E
+ C , L ; % m . R . l '!
,* . /
v' )
.. om. n a t -
' \, i 4l l x N , < .>f
,' s. %Ts a i
,, N T f
(
,. n% ;y. s r-
. ,l 1 'f h ^
m ~ ~ J I.) A l h r b 8 l r 1 2 . w .:u. my.a eww~An EMBEDDED FOOTINGS DECE.' A9iR 1972 SM 12 SM 12 1294 J ( 8.)J (0.) + Y,(a ) Y,(a,)
tn which S, = 21ra, i
.............(6)
Ji(a.) + Yj(a ) f. T - s 4 .,.
...... ........... ........ .. .(7)
S=, y g ,
Herein 1,(a.), J,(c.) = Bessel functions of the first kind of order zero and one, respectively, and Y (a ). Y,(a ) =,Bessel functions of the second kind k
@g ,
- [
of order zero and one. Then the total side reaction is, from Eqs. 4 and 5: i. 4 - y.
y w
. ~
N,(t) = G,(S,4 iS )*(f) d;" O.I(S i+ IS )w(f) 2
'.. . . . - (8) h i.3 .
2 w
.e ~ ..
Substitution of Eqs. 2.nd 8 into Eq. I yields the equation of vertical vibrations: m i.2 m
G I
~
s mW(f) + Gr, C , - iC2 + - , ~ (S + IS2 ) w(t) = P(t) . . (9) gu G r, i
m I i.c ,o With compicx excitation: 0 c5 RC.ATNE EMSEDMENT
.(10)
(, P(t) = P exp (iwt) = P,(cos wt + i sin wt) . . . .. .
the steady. state response is Fig. 3.-Comparison of St'finess increase witr .
w(t) = wexp(iwt) ....... . .. .. . .... ....(11) -
I in which P = real force amplitude and w = complex response amplitude.
Note the frequency dependent stiffness (spring) constant:
" b G I .(12) :
k = Gr,(C ,G +r,, - S, .. . . . .. . -
i .. -
and the frequency dependent damping constant: , .g {
' d Gr. C+,-S G I -(13) I '
I 'l' s x
c= 2 2 g ;
w G r, Then the real part of the vibration is l' l
... ........ ... . . .(14) '
/
w(t) = w cos(wr + 6) ...
in which real amplitude /
/
... : I P. P, 1 w =- (15)
- o = V(L - mw;p + (cw); L w -2'- w -
,~
1- -
+ 4D -
w, w, >
.. u n e. n Q m'u ew phase shift 6 = -a tan ..... .. . .(16) k - mw 2 Fi;. 4 --Theoretical Response Curves ci Foetin Lines) and Side Fill (Dashed Lines)(Rigid Base Stra c
damping ratio D= ..... . .... ... ... . .(17) f 2mw,
( and natural undamped frequency of an embedded footing
' ' ~ ' " ' ' ~ ~
" ' ~e
- - . - , - - ,,-., .~.m y yr: ~. ~w*:' ~~ V'"
f J 'lir/ , 't J n
s*
i ls,, % w A' A 9 Jts f a g T 9 * ' 4/ st- s_* a
. mi m = 1. L A m h - - " -
,,,_.m.,
- _ . _ . = - 2 h are . - a m_m. ._.,i,_.u 1295 SM 12 EMBEDOED FOOTINGS
( ICEMBER 1972 SM 12
,,if a.) + Y,(c.) Yo (a )
... ...... .(6) lito.) + Yj(o o) l. 7 -
............ ...... .... ..... CD .
, . o.2,
[ s_ t8
.e wt functions of the first kind of order zero and
~
- ogo
. ,r. Y,(a.) = Bessel functions of the second kind @ #
=
n the total side reaction is from Eqs. 4 and 5:
y L4 -
p.05 m
. its d: = G,l( S, + iSy w(t) ... . . . (8) {g ,
E_
E mio Eq. I yields the equ: ton of vertical vibrations: ,u m 4 y
G*-(S, l + iS ) w(t) = P(t) -
k(c) " " Gr.
(9) *t G r, 0 2.0 0 05 IO f.S
- os et + isin wr) . . . - -(10) RELATrvE EMSEOMENT 3 . r /r, L
Fig. 3.-Comparison of Stiffness increase with Finite Element Solution
-........ ...... . . . ,(]j) ampfitude and w = complex response amplitude.
ent stiffness (spring) constant: , ,,
g
,( ... .. ... . .. . . . . . .. . . . .(12) m =
'b at damping constant:
j g s .. ,' u. . n y
\ { ,
l........... .. . .. . .... .(13) '
brition is l '
/ . ~ ~ -
l
.............. . . . ...... ... .(14)
/
Pe 1
=;:- - - - (15) .-
P & r. . 2 2
- c. .. .. .. .. .. .. .. .. .. . . . . .
-m a- a." - W
- m .8
.,*** - - .... . .. . ..(!6)
Fig. 4.-Theoretical Response Curves of Footings Embedded in Undisturbed Soil (Full Lines, and Side Filt (Cashed Lines)(Rigid Base Stress Distribution. B, = 5)
....... . . . ...,, , , ,(3 7)
'cacy of an embedded footing
~3- - C m % W .~
N 1 M C W *i 5 M W M .t W M :c. W
- W S m ?n s t w n. w.1~85' C " % e. W' T c.s,..t . .
- 'gh-ht- M, g,, ,
b ,
-7J<.0 5f4,g.j f -
- n,,
,a, h,j&.%Nr,' h,. Al
- w J [ .4 % * ..
4 ,* #
/ '
. *. s s,% , i / ,/
s ., s- ,
.- . , % > p +
., n y
,.M ff
.b *% 4 *
- r,4,% .veW< .* =T /
( ,/ ,,'eh . h I [ fr r;, W'#.'.,.aT,;r"f'.,j A .
M Lil 5 *$ .
l* (L.O@,',#.Wysi+;&3M.t.;,invFLsCc.r-exWu>m-11m>.
I EMBEDDED FOOTINGS SM 12 SV 12 DECEMBER 1972 M96 f in which e = eccentricity of rotating mass; it is sc introduce the dimensionless amplitude. A = w m/(
- (C,+ S)..... .. ....... . . . ..(18) ,
w,= =
m G r, m EXAMPLES OF NUMERICAL RESULTS (The natural undamped frequency must be determmed from Eq.18 by trial From Eq.15 the response curves of embeddec and error approach because it appears in both C, and S,.)
with various assumptions concerning both the b.
pertinent stiffness parameters C, and S,, and dar
- - ,. ioiwswaca m : are shown in Fig. 2 and given in a polynomial g,
F - s.o rs c sa:= rim l to
! t s
4 k w
,o e .
g re- *
% ' ~
3ee .
k
- #.y' '/ :
w 1 / -
307 .
f / ,
I 3
/ /[o
(
- E i.S -
[/ - ,/
s /
'/ \
- , oe - N
/. 7 -
- - /
a / os -
! [ / e' '
.6 ,
s s /,.-
y j ,' /
S o4 ,
3 2
5 a g[ E asEtistief s .81 Moo FTD mas 3 P - g.- [,s t
Fig. 7.--Resonant Amplitude datio Versus Embedme Fig. S.-Oimen*ionless Rc onant Amplitudes Versus Mass Ratio for Various Embed- Mass Ratio. Undisturbed Soil a = 1, Side Fill 3 = 0.7 r.'ents 6 = l/ r,(Undisturbed Soil and Side Fill with a = p,/p = 0.73) 8
' t. 7 ' # h; =. D e'
- is Y #
= i f
}
- g ,, . ! i 5..#pe e
. o ,
a.
1 r
5 oS
-(~'g pp.--- , .g y3 g
,e
?
l..'
=9 g/
o ,,,
23 E
S 12 - ev 'dj% c . - V."
g on O .S to 6.S a0 , , '
., I tweECaspat stario 3. sfr.
8 3',s Fig. 6.--Dimensionless Reso. tant Frequencies Versus Relative Embedmontfor Various ,-
to M ass Ratios B,(3 = p,/p) o oS
'" 8 5 *""' ' " ' ' #
in most practical problems, the excitation force. F(r) is caused by rotation of an unbalanced mass, m*, and in Eq.15 the exciting force amplitude is: Fig. 8.--Resonant Frequency Ratio Versus Embed
. ... . .. . ....... ... ..II9) Side Fill) y" m,e tal ..... ..
N- . g 3% ' w -, .- . " -
~ - == m w -4g ...- , ,-
s-
- '1
_TJ
.mm.- - - ha - -
- '"- ~ " """ ~~'"^~ ' "
- m. __ - -
~% m.
EMBEDDED FOOTlNGS 1297
- in which e - eccentricity of rotating mass; it is sometimes convenient to also g* g introduce the dimensionless amplitude. A = ow m/(m,c).
. - S,) .. ..................(13)
G r, EXAMPLES OF NUMERICAL RESULTS iucney must be determined from Eq.18 by trial 4 appears in both C, and S,.) Frca Eq.15 the response curves of embedded footings can be computed with various assumptions concerning both the base and side reactions. The M'" pertinent stiffness parameters C, and S,. and damping parameters C2 and S.
- '"' are shown in Fig. 2 and given in a polynomial form in Tables I and 2 to Lo
- s. ,,s** ym d tt& Q ' if $**
j./ -
,, 3,4 it 34 q* / #
hoe .
,s ,s', s
/ s
', s' yo7 '
r
,s's'"o ';
'*or s
/ ,-
s' ,
,). e *'.
,a s" h.
5 009
'# 'O 3$ , _
/
- a e S o4 iS to S
0 o5 Io 8a* h h.a EMBE0 MENT B
- 3 /r, n ptrtudes Versus Mass Ratio for Various Embad-Fig. 7.-Resonant Amplitude Ratio Versus Embedment (Approximately Valid for any W and Side Fill with y = p,/p = 0.75) Mass Ratio; Undisturbed Soil q = 1, Side Fill y = 0.75. 0.85) gs -
3 '/ /
$ LS s,. .
/p/ ' --'
-~~~695- 5
/,/ /
,/ ',
,s,
- t A tJ '
',o, . o']' p , l e .o rs
- 'e- e .,y .t
- v - -
, ts to g3 3 't -
e
- ,3y,.
cm uno . . ,,,. E ,"
it Frequencies Versus Ref ative Embedmont for Various '
.r.'-M Lo 20 2$
o oS so aS g excitation force. P(t), is caused by rotation eweco. car nano a. rir.
M b Eq.15 the excitir force amplitude is:
Fig. 8.--Rasonant Frequency Ratio Versus Embedmont ( Undisturbed Soit, -
......,,,,,,- .(19)
...... .. Side Fill)
~
U N . W T W A',*kiurw g y W s p M N M & M S w @ .W '4i b. 9 ' ] T-" .'-
[
, - '. % W M d. 9 4w 4
- .! pp.9,f y sc as ^* yg? ,sR ,& $&5,f .
. a p.
bS/ .
h.
o Yr- , r M*~ * *, ./
woe Is eN ./ ** '*
, { r
. / J -,
g . .
.* ., f t' e, !
,. ,. \ ,s [ rl
.g N b ( /
0 r 'O
- p
. % % QV- 4 ,e s T w so.f: 4.
p
, g Y:2 *-
.Wm ~ i 5
- ) i > & d , N . C h" % 5ic' a y ;a- M c 4:U.@ XScio= S M*D aim:ts -
EMBEDDED FOOTINGS SM 12 SM 12 1298 DECEMBER 1972 i
Ipyer is independent of the underlying half-space a' facilitate the computation. Curves concerning strata were calculated with v = In Fig. 3 the static stiffness increase found by is O.25 from Warbarton (20) and are shown in Fig. 2 in dashed lines. Curves stiffness increases obtained from Eq.12 with C for a rather representative frequency a, = 0.8.
=~ j embedment,5, k(0) = stiffness for surface footir element method and thus, Fig. 3 yields an idea abou
! ]" of the approxiraate analytical approach. It may b If l\ '
tend to overestimate the rigidity increase with 5 g
si
'g i l\
\ "- E shown later herein.
- Embedment in Italf. Space.-In Fig. 4. the the
- l 5 / s e - ~.
given for a rigid base stress distribution.The curvc
'2 .
/\ / \ a situation where the footing is embedded in an
[
\
\ * "- % " " are in reasonable agreement with the finite clerr
} /\ j# \ 's
\ ' Kuhlemeyer (13). The dashed lines show a differ.
i ,,,
/
i \ is surrounded by a side fill whose density p, =
" ,(,
.' ,' ' , , was assumed that G,/G = (p,/p)'.]
)> #,./@/
' e i" -
_ _/ c ' The curves shown in Fig. 4 were computed fi
/ the effect of embedment very distinctly. The vari
(. */ .&
! ,,I 7 d ,',[*'g N# Y,'b's.., with embedment for other mass ratios can be seen amplitude ratio R, = resonant amplitude with emt
- N[
/ 4' ' ' i of surface footing, was found practically indepen u 7 can be used for any footing embedded in a half-
..... .m . . . . r/rr amplitude reduction.
The natura! frequency (m.) variations appear Fig. 9-Theoretical Response Curves for Vertical Vibration of Footings Embedded ratio too. However, the resonant frequency (freq in Elastic Stratum (Undisturbed Soil, Various Embedments, and Various Thicknesses ratio R, = resonant frequency with embeoment 8.
of Elastic Stratum) footing, highly depends on mass ratio (Fig. 8).
Embedment in Stratum. With equal case, the
- 3. - m,, in a stratum can be analyzed using the proper f
., , { - the stiffness and damping parameters C,,. Sid ni the same as before.
, " Examples of theoretical response curves are sh 3
- " ~
of relative embedment & and two values of m
- = -
of embedment aswellasastrong variabiliti ef the I U.$ involved, are obvious. The variations of resona
',' * *'f,[,',
["
and stratum thickness are plotted in Fig.10.
embedment considerably reduces the dependanc j s graphs shown do not apply for other mass rat.
trends to be expected. Any particular situstion ca
,, ,". ' ', , , , ,.,,, \y' I ..
as long as the base reactions are available.
:.-.e
". It can be seen that the omission of emM iv.. % .
c ..c a ..,
4 . . , .*
overestimation of amplitudes in the case of a omission of layering can lead to substantial und Fig.10.-Variation o' Theoretical Resonant Amphtudes of Vertical Vibration of Footing e mbedded in Elastic Stratum with Thickness of Stratum (Undisturbed Soit, Various SIMPLIFIED DESIGN ANA1.YSIS nbedments)
Calculation of amplitudes and resonant freque concerning half-space were computed from Bycroft (4). It will be noticed that fiedif stiffnessparametersC ands aretakenasi i i S, = 0 for a,-.0 which results from the assumption that the base overlying
- ." " , " . " "c . " ' ' , -~
'"'-r ,%"
=ww a v- i =- MW
--3 - - i.s m.ume- - ynsi .r bn- -'
~ l
._.m ~m "-- --
jm . w A w 2'u e L ."-- - - '- '- --
i SM 12 SM 12 EMBEDDED FOOTINGS 1299
'CIVBER 1972 l'
_ .. ,,. - raing strata were calculated with v = layer is independent of the underlying half. space and has an infinite spcn.
In Fig. 3 the static stiffness increase found by Kaldjian (9)is compared with
'[are she n in Fig. 2 in dashed~ lines. Curves stiffness increases obtained from Eq.12 with C, = 4/1 v and S, computed for a rather representative frequency a, = 0.8. [ A(6) = stiffness with the 5 cmbedment,6, kl0) = stiffness for surface footing.) k' Man used the fine I ! efe ent method and thus,. Fig.J_yigjds an idem enn' _vbr 'ccurev wo W'2'icm
}" m_nximate m'o;ca mem4 It may be noted that all the theones
- b. p l; . o r ,w r g l! l\ h, tend to overestimate the rigidity increase with small embedments as will be I
, j l\
'" i shown later herein.
ii ,
s i.e Embedment in Half-Space.-In Fig. 4, the theoretical response curves are given for a rigid base stress distribution. he curves shown in full lines describe 8
l \
r
/ \ _ $ "."~' a situation where the footing is embedded in an undisturbed soil. The curves
\ \ * * * =. " "
are in reasonable agreement with the finite element solution by f_yrer_ and
[i / 's / \
a g#
Euhlemeyer (11). The dashed lines show a different case in which the footing
\ ',< .
is surtnunded by a side fill whose density p, = 0.75 p. (In this example it
/,__gN was assumed that G,/G = (p,/p)' .]
Y - T%e curves shown in Fig. 4 were computed for a high mass ratio to show
.[..,g h 's... P the effect of embedment very distinctly. The variations of resonant amplitudes
- , , ,Q d' with embedment for other mass ratios can be seen in Figs. 5,6, and 7. Resonant Nf amplitude ratio R = resonant amplitude with embedment 6/resoncnt amplitude
' ' ' ' ' ' of surface footing, was found practically independent of mass ratio. Thus Fig.
- *. . . *E. .[ *... . ,, " 7.93.hemid fc.t 3Aylockag.4mbedded in Shalf4pags ID niima,ta.3c nsanant unpluudesc h aon Cur vn for Vertical Vibration of Footings Embedded The natural frequency (w,) variations appear highly independent of mass s Sod, vanous Embec nents, and Vanous Thicknesses ' ratio too. However, the resonant frequency (frequency at maximum amplitude) ratio R, = resonant frequency with embedment 8 / resonant frequency of surface footing, highly depends on mass ratio (Fig. 8).
( ,,,,
Embedment in Stratum.-With equal case, the response of footings embedded in a stratur. can be analyzed using the proper functions f, and f2 to compute
, {
,* rS the stiffness and damping parameters Co. Side reaction parameters Su are
', T L f"~T the same as before.
~~ [ Examples of theoretical response curves are shown in Fig. 9 for several values of relative embedment 8 and two values of mass ratio. The essential effect
[.".D of embedment,as wellas a strong variability of the response with allthe parameters
- involved, are obvious. The variations of resonant amplitudes with embedment 7.J.'],',,',
and stratum thickness are plotted in Fig.10. With higher embedments, the embedment considerably reduces the dependance on the stratum thickness. The
\ graphs shown do not apply for other mass ratios; however, they indican the trends to be expected. Any particular situation can be analyzed without difficulty
- - -- .h as long as the base reactions are available.
.i..n.*....'.,,'
It can be seen that the omission of embedment can result in unrealistic overestimation of amplitudes in the case of a stratum. And, contrarily, the omission of layering can lead to substantial underestimation of amplitudes.
W Resonant Amplitudes of VerticalVibration of Footing M Diciness of Stratum (Undisturbed Soil, Var;ous SIMPUFIED DESIGN ANA!.YSIS coeruted from Bycroft (4). It will be noticed that Calculation of amplitudes and resonant frequencies can be considerably simpli.
suits frorn the assumption that the base overlying fiedifstiffnessparameters C i ands aretakenasfrequencyindependent(constant),
i M' M M M*, % F W m w y m v*r1In* W h h w ,c % c N i W," o
} ,-"% w '
/
. > c . 3. ;fte** 0...,s n+
.s. #
- temp A(
} re" =
j f ff aaam % ,, s a .*
g m * ~ ,'
g4,* ,.*w' . w ,
I ( %z s ., .
1 ; y~ ..
~
2~. G q D e r.
/ -
l "5
) l>
l "s A
4 i
1
(
l Table 1.-Stiffness and Damping Parameters for Half Space and Side Layers
%'.=., .
Constant Validity sange p, ,.',,.
Values parameters (4) q i~ ,1 -p g *
, (3) * , [4 (tl (2) is) Half Space f {h y.N f;l',/% .
- if d s..
4, C, = 3.90 0 s a, s 1.5
~3 C, = 4.00 - 0.04156a, + 0.6146a', - 2.f00aj + l.801 a) - 0.3646 aj Q N 8't g ~1r f %
00 K $". ., b ,, g^t %
C, = 3.418 a, + 0.5742aj - I.154 aj + 0.7433 a*
C, = 3.50 ?3 ..
p8Q* 'k. ..
\.- . s C, = 5 20 - .
i C, = 5.37 + 0.3f>4 a, - 1.41 a j $ "\
0.25 C, a 5.00 fs.
C, = 5.06 a, y I C, = 7.50 0.5 C, = 8.00 + 2.180a, - 12.63 a' + 20.73 aj - 16.47 aj + 4 458aj C, = 6 80 f -
C, = 7.414 a - 2.986aj + 4.324 aj - 1.782al u i j 1-(b) Side Layer 0 s e, s 2.00
- 3, - 2.70 =
any S, = 0.2153 o, + 2.760s./a, + 0.06084 S, = 6.70
] _
qr
- S, = 6.059a, + 0.7022 a /a, + 0.01616 -
L.
h o ., -l Ae' s' C t A o' r t * * - C
{r , e. f r
' i.
g.
m 1 '
i, s %-
o A V e 5 4 > ?
.t
]
& A[,
.J ; ; Table 2.-Stiffness ar.d Damping Parameters for Stratum
- e _ = = .======-==-.--
=_==:--.;_..__ range parameters g,
Stratum , = 0.25 (31 (4) ,
I h/r' (2) 0 5 a,s 1.50 g -
gy U = 10 0 2
l0 C, = 12 2) - 1.178a, - 0.3056aj - 1.177aj + 0.41(aa; C, = 0.30 {
O I
' C, = 0.2395 aj + 0.5646aj + 0.0227oj - 0.3401o; 4 0.20)al C, = 7.00 0 s a,s 3.25 ,
C, = 8.I1 + 0 8516a,- 3.664 oj - 8.289aj + 11.18, j - 3,978aj C, = E45 y 7 2.0 z C, = 0 (nm44 a, - 0.7386aj + 13.27aj - 39 61 aj + 49 8aj C,=5.5- 26.95 al + Sm9aj 0 s a,s 0.88 c3 M
,I ..- ,
ase, sit A C,=3w f7 00 C, a 4 00 - 0.ONI% a, + 0 6346s', - 2.600aj Al s' - 0.364Aaj ,
V Q !
C, = 3.438 a, + 0.5742aj - 1.154 sj + 0.7433 a' C, = 3.50 r
. m -
,Q
'.* 0.25 C, = 5.37 + 0.3M e, - 1.4I e! C, = 5.20 (
- C ,= 5.06 a. C, = 5.00 "g
\ 0.5 C, = 8.00 + 2.180s,- 12.61aj + 20.73 a' - 16.47a* + 4.458 aj C, = 7.50 F C, = 7.414 a, - 2.986 aj + 4.324 aj - 1.782 al C, = 6.80 r f
- . (bl Side Layer p g
'3 any S, = 0.2153 a, + 2.760s /a, + 0 Or.034 3, = 2.70 0 s a,s 2.00
"'"* ' (h S, = 6.059a, + 0 7022 a,/a, + 0 01616 3, = 6 70
(
F, y
M L I
) u -
$ 1 I {
\' l.
J' un \
h q;-.
if C ,
M
$l ,
1 t b '
(.7 $*3 t
'- '( Table 2. -Stiffness and Damping Parameters for Stratum p f
Constant Validity g'
Stratum y = 0.25 parameters range f 7 h/r, b g }1 (1) (2) (3) (4) rvi r j
>< , , s*e , i, 4 C, = 10.0 0 s e, s I.50 $
g
- e, 1.0 C, = 12.13 - 1.17B a, - 0.3056aj - 1.177 a' + 0.4160a* [
T v/g C, = 0.30 y g %,,g I / C, = 0.2395 aj + 0.5646ej + 0.0227a'- 0.3403aj + 0.203 a' e s e,s 1.25 C, = 7.00 !
.4 ,-8 ; 2.0 C, = 8.13 + 0 8516a, - 3.664 aj - 8.289 aj + II.18 o' - 3.978aj O
C, = 0.45 P
- YPO C'N l L' C, = 0.004044a,- 0.73R6aj + 13.27aj - 39.61 a; + 49.8aj - 26.91a* + 5.069aj -
4 s
- .4 s '
3.0 C, = 7.04 + 0.4659a, - 6.989aj C,=5.5 0 s e, s 0.81 $
l / :
h*J h ).M'e > . C, e 0.7361aj - 1.462al + 3.573aj C, = 0.65
- C, = 4.30 0 s e, s 0.62 ;-
4.0 C, = 6.579 - 0.2422 a, - 0.3889aj - 29.69aj + 7.7II aj + 76.44 aj - 77.42a' p y {k. , Af C, = 0.02804 a, + 3.02 a' + 7.458 al - 184.2 aj + 655.7a! - BG4.9a* + 314.2 aj C, = 1.00 ,
(' glys -
r (
' k.
~ (
l _
l
.9,,, ,
e g .* 1
- '* .. p,*/
~s ,
?' i
.fb
. ? . . ? ,.[,#-
. t- 5 f.
v' , . ' t*, %
" ,,@ % '${. / ,
4
/ t< .' i N),4, =%
C, .>.
C. .
a d(' #Q*)
.g. gq. _3g
.hh & f
'{ } gy,z.jg..y M > w s..
EMBEDOED FOODNGS SM 12 ' SM 12 DECEMBER 1972 1302 or the damping ratio and damping parameters C, and S. as proportional to dimensionless frequency. p' G' o,(Fig. 2).These assumpti< ns seem well justified for the embedment parameters l; S, and S, if a, > 0.1. as they do for C, for a half space; C, for a stratum I c, + 3, rl, pG
~'
is less linear but very small, thus adding little to the total damping. The conitancy D=
of C may be generally questioned (see Ref.12);nevertheless.it seems acceptable 2 6' G, I i
in the shown frequency range. Altogether. the simplifying assumptions are I
d , + G r,- - 3, consistent with the generally accepted practice in design analysis of surface
.in which mass ratio 6,= m/prj.
f otings (14 Then, the vertical amplitud2 is obtained from E Therefore. assume that C, = C, ar.d S, = 3, and that:
. . . . . .C0) interest.
C, = C, a, . S, = 3 a . . . . . . . . ..... ... .. .. The amplitude at natural frequency w, (slighdy
= constants whose values can be readily established arnp!itude) is simply:
in which Cu and S u '
P, I i
w,(w,) = k 2 D
-......: .....c c es .
_ c w .., .... cts.: or with frequency variable excitation (Eq.19):
so- i m,e I WO (Wo)" m 2D
( < sc y [
- as in any one degt
- e-of-freedcm system (Voigt me Values of G can be determined by tests. or f.
F 5*
! 8 17.
In Fig.11 exampi of response curves ccer
- / " " constant parameters art shown for embutment in i /
7 The differences decrease with increasing embedm I io . appear to yield an accuracy sufficient icer practica
! [
ee. COMPARISON WITH EXPERIMENTS In order to assess the practical applicability .
" solutius, the theoretical results have been compar e
' [ " '* laboratory experiments (51. and with a series of
$.,s $cs: ',E,'c, ,, University of Western Ontario by the second wri in the latter tests. two concrete blocksxwere 1.2.
Fig 11..-Comparison of Response Curves Computed with Variable and Constant x 27 in. x 48 in. (0.686 m x 0.686 m Farameters (l/r, = 0.5. b, = 8.1. p/p, = 0.75 and G,/G = 0.5) x 38 in. x 48 in. (0.483 m x 0.966 m x 1.2' e nerete block was cast directly in the foundatior from Fig. 2. Tables 1 and 2 or any other suitable formulae for /. Several ' nsisted f a mecharucal scillat r LAzAN at suitable values of C,3 and 3, . are given in Tables I and 2. e upled wi h a kopp Variator.
brained by care With constant stiff ness parimeters. C, and 3, substituted in Eq.12 in place .entsides embedments were depth. The rr th?. I ting to the appropriate of C, and S,. the frequency independent stiffness constant is obtained. and with it, the natural undamped frequency w at the test site are:(1) De -ity of undi' .:,xd soi o directly follows from Eq.18.
Substitut:on of Eqs. 20 into Eq.13 yields the frequency independent damping (2) Pogson's ratio = 0.30. O) shear modulus .
constant for embedded footings kg/cm-); (4) equivalent radius of footing = 1 of footing = !$.48; and (6) Lysmer's modified rr p' G' The experiments are described in more deta
............... . . .f21) e = r \/p G (C, +
2 o
3,P O 'l the response curves in rocking modes are also g
_,,-h- -
y 7 BEgh a ,
- T" 5 'N"We M , -gs ,
. / f
\
[C c ) .. v o
_--e .
. __ . _ , u_ w_ _ m, _ .,___m _. m . m__ m _ _ _ _ _
1303 ACEMBER 1972 SM 12 SM 12 EMBEDOED FOOT 1NGS arm! S, as proportional to dimensTo~nless frequency, or the damping ratio
., seem well justified for the embedment parameters fc' + 3* r.p G/'\
he> do for C, for a half space; C for a stratum P' hus aJJing little to the total damping. The constancy I \
.... ......... . M
.mcJ tsee Ref.12);ncs ertheless. it seems acceptable D= , g**
nge. Altogether, the simplifying assumptions are d ' + .g' , 3' y accepted practice in design analysis of surface Gr.,
in which mass ratio 6, = m/pr'.
= d, and S, = 3, and that: Then, the vertical amplitude is obtained from Eq.15 for any frequency of
........... . . . . . . . .(20) interest.
The amplitude at natural frequency w, (slightly smfler than the maximum enstants whose values can be readily established ,
amplitude)is simply:
P 1 ...(23) w,(w.) = ............... ...... . . .....
.w .e.,
or with frequency variable excitation (Eq.19):
m,e I w,(w.) = .. . ... .... .. ............ ..(24) as in any one degree-of freedom systam (Voigt model).
f
[ Values of G can be determined by tests. or found in literature, e g., Ref.
4 17.
In Fig. II, examples of response curves computed with both variable and
( ', -
constant parameters are shown for embutment m a stratum and in a half space.
The differences decrease with increasing embedment. The constant p rameters appear to yield an accuracy sufficient for practical purposes.
l' COMPARISON WITH EXPERIMENTS In order to assess the practical applicability of the approximate analytical solution, the theoretical results have been compared with earlier field tests (14).
u e, ,o ,, ,, ,, ,, ,,
o"ouss ram cv Iaboratory experiments (5), and with a series of field tests carried out at the University of Western Ontario by the second writer.
oase Curves Computed with variable and Constant In the latter tests, two concrete blocks were used with dimensions 27 in.
P/s,- 0.75 and G,/G = 0.5) x 27 in. x 48 in. (0.686 m x 0.6SG m x 1.22 m) (square base) and 19 in.
x 38 in. x 28 in. (0.483 m x 0.966 m x 1.22 m) (rectangular base). Each concrete block was cast direct!y in the foundation pit. The excitation equipment oc any other suitable formulae for f,, . Several consisted of a mechanical oscillator LAZAN and a 220 v. three-phase motor
, are given in Tables I and 2.
ameters. C, and 3, substituted in Eq.12 in place coupled with a Kopp Variator.
independent stiffness constant is obtained, and Different embedments were obtained by carefully removing the soil around frequency w, directly follows from Eq.18. the. footing sides to the appropriate depth. The major soil and footing properties at the test site are: (1) Density of undisturbed soil = 103.0 lb/ft'(1,650 kg/m');
Eq.13 yields the frequency independent damping 2 SS (2) Poisson's ratio = 0.38; (3) shear modulus = 6.6 lb/ft x 10' lb/ft8 (3:0
--.-- kg/cm ); (4) equivalent radius of footing = 1.26 ft (0.344 m); (5) mass ratio 2
8.f G, .
of footing = 15.48; and (6) Lysmer's modified mass factor = 2.40.
p Gj **** -. . . . . . . . .(21) The experiments are desenbed in more detail in Refs. 2 and 16. in which the response curves in rocking modes are alse given.
&&W%*M%W%h%WWpMaygy*pymn;r ww- -
C- n.w :
c . .
-[ .
"k
(; -
nd, '
. t~ ,.%.Q r . m #e h,[.@
2 G..
&,f.p
. nV .e
. . %Ae a
> . < f h.s 3p';tf, W.'.nk , ' a V
. ' '% . , *fbs , d )/ ;
- -s- < ,
. % ?J J, a
- *v.,f } *$ j* .
N t'a f,q~S"[ f-( 11.,f.1.%op l- aX Q s
'$ *** e '?* ?* *
, u-m,4y gj p f (a'4,%
y i-
-%I 1 s
1 D ., p, ar,s'$,jy',A: . . Mwe& p;;,yhm.yWiM4,1,g.twmn
, m.Y..,
l 1304
- DECEMBER 1972 SM 12 EMBEDDED FOODNGS SM 12 8 5,,, , ,. .. , The comparison of the theory with the experi e o . r. nonlineanty which makes the nondimensional reso cies dependent upon the intensity of excitation.
e I" ,m. ,,, ,,
can be analyzed using a procedure given in Re' p, + ca.a .... inevitable scatter in the comparison of experiment.
e
.e* -
t
- o ..e.. ee, j,.. .. .u.,s,..
, , , , ., . e e se i ,, _
.N-~ ~ ~ ' q. ,
g s
+
.. . N s s' ,, s
[** - g ,,,.... .
e.. -
s N
s s
's:s
- *
- s s's,
- ; s s a g..
e N
ss s ,
es - .
g N
- s sE N o .
. . j t I 3 es y
( a.
- ** 6. ,,
'* a a. .. J. l,
......r.,.,,,, ,
. ... .. , . e s -
F8g.12 -Comparison of Theoretical and Erna ; -- -' es nani Amplitude Rano p;g.14.-Comparison of Theoretical Resonant Ampi Vanations mth Embedment in Undisturbed So,l (The retjcal Curve Approuirna*ely ophes for all Mass Ratics) of Footings Embedded in Back Fill Having Density R i
to -
, si f so arct.o.os r) so arcr .o.oo. a ) , a o o i.ri
- D e ri 9,, , e e sao
,y
- 2 ss , , S e, e e 3 ,o 4 y a no. , o ,yo,
- is
- e s e, . >a .
A h otros ,, y 9 3 .*
a ,3 o e,2seru Lysuge ET ALosts) 9 $'
l' y*
i g id .
4 / .
5 I1 '
e y
.o o
> se
!a lo
^*
- *~
o'S ~
~
o 'to A 0 5 twetDwffit parso a g
o I 12 ge Tig. *.5--Comparison of Theoretical Resonant Fre Footing Embecced in Back Fill Having Density Rat'<
E' A second difficulty is due to the fact that the
'o o of the surface footing are consistently found t
- 'O 8; zo ,3 twetemt envio .'"* Ereater than the theoretical values. {This was
- # ' "* * *
- Y "*
Fig 13.-<omparison of Th M er W aces W h e , W W M N
- Vanacons with Embednent in "8nt eamn Ratio bue, and partly by the shape of the fooung ba5 sturb d SOU However, as far as the effect of embedmer r-g y- ga
- NU EQ " W WIW
-8't g
f) *k l
- NN
__ __ _ m. _ _
_.m, _ m .m c _ m m - - u- - .. m .m. J - - - - -
DUEMBER 192 SM 12
_ SM 12 EMBEDDED FOQ11NGS 1305
.u, The comparison of the theory with the experiments is complicated due to O o. a * --> nonlinearity which rnakes the nondimensional resonant amplitudes and frequen.
. au cies dependent upon the intensity of excitation. (Nonlinear response curves l l can be analyzed using a procedure given in Ref.15.) Nonlinearity causes an
- * * *,* ' ' , ' , ' , ' inevitable scatter in the comparison of experimental results with a linear theory.
.....}...o.....
. .. . s . a ni i . ..<>
,, __ ,,,,' ....u,...
. a e r.
\ N e, 's
- 's s ss
. a
. N N * . .u i ,.r. .
l r.,*..
- e.. .
\s s's s
- 2 N
's s . s'~s .
[.,
e s , N'0'is;,.
. (. . . s's .
c...
~~s .
~z' :... , , , ,, ', _g -
a j . .
- a ..
r
' ' ' 2 t ._
,~~~_,,
,. as as "t . ..
.. .. a.
as
. .. ., o s . e n.
. u .., . c - e u.
.orcut and Expe imental Resonant Amphtude Ratio , Fig. It-Comparison of Theoretical Resonant Amplitude Reduction with Field Tests
'isturbed Soil (Theoretical Curve Approximately of Footings Embedded in Back Fill Having Density Ratio y = p,/p = 0.765 I so nect .g ri...n) a o o tri a g,, , e a sao
+- . e 507 g , a A seown (1370) ,a
., W3 .
5 .
,,,
- g ta -
o
- <#g e
<>q'
'5 r8 '
aa
. 5 . m
> en Eo ot 5
, o s@ to ts ao 2.s ewecosse=r nano e .gfr.
E* Fig.15.-Comparison of Theoretical Resonant Frequency Ratio with Field Tests of
, Footing Embedded in Back Fill Having Density Ratio a = p,/p = 0.765 A second difficulty is due to the fact that the measured resonant amplitudes
,o ts ao as of the surface footing are consistently found to be two times to three times ewecome,ir mare e .'if,, greater than the theoretical values. [This was also found in previous experiments
, (14).] This difference may be caused by the reflection of elastic waves from o'*ticat and Experime,tal Resonant Frequency Ratio layer interfaces and fissures, variations in the stress distnbution in the footing i Uade.rted Sail base, and partly by the shape of the footing base.
. However, as far as the effect of embedment is concerned, the differences I( ,P c mmmswww1:A5eN'VwW.NIw-vwve ..9 ,
, _.9 e, -eM .T. 4I- .
. .a 9 [.%,b f,[,
s -
f 5.s x .pe g% %. f:.d . ,
(;l} ()
,ra L34 y e 4 %w "',y; IA. , V : '
j' (~},m
- o. ,
~~ a- .
/,G % gy.;
j)F.n. % ',-wy [ ss[ t..
.6
.,y
." y. n.s' a rc, e-A W **\ %,
y Qn Q s A ..
- f h e '
, .l. 5.; t'. " ;
wnA: A m ,, w w = g m w w m ~
l 0 06 DECEMBER 1972 SM 12 SM 12 EMBEDDED FOOTlh in absolute values of amplitudes can become less substantial if the relative 2. The real effect of embedment depends on e variations are compared. These relative variations in resonant amplitudes and difference between undisturbed soil and baci frequencies are plotted in Figs.12 to 15, irnportant in the case of a stratum, in Fig.12, the relative variations in resonant (maximum) amplitudes are shown 3. The theory and experiments agree qualiu ve' us relative err.bedment depth for undisturbed soil. Experirrents with both in resonant amplitudes and an increase in resor square and rectangular bases are presented. Lysmer and Kuhlemeyer's results embeament depth and increasing density of the of their finite element solution are plotted also. They indicate a good agreement 4. The experimental resonant amplitudes at with the approximate analytical solution. The field experiments are also in good theoretical values computed with a rigid base agreement with the theory except for the highest embedment at which most of two to three in the described case.
experiments exhibit greater amplitude reduction. Also, the strong effect of 5. There is a reasonabic agreement between t nonlinearity is visible. Usually greater strength of the upper cohesive soil layer relative variations of the resonant amplitudes a may contnbute to a greater amplitude reduction at full embedment. Chae's with the embedment in the case of the undistu laboratory experiments feature a much stronger amplitude reduction than both increase in resonant frequencies is much smalle theory and fic!d tests, probably due to the conditions of laboratory tests. of resonant amplitudes are in better agreement.
In Fig.13 resonant frequency ratio is plotted against embedment. In most 6. The resonant (maximum) amplitude reduc experiments the frequency increase was considerably smaller than the theory of the mass, while the corresponding frequency predicts, particularly with the rectangular base. The reason for this may have been that the bond between the soil and the footing was not perfect, as the ACKNOWLEDGMENT theory assumes.
Figs.14 and 15 present an analogous companson for a footing surrounded This study was supported by a grant.in. aid by a backfill. According to Fig.14, the theory tends to overestimate the effect Research Council of Canada to the senior v of embedment, especially for small embedments. This seems to indicate that Commonwealth Scholarship by the Canadian Fe there is a lack of bond between the footing and the backfill. Perfect force writer.
transmission is assumed in the theory, which can hardly be satisfied with shallow embedments where the horizontal soil pressure and the frictional bond are very APPENDIX l.-STIFFNESS AND DAMPANG Pt low.These conclusions are further substantiated by Fig.15. In this figure previous experiments (14) agree with the theory very well. The backfill (loess loam)
Stiffness and damping parameters presented i was compacted very heavily and possibly a better bond between the fill and from Eqs. 3,6, and 7 by curve fitting to f acilitate the footing was achieved. Because the bond depends also on the roughness of the footing sides, this seems to be a further factor affecting the quality is sufficient. Functions f were taken from of the force transmission between footing and the backfill. reactions and from WarbYrton (Ref. 20) for si reproduction. (The sign convention of these w Froic. Figs.14 and 15, the conclusion can be drawn that for backfill the this paper.) Some of the parameters are also I effects of embedment upon both the resonant frequencies and amplitudes can be considerably reduced. This reduction depends on the soil used, i.e., on its stratum reactions can be found in Refs.10 ar can be adjusted according to the frequency rant compaction and roughness of the footing sides. Therefore, with footings ast into forms, an even smaller effect of embedment can be expected than that observed in the experiments described. The reduced effect of the backfill can APP M R 4 N m be accounted for in the theory by considering p, < p and G, < G. , ,
tion." (in Russian) Vorrosy Dynaraiki i Prochnc
SUMMARY
AND CONCLUSIONS of Riga.1967, pp.195-209,
- 2. Beredugo. Y. O., " Vibrations of Ernbedded Si r The effect of embedment upon vertical forced vibration of a rigid footing $'h'e js '$'b"d gre re rt ra was investigated both theoretically and experimentally in the field. The conclu- 3. Beredugo. Y. O. and Novak. M.. " Coupled H+
sions can be summarized as fo!!ows: Embedded Foctings." Canadian Grotechocallo.
4 Byerof t, G. N., " Forced %brations of a Regid Cir
- 1. The approximate analytical solution compares favorably with the finite ha[) 9Space na Ea ic S ratum -
element solution and can easily be applied to the analysis of footings and structares 5 _
- 5. Chae. Y. S., "Dynsmic Behavior of Embedded b supported by embedded foundations. Research Record. No. 32),1971,pp.49-50
- ' v %m s- --. - - - - , - - -.-** . m ' gr ei. n N . " .TM WN C TWV*'N W"-
."'"C.M '
pr O 3va g*- '
+
[J 9 //, G GCh LJ-
[,a, a .
- - .- - - w ~ y mn - ~ w w w~ ~ ~ ~ - e w --n < -
DECEYSER 1972 SM 12 SM 12 EMBEDDED FOOTINGS 1307 mplitudes can become less sut stantial if the relative 2. The real effect of embedment depends on embedment depth, with a marked nese relative variations in resmant amplitudes and difference between undisturbed soil and backfill, and becomes particularly g a p.12 to 15. l important in the case of a stratum.
.apanons in resonant (maximum) amplitudes are shown 3. The theory and experiments agree qualitatively that there is a decrease at depth for undisturbed soil. Experiments with both in resonant amplitudes and an increase in resonant frequencies with increasing nes are presented. Lysmer and Kuh!emeyer's results embedment depth and increasing density of the backfill.
twtaan are plotted also. They indicate a good agreement % 4. The experimental resonant amplitudes are consistently higher than the
,lpal solution. The field experiments are also in good theoretical values computed with a rigid base stress distribution by a factor
<3 encept for the highest embedment at which most of two to three in the riescribed case.
.rer amplitude reduction. Also, the strong effect of 5. There is a reasonable agreement between the theoretical and experimental
.uany greater strength of the upper cohesive soil layer relative variations of the resonant amplitudes and of the resonar.t frequencies ater amplitude reduction at full embedment. Chae's with the embedment in the case of the undisturbed soil. With the backfill the cature a much stronger amplitude reduc' ion than both increase in resonant frequencies is much smaller than predicted. The variations
.+ ably due to the conditions of laboratory tests. , of resonant amplitudes are in better agreement.
equency ratio is plotted against embedment. In most 6. The resonant (maximum) amplitude reduction is essentially independent cy incrsase was considerably smaller than the theory of the mass, while the corresponding frequency ratio is mass dependent.
b the rectangular base. The reason for this may have een the sod and the footing was not perfect, as the ACKNOWLEDGMENT ti an analogous comparison for a footing surrounded This study was supported by a grant.in aid of research from the National to Fig.14 the t'cory tends to overestimate the effect Research Council of Canada to the senior writer and by the award of a y for small embedments. This seems to indicate tha' Commonwealth Scholarship by the Canadian Federal Government to the junior bet.cen the footing and the backfill. Perfect force writer.
1 the theory, which can hardly be satisfied with shallow xuontal soil pressure and the frictional bond are very e f der substantiated by Fig.15. In this figure previous APPENDIX l.-STlFFNESS AND DAMPING PARAMETERS
.i e theory very well. The backfiU (loess loam) i and possibly a better bon..' between the fill and Stiffness and damping parameters presented in Tables I and 2 were obtained
- 1. Because the bond depends also on the roughness from Eqs. 3,6, and 7 by curve fitting to facilitate the calculations. Their accuracy s seems to be a further factor affecting the quality is sufficient. Functions fu were taken ' om Bycroft (Ref. 4) for half-space betseen footing and the backfill. reactions and from Warburton (Ref. 20) for stratum reactions by mechanical l reproduction. (The sign convention of these writers have been maintained in
, the conclusion can be drawn that for backfill the on both the resonant frequencies and amplitudes can this paper.) Some of the parameters are also plotted in Fig. 2. More data on This reduction depends on the soil used, i.e., on its stratum reactions can be found in Refs.10 and !!. The constant parameters
. of the footing sides. Therefore, with footings cast can be adjusted according to the frequency range of interest.
Ser effect of embedment can be expected than that .
ets described. The reduced effect of the backfiU can i APPENDIX l!.-REFERENCES
,cory by considering p, < p and G, < G.
I. Baranov, V. A., "On the Calculation of Excited Vibrations of an Embedded Founda-tion " (in Russian) Voprosy Dynamiki i Prochnocti, No.14. Polytechnical Insuture US40NS of Riga,1967, pp.195-209.
- 2. Beredugo. Y. O., " Vibrations of Dabedded Symmetric Footings," thesis pre:ented t he Unhusity oNestun Omo. at London. Canada,in tm. in parnal fu!Mment ett upon vertical forced vibration of a ri II'd footinE xetscally and experimentally in the field. The conclu- of the requirements for the degm of Doctor of Philosophy.
u foUows: 3. Beredugo. Y. O. and Novu. M., " Coupled Horizontal and Rocking Vibratien of
/ Embedded Fooungs," Canaian GeorechnicalJournal. Nov.,1972.
- 4. Bycroft, G. N.," Forced Vibrations of a Rigid Circular Plate on a Semi.infimte E!asne alytical solution compares favorably with the finite lalf. Space and n an Elasuc Stratum," Phdosophical Transactions of theRoyat Sociery.
isd) be applici to the analy sis of footings and structures London. Senes A Vol. 248. No. 948.1956, pp. 327-268.
Amdations. 5. Chae, Y. S., " Dynamic Behavior of Embedded Foundation-Soil Systems," Highay Researth Record, No. 323,1971, pp. 49-59.
m w j' -' w -
. t ,.f 3 %,p D .
( " / W,,4 M f . :
Y. ye*aA.p.M,. rv.yv;%f- .
nG n - -
)
p%c .
t ( J (J
- 4*t w 9 /
/
- 1 &;5#
- j,
'a,
'u'd.*t
- t ,5y.' .
- Q ?, )
- - . .a -Q }
I L.,o%,%L. w.Ne 1
a
- .J W.[hy '. .
,. 1:
) :s. $s % h/
-> Nik[NkD d [ h m m . m p m #..mm l
1308 DECEMBER 1972 SM 12 SM 12 EMBEDCED FOOT 1N
- 6. Elorduy, J., Nieto, J. A., and Szekely, E. M., "D>namic Response of Bases of b, = m/pr', = dimensionless mass ratio-Arbitrary Shape Subjected to Periodic Vertical Loadmg," Propteries of Earth Mactriafs' C, = half space (stratum) stiffnes. -
Albuquerque, N.M., Aug.,1967. , .
- i a frequency independent half
- 7. Fry, Z. B., "Desclopment and Evaltation of Soil Bearing Capacity, Foundation of meter; Structures," WES Tech. Report No. 3-632 Report 1,1963. !
- 8. Gupta. B. N., "Effect of Foundation Ernbedment on the Dynamic Behaviour of C = balf space (stratum) damping the Foundation-Sod System " Geortchnique 22. London, England, No. I, Mar.,1972, . C = constant related to C2*-
p p.129- 137. e = damping constant for embed
[9. Kaldjian, M. J., discussion of " Design Procedures for Dynamically Loaded Founda- D = damping ratio; tions," by R. V. Whitman and F. E. Richard, Jr., Journal of Soil Mach ' s and ;
Foundations Division. ASCE, Vol. 95, No. SMI, Proc. Paper 6324. Jan. 196 pp. j e = eccentricity of unbalanced rc 364-366. f,, f: = components of Reissner's di
- 10. Kobori, T., and Suzuki, T., " Foundation V brations i on a Visco-clastic Multi-lay ered vibration-Medium." Proceedings. Third Japan Earthquake Engineering Symposium-1970, l G = shear modulus of soil beneat Tek)o. Japan. Nov.,1970, pp. 493-499. O s= shear modulus of backfill (si II. Kobori T., Minai, R., and Suzuki, T., "ne Dynamical Ground Compliance of a g, = weight of unbalanced totatin Rectangu!ar Foundation on a Viscoelastic Stratum," Buurris of the Disaster Prevention Research Institure, Kyoto Univ., Vol. 20, Mar.,1971, pp. 289-329. H = total thickness of stratum;
- 12. Luco, Y. E., and Westmann, R. A., "D> nam.c Response of Circular Footir:gs," h = thickness of stratum under fi Journal of the Engintenng Mtchames Dinsion, ASCE, Vol. 97, No. EMS, Proc. i = N = imaginary uriit- '
Paper 8416. Oct.,1971, pp.1381-1395. k = stiffness (spring) constant fo
\A3. L> smer, J., and Kuhlemey er, R. L., " Finite Dynamic Model for Irifinite Media,"
Journal of the Engintenng Mtchanics Daision. ASCE, Vol. 95, No. EM4, Proc. k(6) = stiffness with embedment 6; Paper 6719 Aug.1969. pp. 859-877; Closure Vol. 97, No. EMI, Proc. Paper 7862, I = depth of embedment of footi
(. Feb.,1971, pp.129-131.
- 14. Novak, M., " Predict 2on of Footing Vibrations " Journal of the Soil Mechanics and m = man of f oting; m, = unbalanced rotating mass; Foundations Dimion ASCE, Vol. 96, No. SM2, Proc. Paper 7:68, May,1970, pp.
836-861.
N,(t) = vertical dynamic reaction alo
- 15. Novak, M., " Data Reduction from Nonlinear Response Curves," Journal of the P(t) = excitation force; Er:eintenn: Mechanics Daision ASCE, Vol. 97, No. EMI, Proc. Paper 8300, August. P, = excitation force amplitude;
.1971, pp. I187-1204.
R, = resonant (maximum) amplitt (6', Novak M., and Beredugo. Y. O., "Effect of Embedment on Footing Vibration " uith embedment b/tesonant Procerdungs of Ist Canadsan Conference on Earthquake Enguncenng, Uruv. of Bnush Columbia. Vancouver, Canada. May,1971, pp.111-125. R f= resonant frequency ratio = r l7. Richart. F. E., Hall, J. R., and Woods, R. D., Vibrations of Soils and Foundation 2 ment 8/ resonant frequency c Prentice-Ha!!, Inc., Eng!em ood Chff s, N. J.,1970. R,(t) = vertical dynamic reaction at -
- 18. Sung. T. Y., " Vibrations in Semi. Infinite Sohds due to Periodic Surface Loading.., 'o = tadius of cylindrical footing; Amencan Society for Testing and Mattnals Special Technical Publication No.136, Syrnposium on Dynamic Testmg of Soils.1953, pp. 35-64.
. or rectangular footing (see R
- 19. Thomson,W. T..and Kobori.T. "DynamicalCompliance of Rectangular Foundations S,,S: = frequency dependent stiffnet on an Elastic Half Space." Journal of Appbed Mechanics Transactions. Americzn to embedment**
Society of Mechanical Engineers, Dec.,1963, pp. 579-584. .
3 i= frequency independent (cons
- 20. Warburton, G. B., " Forced Vibration of a Body on an Elastic Stratum " Joumal embedment:
of Applied Mechanics, Mar.,1957, pp. 55-58.
3: = constant related to S3 ;
s(1) = vertical side dynamic reactio APPENDIX lit.-NOTATION t = time;
- (l) = vertical displacement of rigid The following symvols are used in this paper:
w = complex amplitude of vertica
- = = real resonant (maximum) am; A = w, m/(m,e) = nondimensional displacement amplitude; w = real amplitude of vibration; A. = nondimensional resonant (maximum) amplitude of displace, ; = vertical coordinate;
' ment of embedded footing; a, = nondimensional frequency at resonant (maximum) amplitude *1
- N/ fo = relative thickness of stratum.
of embedded footing; 8 = t/r, = embedment ratio;
'l
- Ps/p = density ratio;
[ a, = r,wV p / G = nondimensional frequency; v = Poisson's ratio;
( B, = 1 - v/4 b, = Lysmer's modified mass ratio;
~
,.,_.n,._.
A .n-,,_. _ , _ , , _ . , _
-,em , -,
A ,
f.n.J ,
,l
i
- a e--. m - - - g-.. -- = u..m a - . _ __ _ _ m m m .%jg _. _ _ _
., . m_
SM 12 SM 12 EMBEDDED FOOTINGS 1309 D(CgMBER 1972 anJ Stetefy. E. M., " Dynamic Response of Bases of b, = rn/pr', = dimensionless mass ratio;
, per.ac Verucalloading," Properties of Earth Morenais, C, = hal. 'space (stratum) stiffness parameter; O s= frequency independent half-space (stratum) stiffness para-Euluation of Soit Bearing Capacity, Foundation of meter;
,,n No. 3432, Report f,1963. f C, = half-space (stratum) damping parameter;
, roundation Embedment on the Dynamic Behaviour of
,,- c,orechnique 22, London, England, No.1, Mar.,1972, 2 = constant related to C2 ;
c = damping constant for embedded fraoting;
, of "Ocsign Procedures for Dynamically Loaded Founda- D = damping ratin; s anJ F. E. Richard, Jr., Journal of Soil Mechanics and e = cccentricity of tmbalanced rotating masses; CE, Vol. 95, No. SMI, Proc. Paper 6324. Jan.,1969, pp. fi.fi = components of Reissner's displacement function for vertical l
" Foundation Vibrations on a Visco-clastic Multi-layered v bration; Th J Japan Earthquake Enginecrms Symposium-1970. G = shear modulus of soil beneath footing; 0,- shear modulus of backfill (side layer);
bu ub T'. "The Dynamical Ground Compliance of a u Vncoetastic Stratum," Bullerin of the Disaster Prevention ;
g, = weight of unbalanced rotating mass m,- ,
H = total thickness of stratum; Unn., Vol. 20, Mar.,1971, pp. 289-329.
una. R. A., " Dynamic Response of Circular Footings " h = thickness of stratum under footing base-is Mechanics Di6ision ASCE, Vol. 97, No. EM5, Proc.
e
( = N = imaginary unit-*
> 1381-1395. l & = sttifness (spn.ng) constant for embedded footing-cyer, R. L., " Finite Dynamic Model for Infinite Media," k(8) = stiffness with embedment 6;
.g Mechanics Di ision ASCE, Vol. 95, No. EM4, Proc.
e g59-g77; Closure. Vol. 97, No. EMI, Proc. Paper 7862, I = depth of embedment of footing; l m = mass of footing
- of Footing VsbraGons," Journal of the Soif Mechanics and m, = unbalanced rotating mass; SCE. Vol. %, No. IM3, Proc. Paper 7:68, May,1970, pp. N.(t) = vertical dynamic reaction along embedd;d footing sides-*
/ l P(t) = excitation force; A from Nordinear Response Curves " Journal of the msaan ASCE, Vol. 97, No. E',il, Proc. Paper 8300, August, p, = excitation force amplitude; r R, = resonant (maximum) amplitude ratio = resonant amplitude go. Y. O., "Effect of Embedment on Footing Vibration " with embedment 8/ resonant amplitude of surface footing; t,a= Conference on Earrhquake Engineenng, Uruv. of Bntish g Rfa resonant frequency ratio = resonant frequency with embed.
anada, May,1971, pp. !!!.123.
L., and Woods R. D., Vibrations of Soils and Foundations, ment 8/ resonant frequency of surface footing-
' :(t) = vertical dynamic reaction at base of footing; in Sem I fi te S'ol s due to Penodic Surface Loading. fe = radius of cylindrical footing; alsa equivalent radius of square ratiar and Marenals Special Technical Pabfacation No.156, or rectangular footing (see Ref.17, p. 347);
Tesons of Soils,1953, pp. 35-64.
. bon.T.."DynamicalCompliance of Rectangular Foundations . S,.S = frequency dependent stiffness and damping parameters due
- e " Journal of Applied Mechanics, Transactions, Amenczn to embedment' rsg:neers, Dec.1963, pp. 579 584. ji = frequency m. dependent (constant) stiffness parameter due to eced Vibration of a Body on an Elastic Stratam," Journal embedment; Jar. 1957, pp. 35 58.
$s = constant related to Ss i
- (t) = vertical side dynamic reaction per unit depth of embedment; ION i = tune; w(t) = vertical displacement of rigid footing at time t; are med is this paper.
- w = complex amplitude of vertical displacement;
- = = real resonant (maximum) amplitude; irnensional displacement amplitude; *e = real amplitude of vibration; imensional resonant (maximum) amplitude of displace- g = vertical coordinate; t of cribedded footing; A = H/r, = relative thickness of stratum;
!imensional frequency at resonant (maximum) amplitude I " I/fe = cmbedment ratio; abedded footing;
'l " p,/p = density ratio; firnensional frequency; y = Poisson's ratio; mer's modified mass ratio; C
^ - - - M ---- ; w s - - r
,.24 M9">W'tY.NM V*t ep p_
b M W i" P N W *c.M M W 8 M "' C
'l R' c.Wg%w%c a4 .. nru i a y e,U & <. ,
b.y s. Qn.s.s . .
V{m%%yl,*f' '
~ '
.>- Q
'Q.g'f q*,I#
.7f '
g .: : n.]NF V ~ ,
~ %,. .
,g'I.L%.k.l.[,t.
/
- s . p *:i { f'A E ' +s.
+s *f. ,%q
- s
s
( -
$ ,(v,*?d N)N'.
%. s l24 <- s, ,;,,..~, ;p.Lii:Onff&t% K-in .
" * ^ ^ YYI & a;,s, s . A p1 m W g
~
s DECEMBEA 1972 SM 12 l $41 1310 DECEMBER 1972 p = mass density of clastic half space (soil);
JOUR:
p, = mass density of side layer (backfill);
+ = phasc anglet I
SOIL M EF w = circular frequency of excitation; FOUNDATIOI w , = resonant circular frequency (at maximum amplitudes); and w, = natural circular frequency.
Bearing Capacity Theory from Expe By A. Verghese Chummar v
l INTP.600CT10N The plastic equilibrium theories on bearing capaci on failure surfaces either assumed or derived fra The failure surfaces observed in the experim:nts <
( soils by Selig and McKee (10), Verghese (12) and the shape of the failure surfaces adopted in thec given in Fig. I between some of the observed ar rescals a marked discrepancy. The existing theorie bearing capacity fairly accurately until it was esta long footings, the plane strain conditions prevail an 6 under plane strain is about 1.1 times the value .
Fig. 2 gives the experimental values of the bearin againstplane. strain values of 6 and the N, values gis theories. The comparison clearly indicates that I accurately tne bearing capacity of long footings. /
herein to determine the actual shape of the failu bearing capacity theory based on that surface. I data, the present study is restricted to the case cohesionless soils.
FAILURE SURFACE The shape of the f ailure surf ace is determined fror by different research workerrby deducing a patt the observed ones.
Note --Discussion open until May 1.1973. To 88" a wrrten request must be filed with the Editor of Tes rarer is part of the copyrighted Joumal of the Soit Me:
Proceedings of the American Society of Civil Errneer 1972. Manusenpt was submitted for review for possMe
' Lect. Dept. of Civ. Engrg., Indian Inst. cf Tech-, >
1311
-p C -
*h a- - -m - , wy,-..
- e., y m -war e,y w y - i, m my & m7 j , , t
" "">T n e== i.r.--
jf f)
J 'r / :- -
. . l r
d f LAR.
- wr. K 2/6 i
DYNA 511C ANALYSIS OF ESIDEDD2D STRUCTURES d }'
E. KAUSEL sonoa wesua urnw.nu Corporanen.
P.O. Das 123. Swen. brauashomas Ull(II U.S.A.
- 1 C
R.V. WillTMAN 3 Drportment of Cnd &parrrent. Alasmhumrs lasterwr of Torknotocy. b Cambemigr. Atauachusetts 02tJ9. U.S.A. t<
F. ELSABEE soonea wesuer Osmmas Corpo anon. Cherry tra. Nr. Jersey, v.s.A. f J.P. MORRAY .', ; i tas Netrar Inc.. no Afonso ery surer. Son Franusco. Catfornoa ulot. U.S.A. ,; .]
11 SUSotARY d<:
The paper presents simplined rules to account for embedment and soil layering in the -;/
soil structure interaction problem, to be used in dynamic analyses. The relationship be-tween the spring method, and a direct solution (in which both soil and structure a e j{
modeled with finde elements and imear members) is first presented. It is shown that for i consistency of the results with the two solution m.thWs the spring method shculd 5: per- 1l formed in the following three steps:
- 1. Determination of the motion of the massless foundation (having the same shape as the actual one) when subjected to the same input motion as the direct solutien. Fct )5 an embedded foundation it will yield, in general, both translations and rotations. jI '
- 2. Determination of the frequency degndent subgrade stitTness for the relevant degre-s j of freedom. This step yields the so-called " soil spnngs" [ft
- 3. Computations of the response of the real structure supported on frequency dependent i I
soil springs and subjected at the base of these spnngs to the motion computed in step
- 1. 1 The first two steps require,in general, finite e!: ment methods. which would make the .,
procedure not attrxtive. It is shown in the paper, however, that excel:ent approximations ]
can be obtained. on the basis of I dimensional wave prop 1 cation theory for the solution i of step 1. and correction factors modifying for embedment the corresponding synngs of j a surface footing on a layered stratum, for the solu. ion of step 2. Use of these rules not j only provides rem.irkable agr,ement with the results obt. tined from a full Gnite element j analysis. but results m substantial sav ngs of computt.' cxecution and storage require- j 1
ments. This frecs the engineer to perform extensne stud.es, varying the input properties oser a wide range to account for uncertaintics, in particular with resNet to the soil pro- ,
perties.
i ti d
}
A.ittd t'i77 y h)
Mf d
b *a r c/ n cJ' rJ k ,-
- m. , - , _ .- , m--m, , mm,., _,.m. --_.,,m ..- ,
o
. . i i
_~m - -
a-. -m. _ . . - _%s A uah , m. mu 3_ w ag__%_ ,
2-
, K 2/6 J
d 1, .etroduction retarded ena j $1geMicant efforts have been directed in recew years to forwulate entineering solations (a frequence
' to be problew of vibration of foundations and selenic response of buildinee. The prohlem is For the
, of t
.3cial interest la the solemic analyets and design of massive buildings such ao nuclear l j becomes lentt
' eestainment structores.
rional, roch leilding foundations, and auclear reactore te particular, are usually buried to some dashots." I esteet beneath the surface of the ground. This embedment has in ment casse considerable 11eearly den l
effect on the dynamic response of the etructure, both to terms of relative freeuency contente '
the founJath and amplitude of the resultims motions.
Because of the complicated boundary condations that PIII '"'
must be satisfied in a theoretical formulation, rigorous analytical solutions for embedded rigid body.
foundatiene are sonazistent et present.
Mence, ameerical (flatte element), experimental and * " '
- epproatmate analytical techniques are currently being used to provide a solution to the **
problem st hand for these complicated geometries. '
An svarenees of the effects associated [
with embedeent, coupled with the availability of asserical solutione and the lac's of rigorous * " * " " "'
solutloos, has been the basis la recent times for discrediting the spring method as a tool for Prpided -
{
analyses, particularly in the euclear power Ladustry. The detraction of the spring method has break the soli been argued by some researchere on the baste of comparisons between the classical half space I' g
method, and more involved finite element solutions. Many of these comeartoone are not nean-ingful, etace they were based on inappropriate values for both the "sprira constant" and the ' '%
- eo port motion." In fat., the spring method and finite element solutione can be shown to be
- 2. detes i
mathematically equitslant; if they are classified as different, it la because of inconsisten-eies la their implementation. desr e, j
it to the purpose of this paper to show the relationship between a more tenerni apring eoil i
' sethod, and the solutions provided by direct fiaf te element procedures, and to present practical rules for use in dynamic analysis. tice th.
i ef the structa
- 2. %e beste *=eerpositten theorre should be ide, taferring to Pit.
1, assume that the generet equatione of motion of a structure- "" ***"
foundation system are given by the matriz equation MO + Ci + KT = 0
- 1) d**PI"8 *I 'h" where M.C K. are the system masa, damning and ettffness metrices; U and Y ere the abcolute . ****"***"
and relative displacementa vish respect to one general ground reference evetm.
De solution "I*"
of this equation to equivalent to the solution of the two matriz equations **"'"'
Mg B+Ct g g+KYg=0
- 2) *** "*C I*' '
MT2 + CT.2 + IT2 **"2 U 1 The first 3) vbese Og = Tg +g U , M g + T2, and Mg+M' 2 M 1 excludes the asse of the struerure, while that the 3-etet M2 fataraction tee
- EI"8** 'h* *a se of the soil. U, 4 some genen11 red ground mottoa vector.
De equive -
the basis of r lance of 2) with 1) is demonstrated by simple addition. la E4. 2, the respense of the asse- tion facto n F g
1eac structure is found first, and will be referred te as the htwaatie interaction. The *I"' "'"
g resulte of this step are the used to Eq. 3) which shall define the inertial internetim, and streement with l whteb is solved by application of fictitious inertia forces applied to the structure stone . 'N88'"'I'
- la the solution of the second step, it is irrelevent whether the oo11 is modt11mi with C* P I* '" * " '
1 flatte e'lemente. er eeutralently, with a (far-coupled) astrie of ettffnese factione modellint"M
- the embgrade, and defined at the sell-etructure interf ace.
These stiffnese functione can be i
, q n r o
- m .
e 3
7 m -
- ~
w- y ug 2. pus c- _- - %
--n _. -
_ m n ... J.m . ~ - mw - w----- mm enea.m.
~ ~
K 2/6 K 2/6 i
renarded as resultian from a dinante condensattom of all the detrees of freedom in the soil '
seineerina solutione (a frequeocy dor ain eslution is implied).
see. The problem is For the particular situatir,a where the combination foundation-strweture is very rigid, it becomes tenitimate to replace the matris of stif fness functions by the overall vertical, tor- j a such es nuclear '
I sional, rocking and swaying stif fnese functions,1.e., by frequency dependent " springs" and dashpets." For this case, the modal displacements at the foundation-soil interf ace are buried to some e considershle linearly dependent and, at most, ein detrees of freedom are neaded to describe the action of gj tre ency contents the foundation. It also follows that the solution of the kinematic interaction phase le com- ,
(
ry conditions that pletely defined by the rotations and translations of the massless structure, which moves as a one for embedded rigid body. Rence, one can replace the massless structure in E4. 2) by a rinid massless ,j foundatka, Wectd to th sama e and exettathn as the orttimal snum. ,;
, experimental and u.o. a . ore careful ema.u of r. 3 will sho. that the .olution T saa be regarded ,,
1,u ,, ,, ,,,
as a vector of disptacements relative to a fictitious support, while the rinid body transla- i ects unociated u o..
uons and rotau e of e n massies. fou e a u . o r,. : ar. ~ equivu ont supp,rt he m o, ,,,,,o ,
Provided that the assumption of rigid foundation is pertinent, it is, therefore, valid to '
method as a tool for he spring method has break the solution into three stepet (also sus Fin. 2) l assical half space 1. detemination of the motion of the messless rigid foundation, when subjected to the same input motion as the total solution. 2his is the solution of Eq. 2). For an ,
. sons are not mean-t constant" sad the embedded foundation, it will yield, in general, both translations and rotatione. ;
2.
s can be sh to be detemination of the frequency dependent subgrade stiffnesses for the relevant I 4J ause of Laconsisten-tas ne m. e step yields t secalled soil "sprinne." '
- 3. computauan of the response of the real structure supported on frequency dependent ,
,,e ,eneral .,r,n, sa u spm... a.d .. ,en.d .t ~ 1..e o, ~.. spun.. to ~ _uon co.-t ed , e,. :2
** 1** "h'* th' *"If P*i"i'" 1"'*1'*d I" this a pproach concerns the d ef ormabilit y 2d to present [f of the structural foundation. If this foundation were rigid, the solution of this procedure should be identical to that of the direct (or one-pase) approach (assumant, of course, {j
{;
.a .t.at d.f an,u.s .f ~ .ou.. aa ~ .am. . a ,r.ced.r.s).
a . , ,.c . .,-
The superposition principle is valid only for a linear system. While the modulus and g g) dsmains of the soil are strain-dependent, studies (6) have shown that most of the nonlinearity are the absolute Mcure se a result of the earthquake motion, and not as a result of soil-structure interaction. g ystem. The solution e, the soil properties consistent with the levele of strain in the free field (i.e., before 'j the structure has beat built) may be used in steps 1 and 2 without further vuodification to act unt for the additional atrates imposed by the structare. g 2)
The first two steps require, in general, flatte element methods, and thus it might appear *a ,
- 3) s that the 3-step method has no advantage as compared to considering both kinematic and inertial ;
.e structure, Atle tatsrection together in a single step. However, reasonable arproximations can be obtained on 1 1
cter. The equive - the basis of one-dimensional wave propagation theory for the solution of step 1, and corree- 1 ,
.s ponse of the asse- tion f actors modifying for embedmant the correspondtag springs of a surf ace footing on a [
nterant% ~
187ered stratum for the solution of step 2. Use of these rulee not on!, provides remarkable * ,;
tal lateracrirm, and eueement with the results obtained from a full finite element analysis. but results in -i is structure alaa. .
- betantial savin e of ecuouter execution and storate reewirements. This frees the eng*neer 11.to undelled with 88 perform entenetw etudies . varying the input properties ever a wide rente to account for es.funct!One modelling
- C8ftaIEtIes, 18 porticular with resp 4CC CS the soll properties. %1so, deviatione fgge aggg{
l ses functions can be ;
l ik O ' G J *t t ,
~.l i
m'
(
== '
4 Eh6 4
I synastry may be introduced 1sto the adel of the structure la step 3 (which er.
masse that a stonal " spring" and " dashpot" asset be evaluated La step 2) and the e effect of ch maae
- 3. and ettffeese of the structure be evaluated Approvinate Solution fer Cirevlar, Rahedded Foundatione without hartas to rerun as n re saalysia.
The celatione presented in the following sections have been obtained with a th elonal saisy'unetric finite element formulation. ree-dimen.
the exact representation of the model ooundary which separates thA fundamental e finite etervent region free wthe semi. infinite continuum (the free field).
ee developed for the plane strain esse by Waae and f,yener (11)This consisten three-dimensional case by the first author (2) , (1), (5). , 0), and vos extemfed to the a virtual extension of the finite element seek to infinity, and can be plIn essen accuracy iss.ediately next to the foundation. aced without toes of e
ta the followint meetion, it wt11 be assumed that the e tion which th before any structure (or hypothetical manalces foundation) h e ground experiences '
means of one-dimenstmal were propagation theory. as been built, can be described be '
earthquake record, viil be assumed to take place at the free sThe control action, i.e.,
- th Motices at other pointe in the free field can be obtai urface to the " free field." #
procese, 3.1 which ma>es use of one-dimensional wave,propetation eory. thned by the so-called Aporenfmettome to the rin satte interic g 1.e third author investasated the kinesatte to arsetton problem in atsdy parametric a
) (8), meing a r?tte of embedeent ratios t/R and stratum ration it/R
( typically found is nuclear reactor design , covering a rasse of values it interaction. , and proposed ruler to approximate the kinsaatic I, j
- , laterring to Fig. 3, e unit harmonic displacesant j deconvolved te bedrock. was specified at the free surface, and
( fonctions u,9 fer the displacement and rotetton of the masslUsing t ransfer k notion at the free surface, were determined. ese foundatien, relative to the i
t functions for the displacement is the free fiald at the elevation of the u a on, and for f
the pseudorote* ion of the free field ta t h.
/ ,5 , "A O g
1=US g
g were computed.
Typical resulte are found in Fig. 4 d and da were then compared for a renas of embedmont and stratThe transfer func approxiaste these functions were avggested (8). um depth values, and rules to 'N below. A simplified version of these rutas to givee # 18 I
I" field Let F cassa, (la most (0) be thethe Fourier design earthquake). treesform of thereeacceleration surface la the free at the f ple 11:14 fe,undation are then giran approximately byThe tremolation and rotation of theIN mass l
ets t
- IFT f(2) - (to .
If f e 0.7 f, i=
tie F(o) . [ 0.453 Jiffa 0.7 f,
~
f
, . - m M ') cuJ
- p -
. t
/ b u_ ,mJ ji _ h_ h sm - - - '
w- -~m a.we - -
1 - -^' -" ~ "
J l
a ub j l
ep 3 (which means that a ter. 1 F(c) . { 0.157 (1 ces { . f/f a l/R3 if f
- af se effect of changee la the 3=IFT to rerun an entire analreis. F(2).(0.257/t ] if f
- f, *
- r. 4 (d to positive clockwise) -
q ebtained with a three-dimen. where IFT stands for ! averse Fourier Transformation: f, to tha fundamental shear beam 4
esture of the program used is frequency of the embedmont region (for uniform soil properties in the embedaent recion, this y w flatte element region fra vSlue le given by f, = C /4E with Ce being the shear wave velocity, and E the depth of embed- g
'I
'lergy transmitting baundary aest. The expression is equare brackets describes an approximation to the transfer functions (7), and was entended to the for the translatice and rotation of the massless foundation. For surface footines. E = 0, sence it can be regarded as fa
- 8", coa 0 = la it follove that no kinematic interactice takee place for this esse.
tan be placed without lose of The procedure described yields satisfactory results for a wide ranee of et > tuent ration, C see for laatance Fig. 5. tt should be noted, however, that the rotational conoonent is senst- j s which the ground experiences tive to the lateral est conditions. and particularly to the flexibilite of the lateral walls.
}
een built, can be described by For flaxthle sidewelle, the actual rotation is significantly smaller, and in the extreme case p ottoa, i.e., the specified of se sidewalls, the rotation even changee sign' Meverthelaas. ehe coctribution of the face la the " free field.= rotational component to the response of the structure is in aset cases not verv significant. q so-called "deconvolutics" For asclaar containment structures, the attact of the rotational component on the structural 'l ory. F**Ponse is .*f the order of 15-10 percent.
3.2 Appenimations for the Stiffness Funettene blem in a parametric study As with the kinenatic interaction, the values of the suberade ettffness functione [
R. covering a range of values (impedance functions) depend only on the gemetric configuration of the foundation and on p
approximate the klaamatic the properttee of the founding soil. These fonctione can be evaluated using enalytteal, f1 experimental, or numerical methods. O 4
fled at the free surf ace, and * ' ' ' " " " '*#* "" ** '"E*""* * "
- ency dependent transfer fourth writer te determine approminate expressiens for stiffness functions of circular, y favadation, relative to the embedded foundations (1). For each particular geometry, the static valuee were evaluated with tuency dependent trenefer two se three meshes (a fine, a standard, and a coarse mesh), and the results were corrected a of the foundation, and for for meeb stas error la a manner sta11er to that described is Ref. 4 The study was lietted to :
the coupled horisontal translatian and rotation (rocking-awevine) of the rietd. circular. ;
embedded foundation la a homogenous stratum. For this particular case, the ftrce dientacanent relationship can be written f
- 3. w . g
'er functione e and v3, ,M J (Kgg Kgg [f, g i depth values, and rulee to where F = the horisontal force M = the rocking moments and u. d. are the correspond.st die- .()
xstee of these rules is giese placemente (ratations). The etenents Exx. Ku. Ed of the stif fness matriz depend on . he frequency ofexcitatico Q et the forces (soments). Since these forces and the resultina dio-
})
.he free surface la the free placemente are generally not la phase with nach other, these alenents are canalex functions of ad rotation of the massless. frequency. Each ettf fnese function la of the form K* (1 + 218) (1 + is c), where 8K is the static stiffmass. 8 a measure of the internal dampist in the soil (of a hesterstic nature). g e g*
1
- sT, and a, is the dimensionless frequenev QR/re . (0 is the circular f requency of the me-ties and excitation.R the radius of the foundstion elab.and f ea reference shear wave velocitp.)
'f a N
. 1 1
d I
I J
1
- e f ~ ~ ~~ , ~w-,,.- _ ,,_ n,.n n n _ _ _ - . a r--- . w - zms - -- ,
^*]
f..lC) w), 1 3
. J
- _ - - ~
w
+M[
I
< K 2/4 J
s
, k and e are fregeancy dependent toef ficients normalised with respect to the static stiffness. O' h 1 M erun..
The coeffittent e la related to the energy less by radiation. h ce the in Using the progras described earlier, the dymanic attffness functions were computed for a e18ple dynamic a range of embedaant and stratum depth raties, and written se described aseve ass teras can be add-frequency domain En
- E*n 011 + 18e * *11) (1 + 218) simpler physical Eq = R*,j (k12 + is ,. en ) (1 + 218) stiffness and da Eff = E*gg 022 + ia , . c22) (1 + 218} "
Analysis of the results obtained provided then the following approximations to the stette A number et valvea E'n* K'ade K'dd e and to the dynamic stiffnese coefficienta kgg, kg, en,eu , c22' the static attffe Static Taluaan MWtem laterei eet1, whi n.mu.}i)n.i!>u.11) -- " - -d 2 =
seixtion of the l E*,g = K*nI I{ - 0.03) (d is positive clockwise) condisin s within the ana'?sia is e g K'jg
- y(1 + h h) (1 + 2gE ) ( 1 + 0.71h ) **
. The spring ei la these formulas C
- the ehear modulus of soil underneath the eats R = the radias saedof forthe the kines foundation, E
- the depth of embedaant, B = the depth to bedrock, and v
= Potason's ratio. and the accuracy a Dymanie Stiffness Caefficients:
I*"UI III I" I kne %22 half space soletion (i.e.. Ref,10) asets these requia coupling between t practical standpoi 0.88 fore,efIg = a,g specified by a bre
'n * !! the direct appr Ralf space solutloo for a, a a,g aents with depth r ties waves) may le conditions, it saw 0.58 fora,ef{ =s2s vicinity of the et:
cn=
(Ralfspacesolutionfora,as2 o l
k12 = k u , c12 * *11 where 8 to the L. tarnal (hysteratie) damping la the so113 C p and C, are the dilatational wave gelocittaa, in the subgradas a,g ando s 2 are the (nondimenaienal) fundamental sheer beam and dilatattoo frequeneles of the stratias. as defined stnve.
Except for the stif fnese coefficient k gg, which displays a somewhat wavy nature, the eaggested appreslaattan for the stiffness and damping coef ficiente provide reseceable sub- -
statutes for the true fianettees. It can be ebeerved that the radiation desping coefficleate eg, e2 are larger for the embedded case than for the surface footingg therefore, the suttosted l Procedure should give conservative results for an embedded structure.
f
/t
/
,s c-i l
me e
---a ~ ~ . -
. . j
.- ,_ _ _ . _ - _ . _ -. -_ ,,m m_ - . _ - _ . _
en.nem. m m. - .. m
. , ' y-1 E 2/4 K 2/4 4 Soft. Structure Interactfen problee spect to the static stiffness.
once the input motion and the base impedances are known. the last step is reduced to a De "stif f ness" and " damping" simple dynamic analysis of a multidegree of freedos speten.
i functions were computed for a terms can be added directly to the corresponding terms for the structure la a solution la the scribed above as:
- * * *** "" "' ** * * ** * ** '** ** "# ' * **
- I I"*****I LS) stapler physical interpretation of the results. B oy require, however, frequency independent stif fnese and damping coefficients. and for the latter, ia addition the existence of normal modes.
is)
>proximations to the static the static stif fnesses waan their complete frequency variation. It is also worth noticing sats kgg, k12 e C11, c12' 8228 that the increase in stif fness due to embedment is very sensitive to the properties of the laternt seil, which may be disturbed. Considering. in addition, the uncertainties in the soil properties and its nonlinear behavior, it is clear that engineerint judgment is needed in the f f) selection of the most appropriate model, and that parametric studies, varying the assumed f
conditions within ressunable limits. are advisable. his, of course. la equally true whether a positive clockwise) the analysis la carried out in a single step (one-pass an' hod) or in three steps as suggested I bere.
1+0.71k) De spring method has the advantage of being less time-coeming when approximatione are e nati 1 = the radius of the seed for the kinsmatic interaction probles. It allows, therefore, wre parametric studies, ek, end u = Poisson's ratio. and the accuracy of each step is subject to better control. Cf partieler inportance is the peesibility in this method to make use of synneetry or cylindrical conditions if the foundation meets these requirements even if the structure does not (which is a frequent situation). De Raf. 10) coupling between the corresponding terms will came in naturally in the third step. From a practical standpoint, the procedure has an additional advantage when the design motion is specified by a broad band response spectrum n't tailored to the soil conditione at the site.
If the direct approach is applied to such a case, demeplification of certain frequency compo-ments with depth resulting from the use of one single wave pattern (i.e. vertically propage-D a,g ting waves) may lead to unconservative estimates for the actions of the structure. Under such conJitions it may be better to regard the design motion as an " average" motion in the g vicialty of the structure, and to use it directly as input to Step 3.
sa2 s ad C, are the dilatational wave
- 1) fundamental sheer beam and a somewhat wavy nature the ents provide reasonable sub- 6 radiation dagias coef ficients coting; therefore. the enggested acture. I 6
I
. n Q .'
) ) l 0 %r C
,m, apse - _. P m _ _ . , -- -_ _ _ - , -
l
. I
_ _ _ _ . - . , ~ , _ _ _ .
M' 4
i K 2/6 I
o 14ferences %
(1)
EL5ABIR. F. 'Statie Stiffnese Coefficiente for Circula.T Foundations Dabedded in an Elsatic Medium." M. 5. thesia. Mass. Inst. of Tech., Cambtidge, Massachusette.1975.
[1]
KAUSEL. E. " Forced Vibrations of Circular Foundations on Layered Media." MIT Research Esport 1974 E74-11. Soile publicatica No. 336. Structures publication No. 384. January,
[3] EACsrL. E., Rorssrf, J. M. WAAS, C.
Jours. Eng. Mech. Div., ASCZ. October,19 " Dynamic
- 75. Analysis of Footings on 'ayered Media."
[4] EAUSEL E. 20ESSET. J. M.,
Mech. Div., ASCE. December.1975. "Dynamie Stif fness of Cireslar Foundations." Journ. Eag.
[5] EACstL. E., actstrT. J. M.,
"sesionalytic merelas.ent for tayered Strata." paper accepted for publicatica in the Jours. Ens. Mech. Div.. ASCF.1977.
[6] EAC3EL. E.,RoF35ET. J. M., CHRISTIAN. J. T.,
lateraction." Jours. of the Geotech. Ent. Div., "'lonlinear Behavior in Soil-Structure ASCZ November,1976.
[7] LY5MER. T., WAA3. C.,
" Shear Waves la Flane It'talte Struc tures." Journ. Eng. Mech.
Div.. ASCE. Ve1. 98. February.1972.
[5] F..
(m.
M. bridge.
- 5. Beste. "The Mass. Massachusette,1975mitRAY.
of Tech. interaction Problem ofJ.Embedded Klaematic Inst. Circular Fm ndatio
[' t (9) 10ESSET. J.M., WRITMAft R. V., DOBRT. R.,
%dal Analvsta for Structures with Foundation lateracLon " Journ. Struct. Div.. ASCE Vol. 99. March.1973.
1 (10] VII. Y.,
Mach. and Foundation " Lateral and Itacking Div.. ASCE. VibrationFe1.
of Footinae." Jcura. Ean. Soit
- 97. September,1971.TELET5 (11] VAA1. C.
/ " Linear Tuo-dimensional Analysis of soil Dynetice Problems in Semi-infiatte
/ Layered Media." Ph.D. Desia. University of California Barkaley.1972.
asassues stauctuar
/
1 -* 0 -* u,
- Og i
/
) 1 I
l a F=MU, h
t I m j l
w-i t
= +
l.[ sosane (ICITaTEh sos mic tactfaTICW NWN W W W \\\\
%\WW%\%% WNNWN\\NN\\\
COMPLETE KIAthiATIO SOLUTION OYNAM C INTER ACTION INTERACTION NN
- h. vnuRE 1 SUPERPOSITION THEOREW Gr '
.. u i m ,,e ar.m ew , n - -, ,, -- - > - - -
.<a.
p, _-
i
^~ ~~^"' - -^ _ __
___ _ - _ . _ _ - . - -.. - -. = m ,m e=__ - -2 - - - -
~. g --'f
~
52/4 E 2$4
,, g e .a _ . - - - . .
f3 '
at Foundations thbedded la sa Cambridge, Massachusetts 1975. ..-
- s. ,
l1"""
g en layered Media." MIT Research
- a m & ~ ~
L ~
d ~' b publication No. 384. January, '
l l Q,..usw d
, + + Aaus.
4 la of Footings on 1.myered Media,* 4, q
T alar Foundations." Journ. Ena. w sww o o owN"" '" ' "*
t for LaTered Strata." papet
.. ASCE, 1977 er Behavior in Seil Structure Novembe r, 1976.
rams a structures," Journ. Eng. Mech. T>g 3
- STEP SOLUTiom
% bedded Circular Foundations,"
w setts, 1975. su am mm u..e el r,
his for Structures with
,1. 99. March 1973.
- f /I 24 *e
- g of Foetings." Jouri. Eng. Seil H, -4,.JJ s
.971.
a ** N //
'/
e
// /
/ /
males Problems in Semi infinite /
, narkeley 1972. .,.,,,r s /. /!
3nwwwwusu us muuuununmnuwwwwwow- s s emac :
rung santaa 't inTtmacTH>s PectLEM
,O ie Q'****"**
l '
- O s se-l F8MU 3
, F g /
- l
/ A w/ As /
t y._ _;
- . w I
e.- s re
,eammates
[_ v. . . . . . . ....e . . . ,e NNNNNN w %
" ] [p m CYNAW C INTEAACTrxl ***
'W *
)*,,*,'E*'
s**
sw
-,.se yt_
e , e e . . . . . O e e e e . a w WG 1307C4 0F anassLESS FOLAcAf tes. Taan3FER FtpsCTous tesi Mi f 'i (/
Jr '
.. ; tj
-w . . ,. ,
e
_ ~ -. --,,..--,n.,--~
e m a w_ s - - - - - -
__1 1 i 1
it 1/6 ,
L ee
) /. se . tv
< .a6 move.
g 5 .. .
5 .
e, .
, i .
d g [ re .vocar4= ... .. 6s 6 .
l 5- -
i %
i
) .... i
~
- Q, es - f-g .', h y Q . ,a st
.. . '/
son, i , , ,
,,iii (heq e,
se es e. es o. er der
- e. se se etnico sic) The F'G 5 forte MQ1lON CF WAS$LESS FOUNDATION, RESPONSE SPECTRA 1% OSCILLATOR DAMPING disri H/R=2. E/R*I (ISC Chan these clude canh indn; I llk i .
I" d-s en e a
du .)
sdenn j , ma6ree.ca butio:
mmimm s s mm un&.:
h i ,,a A-
- ,j
- 6. -
. ... at dd
...-....e, i.
perim
- 4 77
.i ...
reson-
.i
.i imi w en. i i u noem.e
_ rn r As
_ rn e
..,, . .. .. u bcddes
- "A
.. r; damri w ' '.'.1 1 - ins nd el ing m.
- o; ...
j rn r
. .. .. g' ria
x . .. .. .. .. y neuer e OTkAw!C ST1FFhE33 CCEFF;CIENTS, e*I/3. A*0.0S
^
O b'.4/
aD i. U /
i
,_ m .-_ .=w -
- u. -
--w==.. -
-- , -- w n. - - - ~
"ee
~~~
-g. -
EART){ QUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOt 2, ll-33 (1973) dgg
". nau + M TOfeIONAL AND COUPLED VIBRATIONS OF EMBEDDED FOOTINGS M.NOVAK' faculty of[r'gineerent Scnence. The University of Western Ontario,l.ondon, Ontario, Canada AND I I
K. SAC 115 AEG.Telefuden, Tradfurt, Germany
SUMMARY
An approximate analytical solution is presented for torsional vibrations of footings partially embedded into a semi. infinite medium or a stratum. Simple formulas derived for pure torsional motion make it possible to apply a correction for the effect of embedment to the known solutions of surface footings. The solution completes an approach to the anai> sis of all modes of footing vibrations, including the coupled modes. The approach to coupled modes is il!Jstrated by the solution of coupled response involving horizontal translation, rocking and torsion.
Formulas are presented for sti5 ness and damping cocmcients that can be used in the analysis of embedded footings or structures supported by such footings.
Field experiments were conducted with concrete footings featuring circular, square and rectangular bases and sariab:e embedment depths. The experimental results were compared with theoretical predictions of pure torsional vibrations.
INTRODUCTION This paper complements an approach to the analysis of embedded footing vibration which related t 3 pure vertical, horizontal and rocking modes2818 and coupled horizon.tal and rocking modes.* Pure totsional
.sponse is treated herein as well as response in a coupled mode involving torsional, horizontal and rocking components.
Tree is experimental evidence that the dynamic response of footings can be considerably affected by their parttal embedment beneath the surface of the soi!.18 28 " However, a rigorous analytical solution of this effect is very dif5 cult and finite element solutions appear the most promising approach to this problem.13 88 The approximate analytical solution presented in this paper offers advantages of simplicity, the posd'2tity to sche any number of degrees of freedom, and great versatility, thereby making it possible to consider layering and to introduce the dynamic soil reactions into the analysis of structures resting on embedded footings.
The solution assumes a rigid cylindrical footing, linear isotropic elasticity and a perfect bond between the footing and the soil. The total dynamic reaction of soili composed of a reaction acting in the footing i
base and on the footing vertical sides. The base reaction is deri, '
from an clastic half-space which is assumed j to model the soil under the footing base. The side reaction is c. -';ved as a reaction of an independent layer I oserlying the half. space. As well, this overlying layeris considered to be composed of a series ofinfinitesimally ~
thin independent layers which facilitates the analy:is of more complicated vibration modes involving non-gndorm displacements of footing sides. Such assumptions were first adopted by Baranov;8 however, he did n not consider the torsional modes analyzed herein. l l
l
- Professor. '
Received N August 1972 l Rensed 18 December 1972 C 1973 by Jose Wary a Soes.144.
11
{,Jg o.i 7 ] ;
c
.g ,~) ,,,W'-"'F),, -yA' y,,p. . *C# F' '4"
- Tg ' " " ' _e'.
-"0 9')
,- ," O"
, y,,, 9 g.
.. . %"S".. ~ h.5h $,5* ._ #f f - . ( h.
5'. E ?-Y ? l -Y . .- '- -
" ?.E_
~~ _- . ~d fs ~- , / O%
.. : M.'. .*..;m.. . .
[ ; -
- c ' ,* 1* * ,.,#, ,, *, ? - .* ie , Sam ~ ./, , ,,. -hl* ._^ * ' Wi'[* ' > [ '-.* ].."_
I*
_ . _ . - , . _ . . . m. .
r.'7--y;.y-r # ' f.~ y
.a 4]
- ~, g
- T
- ^
~ , ~ *. . ' - ~'
- D[
4
~ ^: .-:^
[
~
- [ ,li ?' _ '_ _ _r e.- - , -
. -_ w - -_ - - - ;- -
12 M. NovAK AND K. sAcus The assumptions are approximate, primarily because the compatibility condition between the soil under the base and the oserlying layer is satisfied only at the footing and sery far from it. Nesertheless, the same theoretical approach to embedment yielded a reasonable agreement with the finite element solution for sertical sibration'* and considerably improsed the agreement between the theory and experiments for coupled Imrizontal and rocking sibration.' Thus, it appears useful to obtain some information on the tor-sional response of embedded footings too because large differences were observed between the strict clastic theory and experiments with surface footings.82 Finally, the approach presented ofTers a way of applying an approximate correction for embedment to the already numerous so'utions for surface footings.8 *J ' 8S88 88 SIDE REACTIONS OF THE LAYER Firstly, dynamic reactions will be determined acting cn the footing sides if the cylindrical body performs harmonic torsionst oscillations with frequency w around its sertical axis.
With the omission of body forces and with the notation according to Lose, the equations of motion of clastic media in cylindrical co-ordinatr, are e' u
( .+ 26)PA 2C 2w,+ 2Gtw, 7 = Pp er r 26 (A v 26) 2A - 26 Ew,+ 2G fw' = p E'r (1) rte c: cr ct'-
c' w
( A+ 2G)EA 2G & (rw,) + r2G 26- ew,
- p ct' c: r cr i ,
',' in which relatise solum, change:
2w A =r crI d (ru)+ 1 Pcr 26 + 2.
an.i components of rotational sector.
1 I fw Pr I a w' - 2 r te 2: .
~ l Pu &w , (3) ,
~
w, ,
t 2 2: cr ,
i
-j 1 1f(rr) 1 Pu a w* 2 r Er r20
- With pure torsional vibration of the layer around the vertical axis, both vertical and radial components j
! of the motion vanish,i.e.
j w(r,6,t) = 0, u(r,6, t) = 0 (4),
e I red the equations of motion, equ: Sons (1), reduce to l i
f 1 d' I fe o e' v E* r (A+2G)pygr+ Gg7-Gp+Gy = Pp (5)l With the boundary condition along the circumference of the cylinder:
c(r,6,t)=r,(,e o
id (6)l the solution to equation (5) can be sought in the form: y e - R(r)C(6) e'd (7),j Substitution of equation (7)into equation (5) yields i R' R' f \ X42GO' (8),
t r' 7 + r 7 + pw't'- 1; - - g 7 i
j I l
' < < c,l 1 m .-
- -? 5 W ~-7. m y t;? v w - y u- 7 % . 7 3- .yyn m :,,; < < ww, w ~; .41 J.2T? ,7 Q.Q- K 'i.*1-L r - & K ,','a._ f f:~L :_. Q;. . .
[](l';2 7'.K(i 5 f -l $ * '. p: w:= Q' f~. % 1~.7?q
~ *5: ;;, g . . :---
n: w r . m v.: . . a +.L:: e qg ~3. :.,. .; x ~: .y - .x _.g y,_ = : --L __ - M - i_ s r. m Z ~'_. _
_ ,_ _1, _ . _ -- T
vismAnoNs or rustDDED FOOT 1NGs 13 l
r- This equation can be split into two ordinary differential equations for R(r)and C(6) se R' R' r f + f ji + (U- I) " "' (9)
[
r *
- r. I 6 t> *
=n 8 (10) 389 ic in which :
A* = 8 a= l g (11) s Solution to equation (10)is ;
& = A,cosnme+ B,,sinn26 (12) l of t
Motion r is independent of angle 6 which, with respect to equation (7), requires that in (12) n = 0 and thus '
C - A ,,. '
Equation (9) now takes the form: h I
r R* +rR'+ s8r'-1)R =0 8
(13) t
. The solution to this equation is ;
R = Chit'(kr)+ DH!*'(kr) (14) i in which H!1', H! = Hankel functions of the first order, first and second kind, respectis ely; C, D = constants to be determined.
From the asymptotic form of Hankel functions (see, e.g. Reference 5, p. 616) it can be seen that with excitation coming from the body (outward travelling waves) only the function Hl8' comes into consideration and thus C = 0. From equations (7), (12) and (14) motion o = AH! (kr)c'd (!5) in which A = a consta it gisen by the boundary condition, equation (6). Comparison of equations (6) and l
(15) yields, with r = r, and kr = a e, the constant A = (,r,/H!8'(c ). Thus, motion at a distance r i e= H!(kr)c'd (16)
(3) YI# I 8 in which the dimensionless frequency, a, = r r.o o J(p/G).
The torsional stiffness of the layer is obtained from the stresses along the circumference of the cylinder. .
From general formulas, e,- e, = r,, = 0 and circumferential shear nts Ev c Y r,,, = G(p g\ (17)
(4) Integration of moment ro r,,, along the circumference of the cylinder yields the relation between the total dynamic torsional reaction Ng of a layer of_thic_kness Iand cylinder motion (,c'd- ;
rgI ,'
(5) d') ~ - " Hit'(a) E - "b#'N b
- O0) or split into real and imaginary parts:
{
Nft) = Gril(Sp +lSp) {,e'd (19) l (0) in which l
,/ !
g Sp = a(2-a,41, + Y, Y \i) (20) '
and 4 4 (8) Sp= y. (21) I:
L c/c a I o ;= ':p .:.=?yzw..ww v.x n:c : erd -r'. Q .,. ~.a..b. :- . ; .1G~::i ^r ".'lia&=.51 - .~
.. n '.. ...?$
T
.~ W. Y._.&~ .?_Y Yb & Y'.- . .S. m.b '_-. rS *. _..
b:.5.$Y'*N _ ;. L Y:~b m . ? k '. 5,!~ .. *. Y m' S Y. .h.. . m'[b. ' .'
I k
.,k_h., .D .h. _ - . . ..
_$ . O. b ' [... ,".2 I- [_ "
- 2 ~~ i[-b-
- *. .II
~g gh a r-g M ghm -I. '-Aa 2' ep $ AM h 2 _ m C T 3.,,,,M '(%, hh A_" M - ' - -'TA6 m h4# w e L-- - ,6g,,hh_ -^
e f.
M. NOVAK AND K. SACHS 14 I- ir 1
Here.1,(a.).1(a.) are Bessel functions of the first kind of order zero and one respectisely, and 3 o li(c.),
are Bessel functions of the second kind of order zero and one.
unction S. has a meaning of a frequency dependent stifTness parameterawhile S represents a frequency e
dge1 aI,p;pgapmeter.%7se'pa'ra'nt~eis~areTnTw^nislu'rictions of a. in Figure 1. In Ap approximate polynomial expressions are cisen which can faci'itate the computatic. is. V D
12 -
S' I in o tt g
4
, 10 -
m g
a- -
w 84 U 8 s
$n. Cgi h /r. = 0 5 at E6 . cic h/r . to E -
s /
g -
Cgi hir ,
!* ___C'_T_*".'______________________________________
y _
z c;, cean ) (g,an.) ___ , co ,
ai g2 :._______________ ___ ______
,,,,,,c tz
..,_____*),+\~_-
___ ----- ---.fc____
c o . ________
_____ ___________~_______ 4 c lechl cn -
,cz z n/r.* 05 LS 2.0 37 CL5 LO 0 e,r, @
DIMENSIONI.ESS FREQUENCY os Figure 1. Stiffness parameters Sci. Cg and dampingt parameters S ., Cg, for side layer, half-space and strata 7; wi thickness Air = 0 5 and 1 (hlr = m denotes half space) as PURE TORSIONAL VlBRATIONS OF EMBEDDED FOOTINGS s) at The equation of torsional oscillations ( of an embedded cylindrical footing about the vertical axis of symmetry is _
(22) c,.
Ig ? - Afg (t)-Rg(t)-N g(t) ar in which I g= mass moment of inertia of footing about vertical axis Z, Afg (t) = moment of excitation, Rg (t) = torsional reaction of soilin footing baseg and N(t) = torsional reaction oflayer adjacent to footing <
ca vertical sides. ur The base reaction may be written in terms of displacement functions fa and fp (taken, e.g. from g
References 6 or 16):
i (23) p '
fu Rg(f) = -Cr'/p + tfo ((t) = Crj(Cp + iC ) ((t) I A;
in which stiffness parameter Cp and damping parameter Cp are '
C
-f'i C fe.
(24) ar p = lis+fis, p=lis*/p, r.
l The side reaction is, according to equation (19), '
5 Ng (t)- C,rj/(Sp+iS p) U:; (25)
'I '
l U4/ f )
4"
-~~-~-.-*v--~--. --
f. .a.-
c -_ n -
- v. s -r ;,r x <r .-m r 4 - % ma w s . . m. . w, ... -::: x_. - . n n . .>. c::: s -
.~ ~u~ 3.,__;_,, .. y.n m.
- c_
= =..y: _; w:=-.~z - -m g-;, ,u.. .. r. . . ; .
w,2;-. ,.q; w. - , ; ._ . y---=--e, _ ,~ .a.w.p s g . . , - - , -
,- :5:-
3 .. . v. - .
l
- '- }[
v .~ ~ _. _. _ ,
W - ~ L a. _.,
.n - - - - *.. _ , _ _ _
5, viaRATioss Or mar.DDI.D FOOTINGS 15 in which G, = shear modulus of the side layer and Sp., are gisen by equations (20) and (21). Substitution j of equathns (23) and (25) into equation (22) provides the differential equation of motion:
l G >
I g((t) + Grj Cp+g,1 S p+ 1 C p+ G,1 Sp ((t) = Afg(t) (26) ;
With complex excitation: ; ,
Af;(t) = Afpe"d = Af p(coswt+/ sin wt) (27) in which Afp = the real moment amplitude and w = excitation frequency, the particular solution describing the steady. state motion is ((r) = (,e"'in which {,- complex displacement amplitude. l The real part of the motion is
((t) = (,cos(wt+ 4) (28) ;
With the notation of the frequency dependent stiffness (spring) constant kg = G' C+ p Sp (29) and the frequency dependent damping constant: ll Gr GI \
eg= V(C
- p + g Sp ; (30) the re:1 amphtude of torsional sibration ,
s Afn Ain i
=- '
Co = (31) s [(Ag -Ig w')'+(eg w)'] kg g([1 -(wiw,)2]2 + 4Dj(w/w,)2) j v
and the phase shift: ,
cw c
4 = -a tan
- kg -/gw 8 In equation (31) natural undamped frequency w e and damping ratio Dg are ,,
, w, = d(k g//c), Dg= eg/21;w, (32) I.
The natural undamped frequency of an embedded footing must be determined by a trial and error approach 4 as it appears in k gtoo. Equations (31) and (32) are formally equal to those of a single degree of freedom system (Voigt model). Often the dimensionless amplitude A
}
g - (,Gr!/Afn is useful. This dimensionless j of amplitude depends only on inertia ratio B gand a, as can be seen. .
In many practical problems the excitation is due to rotation of an unbalanced mass m, with the eccentricity I
- 2) e,,, acting at a distance r, from the axis of the footing. The excitation n.oment amplitude is Afp = m, c,,, r, w2 ; i and a dimensionless rotation amphtude may be introduced,g A = (,Ig /(m.e,,,r,).
n*
The predicted response depends on functions Co ., substituted into the above formulas. Four typical cases , ,
can be considered: the footing can be embedded in the half. space or in a stratum and surrounded by m
undisturbed soil or backfill depending on the type of construction. >
\ r Embedment in the half space I 1
Stiffness parameter Cp and damping parameter Cnwere computed from equations (24) with displacement p
.3) functions /p., taken from Bycroft.' These parameters are shown in Figure 1 in full lines and are given in Appendix ! in a polynomial form to facilitate the computation. I j
The stiffness increase due to embedment was computed from equation (29) with frequencies a, = 0 and 1 -
- 4) and is compared with the static finite element solution by Kaldjian' in Figure 2. The agreement is quite .l reasonable.
Examples of theoretical response curves computed from equation (31) with various relative embedments l 25) 3 = l/r eare shown in Figure 3.
(, a. 6i
's
.. 4 2
_, , ,,,,-s .
.,- ' c -- .-- -2., r- - .
3= - ^ r. !.*fG ---_- '-
- 7 ' 5 FM 7 ' O ' ' ' ' - ' E.
- .~ --pfy-w M:'U.:.~ .' Q~C + ' ~"; a:~ . -? ^
, .m; .D r. .J ~; . . c.
s: p-&. f =&[).A L :~S Q MT:';;.mY,c.'
% =R.s.2 S H d~;?-!.
.:q!:~ N2 W ~Mi W = S. W ) 2i d ' *:A & .
n.; w 2_,:~:. M..=-: :,r- w y M.. w , = W.~ -u: w =_ 4.+
.. .w .... & : ._ .a. =.E M. ._"
- 1# * " " ' I# "" "" *#
. g f .m -- . at t m -
%
- L - + - =
u- _
1 i
i l'
1 gg u. NovAK AND K. SACHS S- ,s e
o.
~ ** ,'
o - '
a s i
." 4 w .o ei m
< P*'.'
8 w g 5 ' e*
12: P '
T I
- O; ,I /
'lIt
- , i::; 1 Yh <
.3 pi,1 ; ,
/ /' . . ,
< 2 -
!#ft , I ' '- 1 %),'j J
- r. r. '
G tr O
>=
e a a a
+ s e i g
IS 2.0 0 05 to t
RELATIVE EMBECMENT 8 e I/r.
Figure 2. Torsional stEness increase due to embedment: (A) approximate dyr.amic theory, (B) Kaidjian's' static f element solution 23 0 - J 15.0 -
1 80 0 -
1 u
w
- 8. 50 J
3 1 -
2 <
4 ;
g 30 -
m W
W 20 - o. o' ge _
o O %
I.S -
5 E l .0 -
o 4 i i
1 OS to 15 20 02 05 DNEN5iONLESS FREOUENCY o.
1 ntio Figure J. Theoretical response curves of torsional vibrations of footings embedded in elastic half space (inerJ i B;= 10,p.Ip = 1 quadratic excitation) <
s 5
D ) . s.e
+
~ 6\ / <i J ,
Y ~ ". .'.tM..rWK_ , .,;
- h.dT 7.i-Wy.. w-,
- s.- gi g,,wy .~ . .~ gg . ,7 e.y g
-- .- . .% q-%.,,.ww _ ..
p~y2*1R.h = ;' :?Lp 3j
.~. . . . - _ , , , _ . ,
.._7.
.n..
. w
- . ~ : .m g-i* c.~
. 7. c . . . ,,..
-.':P 4 7~.~i.,.=-.
- ,~ .:g.'
= -. %. -- ~. . ..". .72 ~~- .% n ~~:* r:.~. q . ..
.:. .ec- ' ,- ,
~~'!_ .. ;
w
,: 3 ,;, ;... ._.ua s .,4 ; - a
- .s== . .
3.
. . . .c r. . ,. . y _.y ...n ;;~'.A _. . ,.,-- ~ 3 3 w - y . .,, _4, ,, . ,.~
s- .u , . .
~- ,.. *.,. . _ '.. -.s. 3.
.,7
- ~
,_,,,g..--.
- = , ' =mem,a,s ,
?_w. 3 --- ~ _
_---aa Q u _ _ _ _
,, ... ., ,a om _
, i-- . . _ _ . ,,, , , ,,
I C
~
. j VIBRATIONS OF TMBEDDED FOOTINGS 17 y e
It can be seen that the theory indicates a considerable increase in the resonant frequencies and reduction J.
in the resonant amplitudes due to embedrnent. ;
Weissmann" approximately considered slip in regions of high shear stress and included hysteretic damping E.
of soil. Functions Co.s derised from his solution are shown in Figure 1 in dashed lines for both granular and cohesise soils ~respectacly. These functions considerably deviate from the strictly clastic solution, howeser, but may sometimes yield results closer to reality. i Embedment in a stratum 1 Embedment in a stratum can be considered with equal case. The functions Cp and Cp for two strata ,
featuring relat i'.e thicknesses h/r, = 0 5 and I are given in Appendix I and are shown in Figure 1. They were /
computed from Bycroft's functions /u * ("
Base reactions for strata of any thickness can be obtained from Awojobi's solution.1 The corresponding stiliness and damping parameters are also given in Table II (Appendix I).
Examples of response curses obtained from equation (31) with Awojobi's base reactions are shown in i Figure 4 for sublayers having relatise thicknesses h/r, = 1,2,5 and ec. In Figures 5 and ( variations of .
30.o -
) - -
l 20.0 - I I r
[ ' l- J I L -
m;' -
o ite 9u =
a . r4,
- 10 0 -
,3 L [,
W "
- E
\ .
8 4 <
"'s i!
k .
> pr
" B g 50 - Ff
" t
~
9 -
s'
]
f y lo s 4g - , y B, %_ - N '*
. s s N g N, ,
o o .\ a * $
.: p,.? sys $ =~ %{fe,~~~. 1
.lo*l oy %~ ~ ,.
to
.6 1.0 t5 2.o DWENSICN'.[SS FRE0VENCY a,
,r Figure 4. Theoreticat response curves of torsional vibrations of footings embedded in strata with Alr = 3,2,5 and cc.
(Embedment ratios & = 0,0 25 and 0 50, Bg = 3 5, p,/p = 1, quadratic excitation) li l
resonant (maximum) amplitudes and corresponding resonant frequencies with stratum thickness and s I ;
embedment ratio are shown for inertia ratios Bg - 2,4 and 10. These graphs were also calculated with h
Awojobi's base reactions. With 6 - O the solution corresponds to that for surface footings, i.e. to that of Awojob! or Bycroft according to the base reactions used. There is very little diTerence between Bycroft's 6 .
and Awojobi's solutions with h/r,- m.
It can be seen from Figures 4-6 that the effect of embedment in a stratum is even more pronounced than k'
f with that of embedment in the half-space because oflittle or no geometric damping generateci through the l ; ,
base. Figures 5 and 6 can be used to assess the cKects of embedmer.t. inertia ratio and layering.
- . t f .p q7/ ,
0 4 ') .. D V.a
, , w' oe #
[ -
-,-..-,,=~~c.[..m--.-~.'W..- m < *.s,.; W ,
- .- m . . .. - - - ,
,'*.1.,... -
- .r.<- , o- ".:.. 3j
.- :, p? * - -~~ - ' ' ' "*' ... - " '. ,.r*,d"'~ . . ;g . .
u .=.,..y. c;,:,,;A.
~~~.
O c: P 4.W*=,,,- , -M,- d w' *E .7 f.,. 7.~ * ~ . .- . f. N '- * *
- -*'E '* *-_ "E - L. '?'.' J -- ' ' ' "
sb' ' * ' 1 Y' ' - _. - ' YIj'.'
' ' ' ~ . ' , - . *=---- , , ' ..
~
< -- - ~: -
x, . - _ .. " _ . ~ 1-
- g s - - . . - _
r--
+ ~- =.--=A.*~===+-e "" "
a' 6
s
[8 M. NOVAK AND K. SACH5 i
1 1
- i 130 r a
80 -s '
s
"*g LO - s T C 8.o s ,. - -
l SO ~s .
- \s ! I g gE . 2
- y. .
40 Ns o .__-
. , 4 N -
,,7 , .e io
- ) -
's, 81,/r 5 ~ \.
T.
- o.20 w
- .,8,.e__-----------
a f,
. 01 Fig 8.o c
,to o
cc 0 8
- ci s02
? ~ ~ - - -
- a.
- (}.
/
m -
" l ani W O6 m po]
p a u .01 4 9 - --
720.--_--________________., ) Of o
"., c 4 .
04 <
2 1 5 - 02 03 -
fr
- ----..____-------_80_4___________________,0_____.
Thi i
' ' Cor C2 -
8=04
.e .i anc
- the 8
- L3 aff 1
I I 1 1 E I c'
f I E the t
1 2 3 4 5 6 7 8 9 to STRATUM THICMNESS h/r, .,l VaII 550 Figure 5. Dimensionless resonan: amplitudes of torsional vibrations of footings embedded in strata for several embedment j ratios and inertia ratios Bg = 2. 4 and 10. (B; = Ig/rp!; circles indicate finite cie"..cn: solution Reference 20) .l P i.i I
.q ' in s f< ' lorts a
r tw k
N i e
4
/ << n.I ,. > ~ > j (1- .W.
- r
- .! g,
.{.
' - n . - , ., n o-, - - - _ - - ;---- , -- -, : - ---'r ~ ~ - T'"~~
,..L.m w-7 %r-s: , . _ ~, : r * ~; :. Gzm-i s ,-. .u=. . . , wG ' Y . . . . -
,4- ' n -- - e.-
W-a M ' r~ ; 5-$?%W . i - -;.. C : .
- "'~
- f. E{.? i..::.-n,
~ ~~25[5 $s $ ~~ ?.Y&51. '. Y $-Y$*A ?$$??.$ Sa 1.W :',Ii'Y!
-L *T '.'^..- ~ y-;.,., .
- g =. ,3 _ . , ,;
- , f. '_ ,- ,Q, . * .~._) <, h "_.; ' :~~ f.. .. , ; ^ k . .% .. _ .
..... : ..._.3
- :T 2. .' T. . .
f:" 4 o -
- m. c 3-
, , , ,_m s . -__-iw' --
p _m. __& ._ 2 a ~ 2
i s
Y'? A ATtOSs OF FMBIDDED TOOTINC5 19 I
, r 6
S Sg
- 2 '
t
<u.n o.-__. 4 . ,
10 Q oa i 6,;. ._
M ._
g : --.----.-_______ d.*J _________________________
&*o N y 14 y
.- ----_______________o.*__________________t______.
i
~~~~--______________o_2.________________________
$'2'----_____________oJ.________________________
'---_________!_*_o'._________)__________cp___.
olo -
U - o4 m .
- o2 o8 -
oJ se O c6 4 6 7 8 9 10 4 2 3 5 STRATUM THlCAES$ h/r,
?
Figure 6. Dimensionless resonant frequencies of torsional t vibra; ions of footings embedded in strata for several embedment l ratios 6 and inertia ratios Bg = 2,4 and 10. (Bg= lg pr*,; circ:es indicate Enite element solution, Beference 20) 1 Also plotted in Figures 5 and 6 at: resonant amplitudes and frequencies obtained for h/r, = 8 by Wass and I.ysmer 20 using the finite element t:chnique. Their results are shown as circles; arrows indicate with which ;j i ana!>tical cune they compare. The agreement is quite good. (The finite element results shown were inter-
, l potated to match the parameters g B used here. The differences in resonant frequencies with 8 - 0 4 are e' cf little practical importance as the corresponding response curses are rather flat.) l o
SIMPLIFIED DESIGN ANALYSIS The prediction of the respense from the above formula is not difficult; howeser, the computation can be ;
considerably simplified if stiffness factors C, and S., are considerigeonstant.,,(ft:gvecqy d indep.endend ,
and_d,amping factors C as propo'rtional to dimensional frequency 9. It can be seen from Figure !
4 , and Sp l:
the degree to which these assomptions are acceptable. They seem reasonable and quite adequate for I applica'. ions with respect to all the physical uncertainties involved.
Thus, stiffness parameters can be considered approximately constant,i.e. Cn n= C andpS= S o, where the constants denoted by bars can be readily obtained from Figure 1 or Appendix 1in which some suitable I values are given. With constant stiffness parameters Cp and o S , the frequency independent stiffness constant is obtained from equation (29). Then, the natural frequency cu, can be directly calculated from equation (32). b >
neat j
> Proportionality of damping factors to frequency assumes: '
(33)
Cp = Cpa, and Sp - Spa,
>j in which C p and Sp are constants whose values can be again established from Figure 1 or Appendix I. '
Substitution of equations (33) into equation (30) yields the frequency independent damping constant for l l,
torsional vibrations of embedded footings:
(M) c; = rh'(pC) Cp + S o f i
?
6 'fi (/
t L? . O (
-IT n- - '
m ,. a -_- .
ww_m'n . . m::;2.: x: ..
s ,';.J.
'.m ..' ~
,- w , -Q ~ ."*' 4 gY ' .7 - .,[, _
g .. .,,
- pm;- C..,;t,, - .g$. .-44 d P-C.$'d O ?_'C.M[y ,,, -d g-f.-Q,c.gr
. b.I-h M %Q_~r -;.'.f.'37
- q ? 'JM._ d.31W.W9. _.h :: :
..v.,-
R . _. _ ,,ygL_s.> -at -s .s ee- > w .e.
- w .m '; ' A w r a_? _ - ~'
Am*^*'_ -
- m - er*% s Q-wA
20 M. NOVAK AND K. SACRs The frequency independent damping ratio is from equations (32) and (29)
<>v in whictLin:rtia ratio B r= I /prj.
n.-&c.M MU (c4.s )
For surface footings, equation (35) yields Dg = 017/JB; and numerical g
results close to those of Richart and co-workers (Reference 17, p. 226).
With D obtained, g
the ' resonant' amplitude at natural frequency wa (slightly smaller than the madmum amplitude) is simply ( (w.) = Afg d(kg 2Dg ) or with quadratic excitation ((w.) = m,co rg(2D g/g). The maximum amplitude is 1/s'(1 - D}) times larger.
An example of comparison of response curves computed with variable parameters and with constant parameters is shown in Figure 7.g (C = 0 5 was used for the half space in this case because it yicided better agreement in the frequency range employed.) The agreement appears satisfactory for design purposes.
As even better agreement can be obtained with a mass adjustment.
So' VARI A BLE
[g PARAVETERS 70 ' / \ ___, CONSTANT PARAMETER $
l \
5.o [ \
! \ ,,
- 7 C[ \
(I s,
h 3o
,- ~
- ,[ g',- ~~' - .
- ., %. '.,~
8
, 2.0 el r; ; , . '~~., ~.
n %l .e/ . ,~~~.'. .~~~~..
y , ,
g l l
p, p z / / - 2 e.
Lo -
l
,1 l f '
_ v
- r ,
3 i 0.7 -
l W'h
-, ~
W 05'o.5 ' !
to LS 2o ol MENS:oNLESS FREQUENCY e.
i Figure 7. Comparison of response curves of torsional vibrations computed with both variable and constant parameters (B; = 418, backfill with pJp = 0 8. GJG = 0 5, quadratic excitation, Bycroft's base reactions)
In practical applications, the effect of embedment may be reduced due to backfill and an imperfect ' band between the footing and the soil. The effect of backftll can be accounted for by considering G,< G and p,< p in equations (29) and (35). Possible lack of bond between the footing and the soil can be approximately taken into account by considering Weissmann's base reactions (dashed lines in Figure 1) and intuitively reduced values of Sc u-COUPLED VIBRATION INVOLVING TORSIONAL, HORIZONTAL l
. l AND ROCKING COMPONENTS i If the centre of gravity of the footing does not lie on the sertical geometric axis of the footing, a coupled motion in three degrees of freedom is produced by a horizontal force Q, by a moment Af g in the horizontal i
plane and by a moment in the vertical plane Af,.The components of this motion are the horizontal translati u(t), rotation in the horizontal plane ((t) and rotation in the sertical plane (rocking) f(t). This motion can also be sc!ved under the same assumptions as introduced for pure torsional vibration.
Consider the centre of gravity the origin of co-ordinates. With the notation according to Figure 8 and E
with the omission of product moments ofinertia and of coupling between displacements and reactions in
- v. . .
p ,nq , ~ pl
~ Dd.
f5 - , .m
.a M. wym.. s-un x -: z.e ~ , , - w=- 5. ' =
"-w
.L ,;3..: ..y:
~ ?Q4 ** w -~~
~r
".= -
.!:s. .a_-- .-- -;.
m . m.. ~-'
~
-f. O ~~Q 7
D "- -.--
~-
- E.[M' i.]-
5.
$ YA T'-__" __
& R h.t OI E -h
% -d - ,Y"u'h I 'd f.a Nde' ABw4 S L'2# skO k- M #dh kh411. ' # ^ ff* -M
l 21 k VIBRATIONS OF EMBEDDED FOOT 1NGS f Ib M ,(fl lf g.
0 3
- 5) _
/\ ,
al 's x , i,
~ --
CG -i
~
"" t U sg I
,'g
- I 2 3c . / ~
D' <(asnwk 4.'n aus. /
nt r r, , r, -
er # ' 2.w f
- s. Y.v r ,
o I
s QIII I 5 ;
j 'c I
, j X.o
.: of i ec L ' t . ]t Figure 8. Notation for coupled sibration of footing in three degrees of freedorn involving horizontal translation, rocking and torsion I 3
the footing base, 2' the equations of the motion are mfi(t) A r.[G(Co + iCa) + G, 3(S,1 + iSa)] u(t)
+ cro (G(C,t + iCo)+ G,3(Sa + iSa)} ((t)
[
+ ri f- G 2 (Ca + iCa) + G,3 .l i3 *.! (Sa+iS,). I f(t) = Q(t) (36a) t r. A .. .I I
. . [ -
I,J(t)+rj G,3(g- u(t) 2 r 2 (Sa + iSa)-G2(Ca a re +iCa) t eters I,
+cr!G,3(3; *IO2 (Sa+iSa)-G2(Ca+iC IO
- 2) ((t) ,
1 j
l ond
,
*Ce + j 3Sn+y3 +pj-6pSa '- P (40) kg, = k,,, re-en G su; kg; = - Ger. :, Ca + j S(:,- { r,6) Sa = ek,,, i G G k gg = Grf pC +f BS p +7 e'Ca + g 5e' Sa ap 7 . int kg,= ek,,,= k,,g h thr d5' k,=kgg g , and frequency dependent damping constants: l ' c,,,, =Gr. - C, + G' 3Sa w G j I To c, g = c,,,, e b Gr* f ' c,4 - - ws:L , CaG!+8(:,- G } r,5) Sa \ Un wa < Gr - G G 8' ' ~' - V of cy p= ?Cg +p' Ca+p 6Sg+g8 + -6 So I , exe l , (41) cp,= c=9 l 1 soil 1 G I* b' Ger* 1 c,g= , :,Co+g 6(:,-ir 6)Sa , cc . - c,e k y ec# = c,c 1 o Gr' e' d G,3e'I I ett " -- Cp+ G,3S 3 p + 7; Cu + 3 fg .: ] } m 2a\ { ~ 4, m. , w.= - -,- w ,,
- m y , , -, -
' .w *-:s 2.u.~*'-, .;-L; - .;. w' ::+rn:;.2':.m:. ~
- y:. - 3 ; -
- M:~
C'.92 - c -%. % .27.1.:- :n.% : : -- .. , u- ;;~v ..~-. un . .:. :z - ~,.i.ny.&a .2 - r - : . ~~s y .b . . - , . .A. e_:?_3 .w c . -- ~ ~ . - y. t -t-s ' . .- ':. --.- . =. . x .--,u,;. . u%; :.,;. s .+ ~ . - -~..,u~ , -%;'. ? 7 , L.[.;-n ? * ':- W *L v c = ;.' .* -~ . .C' - y---- @ M - ^ ~2; 2 ': .
- r; 2 . : w . ~."~-..
+ c ..~'$_ - .. u _y. . ^,- . L -u-3 ,, _ _. - . - - t w - . . y. . __ , _ .m I I V!BRATIONs OF EMBEDDED FOOTINcs 23 , From equations (39), the complex amplitudes u, - ui +iu ,s etc. can be obtained with any excitation frequency. The real and imaginary parts of the complex amplitudes yield the real amplitudes: 7) u,=/(uj+nj), 5, = g(f[ + f j), {, - g(Q + Q) (42) h 8) ( 6) With quadratic exciation, these amplitudes can be made dimensionless according to the formulas: m I ' A, = m, e,u,, A, - m, e,1+z, @,, A= c (43) g m,e,r,(3 ' 9) The natural frequencies can be obtained from equation (39) by setting the right sides and the terms labe!!cd , by / equal to zero and sohing the eigenvalue problem by trial and error. g The reciprocity relations in equations (40) and (41) hold in this approach but need not hold in general a as shown in References 10 and 19. An exampic of the response of embedded footings in three degrees of freedom is shown in Figure 9 for eccentricity e - r,'3 and for sarious embedment depths. Dashed lines denoted au. indicate the first, second ! rond third resonances of the surface footing (S - 0). 1 The first resonant peak shows clearly in all three components of the motion. _ t The second resonant region is dominated by torsion and can herdly be recognized in the translation component A,. s The third resonant peak is completely suppressed in all three components. , These relations, of course, depend on the properties of the footing. Nevertheless, the example illustrates ; possible importance of the torsional component and the effect of embedment on the response in the three J I resonant regions. It can be seen that the response above the second resonance depends sery little on the i embedment. With small eccentricities (e) and medium or low mass ntics the third resonance usually appears , suppressed. I Tne analysis of the coupled response can again be considerably simplified if constant parameters C and S , are introduced as it was shown in the case of pure torsional vibration. The efficiency of such a simplified 0 approach in two degrees cf freedom was demonstrated in Reference 4. With constant parameters C and S ] introduced into equations (40) and (41), frequency independent stiffness and damping matrices are obtained $ that can be readily used in the analysis of any structure supported by embedded or surface footings, as i discussed in Reference 23. l \, i 1 COMPARISON WITH EXPERIMENTS To assess the applicability of the theory to cecular footings and also to footings featuring square and rectangular bases (cross-sections), field experiments were conducted with three concrete blocks at The University of Western Ontario. The blocks '.ere cast directly into neatly cut excavations. The embedment was changed by removing the soil in three steps. The effect of backs 11 was investigated by step-wise backfilling of the soil. Pure torsional sibrations were excited by means of a torsional mechanica! oscillator with s e.tchangeable eccentric masses. I o si) The soil was a brown, sitty clay underlain by a glacial till. The shear modulus and Poisson's ratio of the N soil were obtained from wase velocities measured in the field. These measurements were complemented by !I laboratory imestigations. The following properties of the soil were found: Bulk density of undistuioed soil = 110 lb/ft 8(1760 kg/m8) Bulk density of backnll - 100 lb/ft 8(1602 kg/m8 ) 8 Water content of scil = 15 per cent 'l Void ratio of undisturbed soil -07 ! Shear modulus of undisturbed soil o 113 x 1081b/ft8 (549 kg/cm8 ) , ,,- - ! f i. G (J 7 / / V .). \ i l i i E ',T h Oa, , . , M y 4 b Ehr 0 9 .. - 2 I 14 l= .-5. "* :f/ZJ4 I I-M,: N--h ; #.77.N N;lM : . Q'N [ ' 2 ',;[L . ; , T -7' ~ V t M*j$_z - $C'&
- r Y Q.-p. R & & ; % ll{'~ $.,,l-N R
- j
. .-- - --? $ J E ~ $ ? -- -- g. M'T,[l%'- % K~,~
- . x y ~~gq w ~ . , ~ .mu ~ x -. - LL.hr
1 M M. NOVAK AND K. SACH3 30 0 = ~ < 200 - e . o.s I l r ! o ( * * "# * - ( 60.0 f. $z \ s
- "8 a -
> 3.0 $ e . ars w 3.0 - g,0 + e.. z ' e y y ^ I 5 I.O . l I ' '# l *. ' Of t.S 2.0 01 0.3 s .O DIMEN$iONLESS FRE0VENCY e, 20.0 s.ao 15.0 - . l0.0 - - * - e.oJo e o a .a.a s a Sc. x 8 v o \ ' 9 e . o.s o a 3.0 - e y EO' e o o.ss i z a.eA O j w - z t.0 - ,! ~ ' w /JW-5 s o N 02- 1 i e 8 8 ,ca ' i L ' ' ' L C' DJ 0.S 1.0 1.3 2.'0 i j p- DiMENSiONLC$$ Fm(QUEN0Y e, 1*4 - f, k 10 0 - 7.0 - = Sr - L ~ e r. n
- 3D 1
-o f ., 9 O 24 - O' I h ,.& , 1 - s* , to-w ; sa J 1 ent ) i a ~ z L 90.5- > e 1 ,- f I j
- t. .
5 i t e 0.2 i { i , 1 .. . 3 .- e., a.S 2.o f %- 01 c.s s.0 i ciutNsiONLess FatoVENCY. 8, 4 Figure 9. Example of coupled response of embedded footing in three degrees l o t g translation. rocking and torsional components. quadratic excitation) j
- p. -- '
-7 boa w
- i j
. o
- t. 17 / (. U ]
k- = .W~,~. ,. , ~ %, ;. - 7:,..mW v- -
- ~[;m m k'c. ,_ .a-.;-. ~.. ; . " _- wv *R*
,+ ,.* ~, ,, %_ _;). - .r.. A.
- Y&. . lT..,*;. . - . . - K. . . _W.
, - -W-o=p . .W.:..,.p--<,. -. n. y-,%,,"_". ,_;.,. n, 7. .;. - ~"R,
- t".,.
,j . 5.,.!$ l lf, . ; ~ ' ' '" j--:~_ L~ ; ~-n., l{i$Ya sLiN' 'Q~5:f ; k$5,&' 'z- _"~A- , ,' ').*,n:-Q^-f [ . . ~ -~~; $: h Y} m ::- 2- - - :. . _ 5,-. -
- z(q yhv-3 ; 3% y_--g,
- a. s. -
,]., _. n - i vinxAT1oss or twatooto roortscs 25 , T, . The main data on the test footings are given in Table I. The equivalent radii r for square and rectangular It , footings were computed from the equality of polar moments of inertia of the base. l( Table ! } I Shape of Base area It r, base ft' (m') Ib ft's' ft (m) Bc Circular 3 98(0 369) 545(754) 1 125 (0 343) 8 65 [ Square 5 06(0 469) 77 6 (10 70) 1 284 (0 391) 6 50 g , Rectangular 5 01(0 464) 93 2 (12-88) 1 330(0-405) 6 55 i r A typical example of steady torsional response measured at various embedments is shown in Figure 10. > The foundation base was circular in this case and the excitation intensity was the same for all embedments. l s- , f 5 - 'I j' ko I 2 e4 - y* . O 8 9 e.. , 3 - m h \ ; h s.78 , e . m 2 - z 1 i g 7 42.67 - ^ r , , , o of o2 o3 o4 oS o6 0.7 o8 Di sA E N s io N L E S S FREcuENCY e, , Figure 10. Measured response curses of cyhndrical footing at various embedments (m.cs, = 0 00326 lb ft s'. undisturbed I soil, B; = 8-65) 4 - Comparison of Figure 10 with Figure 3 indicates a good qualitative agreement ia the major effects of embed. ment manifested through drastic reduction in resonant amplitudes and large increase in resonant frequencies. ; 4 The former effect is primarily due to increased geometric damping while the latter effect confirms an increase in stiffness. ,t With other foundation shapes the changes in response due to embedment were similar to those depicted in Figure 10. A comparison of response curves observed with various footings is shown in Figure 11 for i surface footings and for an embedment of one foot. It can be seen that embedment was most efficient with (I the rectangular footing. This indicates the efficiency with which torque can be transmitted into the surrounding ; l soil. The quantitative comparison of theoretical and experimental resonant amplitudes and resonant frluen- < , cies is somewhat complicated by non.linearity of the response which renders the dimensionless aonant amplitudes and frequencies dependent on the excitation intensity as shown in Figures 12 and 13. In Figure 13, , C; resonant rotations are shown in full lines, resonant frequencies in dashed lines and embedment is indicated . i , as 0, I or 2 ft. !l li f,AG, na a ^4, l{ P, m k .[=. M '*e_[ h Q [ .
- 4. "#p* # e, , . d -
N . a.; i. 'g. . - ' % ..< g .--'[*, b *r~."... . - ~ _ - - -- ~ . * * * * .r.. . f,c,_.;;",'f'4a y . ;;g a,:'j. . . 2'.~ - . _ ,' ?.3 g ..y;:- ..~7 ' Y '_- ( . : ,.~~. ._ j ' -j ._:_.- ? - - G ' Q :(~& , G %-Y : "y ~.' .~ ~
- 5 -
', L ' e ' - :.l.\ .- , , - ~ . . - _ p.'. . ,.i,-;; _ f.%.,._X .:~3. ut f* *;*.'1,' -j_*.C j';'[* ';=:~ y*,.~.T2 g ~. ', ., _. .;. --- _. _ . , - ~ . ~ ,u 4.[m ,a - .
- _ - - .O - .. ;_,. __ _ w, ,__g -_,,.a, _3,, ,,
26 8- M. NOVAK AND K. SACHS 8- -- g.o - 3 = 1 F T. A c CIRCLE O SOUARE ~ 77 - & RECTANGLE z 1 p s /p
- I o i -
. c I 4 $6 o l g a = 8 $ % k = ~ sht E* (*g ll \ 9 8 8 o fl 3 h = ,I go l[ \ t e d -g \ ll ii " \ J i I \ \ l N -3 -f g II c g I 5, ,, y i ? 1 ) - ~~~_:c w% m e \ 4 , 1 a -P ? .ds' m / _ F, 0 05 06 07 08 09 g 02 03 04 DIMENSIONLESS FREQUENCY e, k i th Figure ll. Measured response curses orcylmdrical, square and rectangular footings mith embedment0and cx i ft 4 TC C K e 0.00t168 LS FT S2 35 0 0002150 m a 0003264 h, at 5 . O Osl' g~1 1 u s 4 f, 3E la 5 ' % Of k.- e 6 .' lt o oc 3 F th. o j 'k m s , w "i Val " CX o [ a2 - dit z d Co I " it. th. I - g,; e . wit N, M- ' ' i 0
- al -
IS, I .2 0.3 C.4 C.S 05 0.7 08 0.9 + CIMENSIONLESS FREQUENCY C. dd Figure 12. Dimensionless experimental response curves of torsional vibrations measured with three excitation intensities y sht i (square footing. embedment = 0, Bg = 6 5, quadratic excitation) EJ I ^ f! I }. b 'jt ()/ '. u J - I , v - ..ey .-.r . z. - zn .-: ,1;.4 _ m.p:y. c ;.-G~;...w3; ^ cq7._- p-e yru. 3,~ a,.e /,; >. .. , . - ;-
- ~
- m. .m . _. , e ., .9-'_'.
, re . . . . p x &s 3 3 ~s . ~*~,,_ . -.- v m_.:.. ~ri L ,~~ _1=g, ...w ::_.e y
- - = c *; .; %f. % r f yc: '- , -q~ .3.
_ ; y,a~ z; Chy",.y:,f:.p q Q.y,,-. - }.^g M:.L'; ; -C;z; .-~,fi.'[2:b.* cz .Q }gs . : [; i ;5%=Q.g, ~ 3 -_- - w= - _ ;_ - _ t _. , . - = , . . . , . _y,._,,__ . ,. i [ i I V! BRAT 10Ns OF EMBEDDED FOOTINGS 27 ) 4 b oo ur a- l [f 7 - O clRCLE - o.7 g l 2 -- O SQUARE z g g.
- " ,. A RECTANGLE
- n p
. U - ROTATION - 6 - - -FREQUENCY o.5[ ' - -a 2 ' z $ % B'N oS $ $g - O g o
- l m O' w .
w E e s 38 o4 , y4 - % . oo w . w o,~- a o- ~~o'* ' = - - . . l $ 4 o o o3mf' 3- I'}., E 3 l . a pi a 42- - o2 , 2 - ' I 3 S 6 7 8 'O 8 2 2 ExclTATioN FAeToR K a lo3 t it f t s ) ;' Figure 13. Measured dimensiocless re:ccant rotations Ag and resonant frequencies a. versus excitation factor K = m,e.r, l (zero embedment or backfill 1 or 2 f0 l-It appears that there B a trend for both the resonant amplitudes and resonant frequencies to decline with
- y the increasing intensity of excitation. (The measurements were conducted with increasing intensity of excitation.) These changes seem to indicate a damping producing slip between the footing and the soil in regions of high shear stresses as suggested by Whittnan22 for surface footings. The theory presented herein [
assumes a perfect bond between the footing and the soil; therefore, the measurements conducted with ; minimum excitation intensities are used, for the most part, in further comparisons shown in Figures 14 , [ and 15. In Figure 14 the theoretical and experimental resonant amplitudes are compared. ' Vor surface footings, the experimental amplitudes are considerably smaller than the theoretical ones, in , i agreement with observations made by Whitman.22 For embedded footings the experimental amplitudes are , ! J larger than the theoretical values; however, the differences are smaller than with surface footings in the case ' ? of undisturbed soil and are quite sma!! for the rectangular footing. For backfill the agreement is very poor. ! It can be seen from Figure 14 that there is a region of small embedments (I/rg01-0 5) in which the ' ' theoretical and experimental amplitudes must coincide. The comparison of theoretical and experimental resonant frequencies is shown in Figure 15. The q experimental resonant frequencies are consistently much lower than the theoretical predictions. The differences do not diminish with increasing embedment and appear largest for the circular footing. (Of i course, there is a dependence on the rather inaccurate value of the shear modulas.) Nevertheless,it is clear that the torsional stiffness is much less than the elastic theory predicts. The experimental frequencies and amplitudes of surface footings reported here are in reasonable agr.ement with Fry's experiments and with Weissmann's modified theory.21 'l It can be concluded that the quantitative agreement between the strictly clastic theory and the exp riments ' is,in general, poor, particularly as far as resonant frequencies and the effect of backfill are concerned. These differences may be attributed to limited capacity of the soil to transmit the torque from the footing through te shear. It is desirable for any theory to allow for a slip in the regions of high shear stresses. The presented bkh ,'b q 1 -w .-. ...n.,.. - ,:sw,m ~m~~.=- w==- ,27=- + a. w g- a,. E e m . w w w. w m m u - .n .m- m~m . .g ..
- _
- p -- - - _ ,, _. _
28 M. NOVAK AND K. 5ACHS ti l toc THEORY f EXPERIMENTS
- l
' E O CIR CLE o l
- O SQUARE 8 e
a RECTANGLE & ' z too ddL E l dd4 escurate e x I > o i < ^ f e IL rt o A u E SO % d4
- i je
" 40 - N #N, 4 Nf, %s% ~% z 3o O h Ctq 4 1 4 H 20 +7# t g's Auf %,'C(c II H m % . ~e < w - w eIRctg n a z d ; U , $z to - < 3' w _ EL s - f( 5 - , , t . , , , e i , , i os , , . l0 2.0 3.0 0 la EMBEOMENT R ATIO 8
- J / r, l Figure 14. Comparison of theoretical and experimental resonant amplitudes of torsional vibrations (- theory; E' .
experiments) i cf d rc
- 20. 1 I.s - .
$d ! ts ~
- d- 1 T t
e ~ C \gC' / o l4 - St - 8 J* p y t2 - 4 1 w _ 4 s d O lC w U p'A - .i E 09
- A f - 9 j i j a
,08. ..- fe z 40.7 z f #s# . 0 / d " o6 ',- - / / / L$c)*' I ,/ m os -j / WA . ',. w / d y a / lli / s on - , 7
- /
- THEORYI EXPERIMENTS 8
/ g o 'J o 3 / CIRCLE E 3 6 03 4. 7 O SQUARE y , a RECTANGLE A ') - ddd eaCxritt # m' A ' ' t t f f i ' t ' f I I f f % 02 O 10 20 30 EMeEDMENT R ATIO 8
- f / r, .f Figure 15. Comparison of theoretical and exrcrimental resonant frequencies of torsional vibration (- theory; . ...
- cxperiments) 1
.. - ., 4 l b N c/ / I,j [ ' -47Ny- gy- g gm ,.,y . 7 --""V - "g- . % ^ 1Q.
- a
- 'M .e q, gO # *-- - .. . - . r ~ f Q~.vl.t l , _ffe &.J*LC"Q...J.W.7 a. .n - ,. ,-.. ~ . . - s- 9 y -:4:~$-@sh *.'%: if.C..N-r. ;TW:,.iO2,2, !_,. m .... w _~.C ..~ 7 qL. - ww n :m xw ,. ~~.,' _ . ..: .m3 ~ = - :.x,.- :m; . .a xE.1- y 3: _. ,u. ..w ,'~~ .-~. ?.~~,-.' _, .g; 3 Lf..+...,-:
- u. ,.
, :. G ~ ~ ~ . , . ' Q ~m c- : : V.' .g". ,,. - _ ' . _z ~ ~ %..y - 'y' "' '"" >
- m. _ ~ " c - _ _% m , .
, f' i, 29 VIBRATIONS OF EMBEDDED FOOTINCs (' 6 theory makes it possible to include such modified soil reactions in the footing base. Such a modi 6 cation would ' bring the theoretical predictions closer to experimental results. , P' CONCLUSIONS i-Embedment produces a7rastic decrease in resenant amplitudes and a marked increase in resonant frequencies of torsional vibrations. These efrects are particularly pronounced for footings embedded in shallow strata with l! relative depth h/r, smaller than about 2 where the omission of embedment and internal damping can yield unrealistically large resonant amplitudes. With embedment taken inte account, layering appears much , less important. ' The coupled response involving translation, rocking and torsion is strongly affected by embedment in the ; regions of the first and second resonances but depends very little on embedment in the region of the third ', resonance where there is usually no appreciable increase in amplitudes. The approximate analytical solution agrees quite well with the available finite element solutions and requires very little computing time, and layers with any thickness can be readily considered. [ L The field experiments show a good qualitative agreement with the theory in the reduction of resonant f amplitudes and in the increase in resonant frequencies due to embedment but the quantitatise agreement in general is poor. Best agreement is found in resonant amplitudes for small embedments arid rectangular k 7 footings, worst agreement appears for surface footings, and for circular footings in general. f The real resonant amplitudes are smaller than those theoretically predicted for surface footings andare larger than predicted for embedded footings. Only for small embedments fair agreement can be expected. {. The resonant frequencies are considerably lower than the predicted values. Also the effect of backfillis in , ~~~ general much smaller than predicted. f The differences observed can be attributed to the slip in the regions of high shear. The inclusion of this 5: c!fect appears necessary in order to improve the reliability of the theoretical predictions of the torsional I response of footings no matter what kind of approach is used. ACKNOWLEDGEMENTS " This study was supported by a grant-in-aid of research from the National Research Council of Canada. The ' assistance of Y. Beredugo is gratefully acknowledged, as well as the assistance of J. G. L ask and J. Howell. 5 r I L l l d 'l < 'OO
- F b .iT G
/ .OU j n lj Q_ '~ , ,Q ?? , ~,l -:h y ;.glf ~, , ~ C e
- ' '~~C [ ?,
w;- c_ 9 ,,a. -n, m; .. I,wt% 7 . . f'" "'T.: ~- ~: ' '^ Kf : ., , :., - - w '.* r K p :2 :j f ",,_ _- Q *y -2 ,- Q ,_ lf ~;p X*- -. '. QR..;. a:.. v. z i:q.~~ f O '~2L _ ~ T } f .:f ,.; - y Q,[z '*yl' [ Q*Q* 3, :;.- ' :'-Q::z~~, ;,y,j;,;C( w_ mW- rN': .hu:Jr fu+=ns % % w iwy g, n_ _ [ .,,.s,,J.- ,, _ , _ _ . y ,j.- . i 1 e I *.h 30 M. NOVAK AND ~;. 5ACHS
- APPENDIX I Stifness and damping parameters Stiffness and damping parameters are gisen in Table II, calculated from papers indicated. From Awojobi's
! solution,1 the parameters can be obtained for any stratum thickness with a,>r /h when theequivalent ' l dimensior.!ess frequency is introduced a = /[al-rg/A2] (44) I i Table II. Stiffness and damping parameters for torsional vibration Constant Validity Stiffness and damping parameters parameters range 5 i J t Side layer I 8ci = 12 4 0 < a. 50 2 :1 Sci = 12 58- 1-01a.- 5 9120! ' Sci = 10-2 0 2 < a. < 2 0 t Sci = 12 59-1855a.-3 349a!+ 5 335al-2 76ai+0 495a*. Sc - 9 04a! 8c=20 0 C a. < 0 2 3 6a. Sc, = 5 4 0-2 (a. < 2 0 . Sc, = 7 So. O 455 + a. ! Half space' " 5 Cci = 4 3 Oca <20 l l ; Cci = 5 333 + 0-032a.- 1358a!+ 0 7434c.-0-1414a! O < a. 4 2 0 - ; Cc = 0 486a! Cc = 0 7 # Stratum h/r, = 0-5* 0<a,<25 j! Cc = 6-78-0 5377a. +0 4545ai-0 2206aj Cci = 6 5 Cc: = 0 O c a. < 314 l_ Cc: = 0 i Stratum h/r. = l 0* Cc = 5 2 0 < a. < 2 00 l } l Cc r- 5-75 -0 419 to + l 381a!-2115ai+ 0 5927ai s Cc, = 0 Cc, = 0 0 < a. C 157 i , Cc, = - 6 28 + 4-Oo. Co=06 157 < a. < 2-00 '[ Any stratum thicknesst 'jt Co = 5 333 - 10678a* + 0 5607a!-0 2715a* + 0 0739a'-0 0109a;* Depends l a.> r.Ih p on h/r. .Io,<20 b; Cc: = 0 755a8 -0 4223a!+ 0-1631a!-0 0302a' + 0 00204a;5 ' s
- Derived from Bycroft.' ,
1 i i i t Derived from Awojobi.' i APPENDIX 11 C NOIal5011 r A = constant , 7 A,, = function , A, = u om/(m.e,,,) = dimensionless amplitude of horizontal component of coupled vibration - I j Ap = 4,I /(m,c,,,:,) p = dimensionless amplitude of rocking component cf coupled motion Ag - (,I /K g = dimensionless amplitude of torsional vibration with quadratic excitation A g - (, GrjlMn = dimensionless amplitude of torsional vibration with canstant force amplitude A g,,, = (,,,I /K g = dimensionless resonant (maximum) amplitude of torsional vibration a, = [aj-(rgh)2]I = equi'alent dimensionless frequency for strata a,,, = dimensionless resonant frequency (at maximum amplitude) a, = wr d(p/C)(or - wr,/(p,/G,)) = dimensionless frequency of excitation B,,, = function Bg= I /(prj) g = inertia ratio for torsional vibration C = integration ecnstant C,,i Ca = elastic half space stifTness and damping parameters for horizontal translation , C,,i,Cg, = elastic half space stiffness and damping parameters for rocking Cn,Cp = elastic half space (stratum) stiffness and damping parameters for torsional vibration CoC p= frequency independent clastic balf space (stratum) stiffness and damping parameters torsional vibration i l (') *k /,r) nnp <
- i. i i
'I-h. ~q.-;;f9 3 S [Q . b "" .k. E *I a. ,[. * [ i * . z _ W[=g-- m y: 'Is,.a ,, -( b ,' 7 S...
- q. -W;_':
- q>rX,*
f * [::"'l ' W 443=f: 8.gMik}-.C y .s 3_=5M:g: 2m.=<:3%;u-x3W Li:-;r.':.5M-- - -: ;e. 2. T =. . -- n - - 4% _ ~. - .. ~ -e VIBRADONS OF EM9EDDED FOoTTNG5 3I c,j = constant of damping force acting in direction idue to displacement (velocity)in directionf cg = equisalent damping constant for torsional vibration bi's D = integration constant lent D g= damping ratio for torsional vibration c = eccentricity of centre of gravity g e,, = eccentricity of rotating mass I fp,fg = components of Reissner's displacement function for torsional vibration l G = shear modulus of clastic medium (soil beneath footing) G, = Shear modulus of side layer (backfdl) H = height of footing Hj",Hj = Hankel functions of the first order, first or second kind respectively h - thickness of clastic stratum 1, - mass moment ofinertia of footing about horizontal axis i l I g- mass moment ofinertia of footing about vertical axis I i = J(- 1) J.,J - Bessel functions of first kind of order 0 and I respectively i K = m,c,,,r,- excitation factor with quadratic excitation k = constant , k,j = constant of clastic restoring force acting in direction i due to displacement in direction f y k g= stiffness (spring) constant for torsional vibration l= depth of etabedment of footing ! Afg= excitation moment about horizontal axis , Afg- excitation moment about vertical axis Af,, = amplitude of moment Af, Afg = amplitude of moment Afg m = mass of footing m, = unbalanced rotating mass 3 I Ng- torsional rezetion oflayer adjacent to footing Q = horizontal excitation force 1 G, - amplitude of horizontal excitation force r - cylindrical co-ordinate b r, = lever arm of horizontal excitation r, = radius of footing base; equivalent radius of footing base R = function of r Rg- torsional reaction of soil in footing base So ,S,,, = ride layer stiffness and damping parameters for horizontal translation S,i, Sa = side layer stiffness and damping parameters for rocking ' S p ,S. = side layer stiffness and damping parameters for torsion Sp,Sg= frequency independent side layer stiffness and damping parameters for torsion i = time u = complex horizontal displacement of footing; radial displacement of medium u,- complex amplitude of horizontal displacement ' i u, = real amplitude of horizontal displacement u u = real and imaginary parts of u, [ er - horizontal displacement of medium perpendicular to e ' X = horizontal axis L Yu - Bessel functions of second kind of order 0 and I respectively - vertical co-ordinate [' \1, r, - height of centre of gravity above footing base h i 7 2, = height of horizontal excitation force above entre of gravity w = vertical displacer.ient of medium } G f.) 'nT / ')'IO G i y-- ;. . a ~.v.7 ,, h ' u., .3: '.T-w;b...- . , n.s u _i . sWn~~ p'"'~.w-ww.w~==Ww':W-~#:: e .-m w.c-u .-- -';,"~ . ; U L2;~5M : ,e -'" C > .. y-
- v C %. =.- , g . ~. . . u.. ,- e -
qmcp. . - w-SE? cs . f C -.4- 0
- ---?,' ',*., & # S h' . 5 Y b Y " $ E '5*
. a w - .+ ~ 2 . . -2 "?k'. _. . - ~; - v- = ( ___ _ - - ~ _1_M w ,,, ,. A ig . _ m_ s- A . ,- e . _2 . d%.-- &qwe&Nh* -9 j'h -Weitt@ I, . bl. NOVAK AND K. SACil5 32 I a = Constant " A = relative volume change , 3 = Ile, = embedmer.t ratio , ( = torsional vibration 2 (,= (t+i(s = complex amplitude of torsional component (,Treal amplitude of torsional vibration ; 2 ~ (i,(a = real and imaginary parts of (, q = pjp = density ratio 8 = cylindrical co-ordinate I A = Lame's constant p = mass density of clastic medium; mass density of undisturbed soil p, = ma;s density of side layer; mass density of backftll o = normal stress r = shear stress , l @ = function of 8 4 = phase shift f = rocking component of vibration i @, = fi+if, = complex amplitude of rocking component of vibration (, = real amplitude of rocking component l 8 @u = real and imaginary parts of complex amplitude of rocking (, w = circular excitation frequency < w,,, = circular frequency at maximum amplitude ,I wo= natural circular frequency j w,,,,, = components of rotational vector I e REFERENCES
- 1. A. O. Asojobi ' Torsional vibration of a rigid circular body on an infinite elasticJstratum *,
f 369-378 (1969).
- 2. V. A. Baranov,'On the calculaiion of excited vibrations of an embedded foundatioC (in Russian), Vap. Dynamiki t Prochaosti No.14 Polytech Inst. Riga, 195-209 (1967).
- 3. Y. Beredugo,' Vibration of embedded symmetric footings *, Thesis submitted in partial fulfi!! ment of the requirements
- for the degree of Doctor of Philosophy Faculty of Engineering Science, The University of Western Ontario, London,4 J' Canada. August 1971. o
- 4. Y. Beredugo and M. Novak, ' Coupled horizontal and rocking vibration of embedded footings', Can. Cearceh. /.
4 Nov. (1972). 9, 417-497.
- 5. E. Butkov, Mathen<atical Physics, Addison-Wesley,1968. L
- 6. G. N. Byeroft,' Forced vibrations of a rigid circular plate on a semi infinite clastic space and on an clastic 4
~ stratum
- Phil. Trans. Roy. Soc. London. Series A,248, Math. Phys. Sci. 327-368 (1965). 7,1011-1024 (1969).
- 7. G. M. L. Gladwell 'The forced to sional vibration of an elastic stratum , Int.1. Engny Scl.
- 8. M. J. Y e Idiian,' Torsional stiffness of embedded footings *,J. Soil Mich. To end. Dio., Proc. ASCE,97 No. i :
SM7,969-980
- 9. (1971).
T. Kobori, R. Minai, Suzuki and K. Kusakate,' Dynamical ground comp ance of rectangular foundation on a semi.1 nfinite clastic medium (Part 1)', Disaster Prerent. Res. Inst., Kyoto Univ Kyoto, Japan, ,, Bulletin (1967). (In Japanese with English captions.) t
- 10. Y. E. Luco, and R. A. Westmann,' Dynamic response of circular footings *, /. Entng Mech. Dio., Proc. ASCE, 97, No.
EM5,1971,1381-1395. II. J. Lysmer and R. L. Kuhlemeyer, ' Finite dynamic model for infinite media', J. Estng Mech. Dio., v Proc. ASC No. EM",959-977 (1969); Closure to Discussions in February 1971, 129-131. g
- 12. M, Novu, ' Prediction of footing vibrations', J. Soil Mrch. Found. Dio., Proc. ASCE,96, No. SM3, 837-861 i (1970).
- 13. M. Novak and Y. Beredugo,'The effect of embedment on footing vibrations', Proc. Ist Can. Conf. Earthquake Eegg Res., Univ. of British Columbia, Vancouver, B C., Paper No. 7, 111-125 (1971).
- 14. M. Novak and Y. Beredugo,' Vertical vibrat on of embedded footiegs', J. Soil Mech.
i 1291-1310 (1972).
- 15. E. Reissner 'Freie und erzwungene Torsonsschwingungen des clastischen Halbraumes', Ingenieur-Archie. ;
Bert Germany, 8. No. 4,229-245 (1937). 15, 652-662 (1944)
- 16. E. Reissner and H. F. Sagoci,' Forced torsional oscillations of an clastic half space', J. Appl. Pays.
- 17. F. E. Richart, J. A. Half and R. D. Woods, Morations of Soils and foundations,
{ Proc. ASCE,93, No. SM6, Proc. Paper 5568,143-168 (1967). v t '- [naf ,a Oq1 s, . c y l r,. C.Em.b. A. ,,. . -e :: w1 2 2.M..J. ' ~ ; . w- . 7- ~7.'l~V w , --' : rc'u p w~ ~;"' m '. .+. a -,-.a... .N = ac.- - ...~ s, * [ I
- bE g . . . -;-
- ** d' ,3.- ,=.
- . . .- - .. ; ~ . x u.g .-3 _ . 43, y .
= - c .,-~ -.z.: . .3 , ~, - .3 . . , ~ . , - -- J~ ' - u. . . ,-i , , V1BRATIONS OF EMBEDDED FOOTINGS 33
- 19. A. S. Veletsos and Y. T. Wei, '!2teral and rocking vibrations of footings *, J. Soil Afech. Tound. Dir., Proc. ASCE,97 No. SM9,1227-1248 (1971).
- 20. G. Waas and J. Lysmer, ' Vibrations of footings embedded in layered media', Proc. WES Symp. Appl. finire Element Meth. Geotech. Engv, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss.,1972.
- 21. G. F. Weissmann, ' Torsional ubrations of circular foundations *,J. Soil Mech. Found. Dir., Proc. ASCE, 97, No. SM9, Proc. Paper 8402,1293-1316 (1971). (Also private communication.)
- 22. R. V. Whitman, 'fnalysis of foundation vibrations *, Proc. Symp. Vibrosion in Civil En v, Imperial College, London, April 1965 Butterworths, tendon, 159-179,1966.
- 25. M. Novak, ' Vibrations of embedded footings and structures', paper presented to the ASCE National Meeting.
. No.10, Soil ar d Rxk Dynamics, San Francisco,1973. es, 5, dii . ents p den, F
- h. J.
I tufn*, 980 semi- -314 No. 95, i 5 970). nav ] , 9s, [ii rlin, , b 944). 4 2 j; N ,n -. ,e r; b d '/ , / t. 4 m -_7 '.o_,;_.y" 7-*' _- ,,. , . -'35. .- : , %-- .72 *'A'.1'-'ZQ ,,,, . . . - - - ~ '7 f W " J ~:r.: , 7 ;,7 : '; w .. T,' . .*, L ., ~ i;4 ' 4C ~ ,7- ' ' .-~~a. - ~ - - Stl ( '. Ji 3 {,* , ,,,y- ,' **-H S
- e ? -. ** ,'- ,,w.+ ** <'%y, , 4%-. wi * < s L3 __ . a .-. . . . , _ . , , , , ~
5 3'; .- * ?- l'. . * ^* ~ ^ ,[ * . * { ' ,' '.T , m - * * . *- -A ~-_** h,, "' - , , , , h"' ; ^ ,
- f -
_2 , . . , - - - -2 - ~ . - . ; - - .--x ' *r -- -a ' 3~. ^ M - ~ -
- d. O be
. 6 9 et M i M e ' 477 $ Coupled Horizontal and Rocking Vibration of.Enibedded Edoting,! a ,, Y. O. Bratouco ! ShcIl-B.P. Nigeria Limited, Lagos. Ni:eria d . yg 6~ M. NoVAK l ). ,l Taculty of En:incering Scieme. The Umwcrsary of Western Ontario. London, Ontario d Recched April 26. 1972 > Accepted June 1.1972 . The prediction of the coupled response to horuontal forces is of major importance for the design ' of footings expcsed.to d3 namicctTec:s. The theory of surface footings greatly oserestimates the real response and ne:lects the fact that :ne footmes are founded bcneath the surface of the ground,i.e. partly embedd:d. This lauer factor considerably sticers the foo:mg ubrations in that it reduces the resonant amphtudes and mcreaws tr'c resonant treauencies. j An approumate anal)tieal solution w:s dese!cred m this paper and the theoretical resportse curves g were compared with tie'd esrenmems. Closed form formulas wcre obtained that are urnele enough to be d:rectly used in cesi;n. The approumate ana!ytical setunon irbener able to predict the coupled re-sponw of ernbedded footmgs than that of surface foctinp. The response is usua!!y dominated by ,' l the first resonant pen wnh the secend resonant reak entirely suppresscJ. The deriud formulas for equhalcat st:tinesses and dampmg coe:licients due to sod can be introduced into the solution of any struc ture. f . La pred:ction de la rdpense combnde a des forces horizontales cst d'importance majeure pour le calcul des empatte.nents soumis i des e:Terts dynamiques. La theorie des empattements de surfao. surestime la reponse r6eile et negHee I; fait q.:: 1:s engattements sont places sous la surface du scL i.e. ds sont particll ment enfcub. Ce dern:er facicut afic te consderablement les ubranons de l'em. / .' pattemer.i en rdJuisart les ampiauces ce esonance et en au;rcer.trt kurs frdauenecs. , Gae so!ution ana!>tge at rroustr.e est coeb; Tee data est rt;c.e et ks ccurr:s thectioues de rdponse sort compardes au: des mesures etfe;;ue:3 sur le terrein. On ebrient des forrnutes nn.cs cat sont asser sim;'les pour dtre utuisdes d:ree:cment dans le ca' cut d s fondations. Cotte ;oiaticn analy-tique approsimatne est meri!eure rcer rrdJire la reponse combinee des emrtt'c. rents enfouis que 'e ceile des empattements de surface. Le comrorter ent est h&tud.ernent gouurne ar la prem:ere pointe de resonance alcrs t;ue la 3 :ende ponte ci3parait compWement. Pour scrur compte du sol 4 autour de la fondatien. on propose des formu4s ase: ces cctriicients ce rigiJud et d'amortasement , equiu ents, formuks qui pausent servir .: la solution des cakuk rour n'imperte quel;c structure. Introduction first ins estigated by Baranov i!967). who - It has been recognized for many years that formulated an approximate analytical sole-
- , the sibrations of footings can be greatly tion: howeser, he presented numerical results alTected by their partial embedment into the only for pure rncking. T;jimi (1969) analyzed
' soil. N ct rthe! css. very little quantitatis e the response of a structure pattially embedded inf ormation is n ailable because of the in an clastic stratum and attached at its base dimculties associated with a rierous theore- to a rigid half space underlying the clastic tical solut;on and with the generalization of stratum. , experimental findings Uncoup/cd ribrarien - In this paper. an approximate theory is of embedded footings v.as scised by Lysmer used based on the assumption that th: dyns-and Kuhlemeycv ' 1969.1971) :.ho ins e>tigat- mic reactions in the footing base are equal' '. ed scrticai motion using the fimte element to those of an clastic half space and that the method. Novak and luredugo (1971) hase reactions acting on the footing :ade> are pre;ented approximate foimulas for serticai, equa; to those of an oser!ying independent . horizontal, and rockirg sibrations. Coup /cd clastic layer. Tl'is apptcach, employuu first /iori ental cad rodmf irhet;o r of emb:Mded by Haranov (19/J). is evended here to sield footings, most iraportant in design of machme clo3ed fcrm fermulas ana graphs that can be foundations, nod:ar pour plants etc.. was direcij used for de3ign purposes. Alm. the came cowai 9.,n.e. 9. m men .-* - . ~ . . _ - . ~ . _ _ _ _ , . . _ _ . _ ._ _ , _ _ _ _ , _ i l 473 CANADIAN GEoTECl!NICAL JOURNAL. Vol. 9.1972 efTect of backfill is incorporated. The major proximately described if the following as-advantages of this approach are its relative sumptions are adopted: simplicity and great sersatility. The known ' (i) the footing is of cylindrical shape, solutions for coupled motion of surface (ii) the reactions in the base arc the same footings (Richart et al. 1970: Ratay 1971) as those of a surface footing and thus can be are a special case of the more general approach taken from the solutions of the clastic half-presented herein. space or stratum, and To provide more experimental data. field (iii) the side reactions are produced by an tests were conducted with concrete blocks independent laver lyine above the lesci of subjected to horizontal escitation. The block the footinz base. ~ ' ' bases were either square or rectangular in For this layer, various assumptions can accordance with shapes most ofien used in be made. If it is assumed that the layer is practice. The reported test results are com- composed of independent infir.itesimally thin pared with the theoretical solution in order layers, then the side reactions derived by to assess the validity of the theory. Baranov (1967) can be used. Bara n ov's original solution can be extended to yield , Eiluations of Motion formulas, directly applicable in fou nd:uion With the notation gisen in Fig.1. the equa- design and analysis, and to insolse the etTect tions of sibration of a footing ir, coupled Of Soil Id)ering, backfill, and various stress .iorizontal translation utt) and rotation p(t) distributions. about a horizontal asis passine throuch the ~ ~ in accepting the a'asumptions (ii) and (iii). center of crasity are: the compatability condition is satisfied only i ' ' at the footine and far from it. Despite this. l g miitt) - Q(t) - R,(t) - N,(t) the comparis'on with experiments and with [1] finite element solutions for vertical rnotion .fIvo) .tf(t) - R;(t) - N;(t) - ( Nos a k and Beredugo 1972) and torsion indicates that the approximate theory 'vields ~ in which m - total mass of footing. / - mass quite reasonable results. moment of iner tia about a horizontal axis passing through the center of gravity. R,(t) 9 - horizontal reaction at the footine base. w.to N,(t) -- resultant horizontal reaction acting on embedded surfaces (sides) of footing. ~ yj, R;(t) = reactise moment of forces acting at T~ J\ /h T- i f footing base about center of grasity. N;(t) - l
- reactise moment of forces acting on footing f'jl_ llj ';. ,1 p ,
1 sides about center of grauty and t - time. a p~,- egy ~' , , The dots represent ditTerentiation with respect i ll lj C, j vo horizonial exciting f oice h to time. Q( t) - l g7 [ ,,,a, acting at height : If .1/, (t) - exciting mo- L u. -- _ ~~ ment, the total moment of escituion is: }m . t r.G [2} .tf(t) - Q(t):, + .11,(I) '* " ** .. x , m. No rigorous expressions for r: actions R Fic. t. Mathemancal model of embedded rooting
- and N are know n; howeser. they can be ap- in coded motion.
With the above assumptions, the soil reactions can be formulated as follows. The horizontal and momcot reactions in the footing base are: R,(t) = GrjC,,, + i C )g [u(t) - r, v(t)) [3] ' f Rg(t) - Grj (Cei A iCg.) v(t) - Grjc,, + ic ) e[ ult):, - :lv(t)l ' ~ *j G 4 ') ,l )! y -e i I BEREDt;Go AND NoVAK:DIBEDDED FOOTINGS 47'# i t , , in which c, -[' c"' . fis + fi: ne _ Se .T- [=' c~2 , fi, + fi, an I4] 4 i or c,' -lei i Iji +/i. ; 1" i fg , is Cg' - ' in , /ji +/j, by i . 's Here, G - shear modutos, r, - base radius,i V 1, /;, - displaceinent functions of dimen-dd sionless frequency a., - cer,,v'3 and Poisson's ratio r. c; - frequency of excitation and p = an mass density of the medium below the base. Functionsfg can be found, e.g. in Bycroft (1956)
- et . or Luco and Westmann (1971) and are used here with the original signs.
- ss ,
Tiie horizontal and moment side reactions car' be written as e . I c-U )' .gY N,(t) - G,r,J(S,, + iS,,) u(r) - - - . , y(r) .s. ' \- - [5] < N p(r) - G,rjJ \_(S;i + iS;,) + 6: : ::
- th n ,
7-6 + (S,,, + IS,,) v(t) on - - Eds i J ' s + -r. (5 - r. ) (S,,, - + 1S,,2) u (i)}f, -m in uhich G, - shear modulus of the overlying layer. J - / / r, ,- relatise embedment and 1 - embedment depth. U Functicns S; are independent of r and are ' ' ~ ~ 1,(a,)/,(a,) + Y,,( a.,) Yi (a,.), S;' - ir 1 - a" J :(a,) + Y :(a i i o) [6} ~ S#2 l i:(a,) + Y i(a,) . In Eq. [6] 1,,(a ) and Y,,(a,,) are Bessel functions of the first and second kinds. respectnely. of order n. . Functions S,,, and S , depend on Pei> son's ratio r ar'd aie the real and imaginary parts of the complex function: t,eg [7] S,,(a ,r) - G,[S,it a,,. r ) r- i((a r)] . I -- - //j:) (a,)H;2' (x,) + /li:' (x,)Ilj:' (a.,) - a, G 3a* V9 //l,2' (a,) //j:' (x,) .c /IP (x,) //j:' (.7,,) . licre, cf - (1 - 2r)':(I - r). x,, = a.,vj and II', ' = Ilankel functions of the second kind of order n. Baranos esaluated functionsyS and X,, only fer Poisson's ratio r = 0. 5 with w hl:h Eq. [7] simphfies. For seseral other vlues of r these parameters were computed horn Eq. [7} e* ' * *
- O -- essa. w -
_ e --e, . . . - 480 CANADIAN GFoTCCI!NICAI. JoGRNAI VoL. 9.1972 and approximated by polynomials. The results are gisen in Appendix I together with poly. nomial descriptions of other parameters C and S to facilitate the computation. The para-meters are show n in Fic. 2. Substitution of Eqs. [3] and [5] into Eq. [1] yields the 'difTerer.tial equations of coupled sibra. tion of embedded footings: mii(t) + r,[G(C,i + /C.,) + C,d(S,, + IS,3)]utti +r - G r, (C., + /C.2) + C,6 - r, (S,i + 15,2) F(t) - Q(t) i . . IQ(t) + rj C,0 T J - y. (S , + IS.2) - C f. (C,i + IC,2) u(t) [8] < _ / J2 -2 }\ ( . ~' q + r.' C (Cg i + /Cer) + C,J (Sei + ISg2) + - 4 + 2 I (S,, + iS,3) +C~', (C,, + iC,3) . p(t) - Af(t) , s . ) . With complex excitation Q(t) - Q,exp(iot) = Q,(cos at + i sin ot) - [9] 3f(t) - Af,exp (int) - JI,, (cos at + i sin ot) in which Q, and 3/.. are real excitation amplitudes, the par:icular solutions describing the steady state motions are: u(t) - u, exp (int) [10] p(t) - v, esp (iot) where u, and v,. are complex displacement .in.plitudes. Substitution of Eqs. [10] into Eqs. [8] prosides the following equations for the complex amplitudes of coupled motion: [(km- mo:) + inc,,]u, + (loc,; + k,.g)v, = Q, [11] [( A g; - I o ) + incge]v, + (loc,; + A,;)u, 2 - jf,, S,g:0 5' S.2:04' i Swate) lI . >J - 5,3 325) . Ei ' C,tO *a .. d[ 'k....-- . . 4 %. . ." , ,
- 9, . . e;=
2 .e,W ~ , ir_ ....... .se, / s ,, - ^ .. .. , %io c ......%y~~... A, ,, w, i ~ o a.i oIo~iYIii'Es"[ .'s o 3 .o ii i e i 3Zs ev e e.s r a s.cv ......: f ir. 2. Half space snifreo end damprn; p.n ametem Cf f (dashed hnes) and Baranois I.s)cr stifTncss and damping par meters Si . ifuil hnN ti'ow6 catio shown in p.irenthesis) s - - . , ',)/ 16 [) 7 /) . f.) e BEREDUGO .\ND'Nov.\E: Ei!DEDDED FOOTINGS 481 , in which frequency dependent stiffness constants k.. - Gr, (C., + +G s S., ~ 2 ~ [12] < kg 4 - Gr.' C.,i + G' Cg, + G' g JSgi + g J 62 ~' 7+ -J { S. l k,4, - - Gr, :,C , + g' 6 :, i S,, , L 3 and frequency dependent damping constants ' ~ cm Gr* , C.2 + G' g - J S.2.
- ~
Gr'* c44 - C42 +{:'\' C.2 + G' g JS/2 + G' d { 61 + f:1 , f, g 3 . (l3] < ~ ' _ l -JEr, 5,*, , ~ ~ Gr -i G' 1 c,4 - -
- ,C,, + i s -
g 6 {/ s y I S, i and Eqs. [11] are formally equal to the equations of motion of a two-degrees-ci-frecdom system therefore, k,,, ke;, and k,, are the frequency decendent spring constants. and c,,, e cgg, and c ;, are the frequency dependent damping coedicients. Thus, further solution does not r present any difficulties. Ca!culation of Vibration Amp'ituiles Forcea' l'ibration \ Constantfcrce excitation is cons,dered first. Note 3f* a, - kg; - ful- k,; l ' 9. ; 3I* ^ a2 - (c;; - c,j) o Go [1 41 p, k,, - ,i:e/ - -- f*7A s . O D2 - (c,, - -l-- c.;)u t, = vila" - (1,.4 4 ; + Ik,, + c,,cq; - cl;)ni + (k,,k ; - L li) 4 c, l = -(ing; + Ic,,)n' + (c,,le; + eg;k,, - 2csks)w l t i _,.~--. cA .-~.'.I -- _i 482 CANADIAN CEOTECl!NICAL jot! RNAI Vol 9.1972 Then, the compicx vibration amplitudes are from Eqs. [11], a u' - Q* c,, ++ ic, ia, [15] d y,- M,e, D , i ++ idsit, or : i u, - u, + iu, - g , a,c, + a,c, ;- + 10, a,c, . a nc, sl + ti [16] < tl + ei . v, - v, v. av~, = Si, 8: C i 88 ,- 2 + iSf, 82 8: ~ 8s. t tl + ti , l et + ci Th. e real sibretion amplitudes u, and v, from Eqs. [16] are , 1 , u, = 4'uj + ui - Q, "I+"k ' [17] < 'i + Ci 0 ' , v , v'v! + vi - M,Tlcrl + ciAk When the motion is excited by a moment alone. O, = 0 and Eqs. [!7J simplify to: e l ki; + c!:o' u, - M,31 - l, g t ej, [18] < v, - M,3l{k - inc>')'u + c' v>* , s, ,2 The phase angles are: r &, "2 - Arc tan un - - Arc tan "'#2 ~ "'#\ 2 [,19] < a,ci + a,c,/ ^~ &". - Arc tan ' - Arc tan '#2 I-vi (# c2 -+ B2 O './ c c.nd the real motion of the center of gravity is: u(t) - u, cos (at + &,) [20] t p(t) = v, cos (or + c;) J As in the case of uncoupled inodes, dimensionless amplitudes .4 s = v,Cri'.tt, may be introduced to facilitate the presentation and analysis of the results.- u.Cr,,/0,and A Frcquency rariab/c cacitar va, often encountered in practical cases, can be casily introduced into the abose formulae. Awume a frequency sariable horizontal escitation, caused by unbalanced rotating mass ut,. acting at a height 2, abose the center of grasity. Then, [14]-[20]: f; p ._,d b' f3 y /Q BEREDUGO AND NoVAK: E3IDEDDED FOOTINGS 483 Af' Q, - m, e c>t, A1, - m, e c>1 :,, and - :, (2l} 9. - in which e - rotating mass eccentricity. ' dimensionless vibration amplitudes are, in o e case of frequency variable excitation % i,,mlm,e and 4 - y,J:m,e:,. From Eqs. [17] and [18) the complete response of embedded footings in coupled motion can be computed directly and relatisely casily. With the use of approximate expressions for the side reactians S gisen in Appendis 1. the computation becomes esen simpler. a The uncoupled modes of sibration (Nosak and Beredugo 1971) and the known solutions of ' surface footings (Richart et al.1970 Ratay 1971) are special cases of the solution described. It may be noted that an alternatise direct calculation may be used in, which the comptes amplitudes u, and v, are separated into their real and imaginary parts beforehand. This ap- [ g proach leads to four simultaneous equations with real coeilicients; howeser, the computing requires more time. From the motion of the center of grasity, the horizontal and sertical components of the motion experienced by the surface (edges) of the footing can be computed. The upper edge of g the footin: esperiences scrtical amplitude w, and horizontal amplitude u, that are: b w, - r,y, 122) i u, - u, + ( H - :,)y, (In the last formula. the phase difTerence is neglected between u and v.) ! Natural Frequencies and 31 odes l In addition to the computation of the complete response. the natural undamped frequencies and medes of free vibraticas can be of interest and are useful in the direct re:onant amphtude calculation described later herein. The equations for the natural frequencies and modes follow from Eqs. [11] by putting the 8 damping coeflicients ne . c;#. and c,# as well as G,, and Af, equal to zero which yields. in ' terms of real amplitudes. 'k ,, - me>2 kg .[ u, ) - 0 : [23} kg Lu - Ice _ ( v, j. 2 l The two natural undamped frequencies ce i and ej are found from the condition that the de-terminant of the coctlicients must be equal to zero, which sields: t I [24] ce ;,3 - + T - - + 1 From this equation, two natural undamped frequencies can be found by a trial and error i j procedure because coctlicients A are frequency dependent according to Eqs. [12]. With the'c - two natural frequencies ci, () - 1.2) 'he two sibration modes teigensectors) are, from Eqs. l ! [23l: f "' \ [25] ed l \ V /' A de>:) - *C'l - k d e6) i an '
- with j - I or 2. (These equations provide.a quick check of c4. With correct values of ciu p both equations give the same results).
j ' " ~ ~ 4 .. - ) I I I 484 c r. u>ux crortcasicAL Jouns.m vot. . ,r 1 The first mode represents . ... . I in about a center lying under the footing base. In the second ', - rnode the rotation takes place about a point lying above the center of gravity (Fig. 3). Th modes actually have a meaning of radii of rotation. Examples of Theoretical Response Curres Several examples of theoretical response curses computed from Eqs. [17] with quadra , excitation according to Eqs. [21] are plotted in Figs. 4-6. t* St uc0E . 2*NO WCOE d' NT*, . u*- , ' * . v t.we e' -Dage ' , '... y. J .- i ,! s. f. ,. A.1o9. .Y N u. m.. - . i! N
- u. ni '
E :"* t! . no
- a. I .
Fro. 3. Modes of free vibrations. L i 50 , . 40- -3o i 20 I ,e e t.00 g *
- f.12
! so- , E e 0-T
- 5 * * '*
ha x = c' \ { }* \ k^- w 2, .* l' w *o o e & coo ~ / m f e- , . a* Oe
- o. s os
- o. 4 e 02 03 04 OS C6 -7 DME%$i %LESS FRC ul%CY d.8 0,
- r, w pra Fac. 4.
Theoretical revonw cunes
- = 0. Bl = 4.0 B; ~ 4.35, sarious embedments).for borizontal translahon of footing in coupled ,
l.0, monon (1 [!% tJ/ __J7 O \1 fl' . = I I BERCDLlGO AND NOV.sK: EatcEDDED l'OOTINGS 485 i Ond 3, , The 20 " llic ' i 10 8.0 p O. O. D y,o i Sa
- 8.0, 8,, e8470 2. 7, Op
- 2 3 19
- 6. 0 C#'*
- L O S , Ze / r,
- 1. 8 2
. N/r ee 2.0 - c I y 2,40 c '. .{ 200) E 5 \ Z, gp \ g r Zl , t, w- t N U 2.0 L @ J, 1,_ a .vr. ,0 0 1.0 / j $06 'n o / .L ,Y ,9 0.7 -f ~ // Z 06 [ 23 j' k O.S c' l 04 ' e 6 03 0.2 f Oi
- 0. 8 02 03 04 05 06 07 08 09 to II DIMEN SIONLES 5 FREQUENCY e,=w e R Fsc. 5. Theorciical respense curses for rocking comrenent of courted motion H = 1.0 B, = 8.0 sarious embedmentw
- m , CO.
a In these figures the horizontal translation and rocking components are plotted for .arious relatisc embedments J = l r... The mass parameters u<cd are shown in both the standard h, . and modified B, g shapes (Rich' art et al. l970). The modif:cd mass parameters are show n here to reduce the dependence of the results on Poisscn's ratio. Two sets of ma<s parameters were used to indicate their etTects on the character of the response. Figures 4 and 5 illustrate the steady response of footings embedded in undisturbed soil. Figure 6 illustrates the c!Tects of backfill. The properties of backfill C,. p, were introduced using an approximate expressh n (lleredugo 1971) G,.G y (pgp)t Thus. ratio if = p, p - I denotes embedment in undisturbed soil. Sescrat conclusions can be drawn from Figs. 4-6L The response of embedded footings is dominated by the first resonance peak. which is thus of major importance. (For surface footinp this was also obscrsed by Ratay (1971). The response in the region of the second resonance is, in general, much less pronounced and does not sary too much with embedment. , ISee aho ri;. 9. ' .i , e n (} L.. ') ;Q O ~ l. i 486 CANADIAN CEo7ECIINICAt, jot!RNAL. Vof 9.1S12 In most practical cases. the second resonant peak is entirely suppressed: it can onl j cognized in the rocking components with very hich mass ratios (Fic. 5).
- The incicase in the first resonant frequency and the decrease in the corresponding res
} amphi.de due to embedment are quite drastic. Both of these clTects are smaller in the case of backfill. t The variations of the resonant amplitudes and ' resonant frequencies with relatise embedment and mass ratios are further illustrated in Fics. 7 and 8. Resonant amplit.ude ratio R, and reso nant frequency ratio f R represent the relatise variations of resonant amplitudes and t'requen . l due to embedment related to amplitudes and frequencies of a surface footing. These f apply exactly just for the parameters used in the computing: howescr. they indicate the trends l to be expected in any particular case. Furtlier parameter studies rescaled that the resonant i ' amplitude ratio is practically independent of the relative height of horizontal excitation , N and only weakly dependent on the modified mass ratio with any particular embedment. Com-Ii parison with pure rocking and horizontal translation (uncoupled motions) indicates that coupling increases the resonant horizontal amplitudes and decreases the rocking amplitude . and 1971). resonant frequencies. (Similar observations were made with surface footings by Simplified Desi:;n Anal3sis The calculation of natural frequencies, vibration modes and amphrudes of forced oscillatio ' from the abose formulas can be considerably simplified if the stirTness i parameters C and
- i are assumed to be constant and if the damping parameters C., S, are assumed to be propo 90 w*Qo D,
- 4 2. 7, 6,
- 2 3. O 60 I 8a ae o,s ,* e 70 zeir,
- s ca , 2, se, o u2 [
- N /f, ' t. 3 l 4o '- l 1
- l g
e=e o r,l [~ Q W . 1,i T-~ " u. ! ' { l } c H w o 6 Ze 2 o ,, j t ? Io l' s ~ ~s a 80- - (d # r.~ S e lse, I 6.0 m m3 s IO \ 0 ,I it i Y Ao t=1sI lf l a l ' l 0.1 02 03 04 c5 06 o7 08 }.5 e9 to cNENsioNLcS$ FRe XENCY o, e e., r, F K,. 6.
- 0. 75, B, - S.0, B; = 8 '0). Theoretical re pome curses for heruental trand2 tron in coup;cd rnohon of t ,
I n 7n9 i ) / , i. h l 0 I f 8 i , BEREDtJGO AND NOVAK: ES! BEDDED FOOTINGS G7 ! l ,o ,... - l . ---,.on ..o o ; o.. , so , .. O @ ; e ooe-ti" ots_a 5 ., sonrarers reanstaroe , , 20< p" o r* .o, I / $ I. ! ,/ o
- w_ - -
e .'s i i ae. ,s se - eo - ,m o. s",,s "~_ g S , t i ,. e p ec 5 3, 'o ' re 9 5
- ,,s # $03' o g . ~ - ,.
,e, o2 I a s e s a r o i s a s , ,e ,r e rocer< 3 wass maro e,. ',g g '"' " "'## * *
- uo es E8 '
Fac. 7a. Efrcct of nuss ra io on thet rctical resonart amphtude for horizontal translation in cour d motion. . Fac. 7h. EITect of nus, ratio on thcorctical resonant frcquen;y of footm; in coup!cd motion tn - 1.tn. e 24< a 23 22 @rronTAL inAN$ TAT 4Pe g ~ . .
- 2. s 7 a to 9' , o 9 e 420 - - - 3. ors c /
- > l.9 , /
Y yof g i.e r ,.T l6 / / ' s / > > .* b is C / // I E f. 4 / ! l.3 I'< /p
- 8. l
- f/
14 0 CLS 1.o s.S tWater4NT RATIO $ . 4 / r, Fic. 8. Tlicoreneal resona nt ficquency ra no n. embedment for footin g .n couricJ motion. . I to dimensionicss frequency a.. It can be seen froni Fig. 2 that tl ese assuniptions are quite accent-abic for practical appheations except for w:ne side reactions at scry low frequencies. Thus. ; stiffness paranieters can be considered approxiniately constant: e i o ,9~ -- - __ +_ O i sI l i 1 485 CANADIAN GEOTECHNICAL. JOURNAL. VOL. 9.1972 [26] C,, - C,,, C,, - C #i, S , - 5,',, Sg , - fg , and with these values the stitiness constants, gisen by Eqs. [12), become frequency independent. The damping parameters in Eqs. [13] can be approsimately taken as: [27] C,,, - Ca a,. Cg, - Cg ,a,, S,, - S,p,, Sj, - Sgg, I The parameters denoted by bars are constants. Their suitab!c values are given in Tables I and 2 in Appendix !. . Substitution of Eqs. [27] into Eqs. [13] yields the frequency independent damping constants for embedded footines: I l p' G \ c,,= VpGt,i lC,,, r 6I p fS a I - A' cgg ~~ v' W r* , g, r Ca +6 5g, + * ~ [28]j l 62 . :, - h ~ , 3 + ,; o ,a Sa ? . ) b c4 = - VpG r*, :,Ce + 6lp' \p G l ) '~ G 2') "_ With frequency independent stitTness and bedded footings. . vt surface footings, the damping parameters the calculation of vi. constant parameters given in Tables I and
- bration arnplitudes is as follow s
- (i) with 2 yield smaller resonant amplitudes. This n
( values cf C and Si taken from Tables I and desirable for reasons discussed in the next 1 2 (or read from Fie. 2) stifine*> costants. section. (If desired, a perfect tit can also be ^ are obtaincd frem 'Eqs. [121 and the two obtained .'at surface footings by the proper natural frecuencie< from Eq. [2 'j: t,i) dampine choice of constant parameters). coetlicients' are ct ,puted fro m Eqs. [25J The eirect of embedment may be reduced with C, and S, taken from Tables I and 2. and by an imper fect bond between the footing a.#. and e 'obtained from Eqs. [14] for and the soil and by backiill. These erl,ects can any escitation frequency of interest. Substitu. be accounted for by considering G, < G and tio'n of the3 saluc3 into Eus. [17] yields the p, < p in Eqs. [12] nd [2S]. amplitudes of horizontal translation and The_ choice of. equisalent radius r, for rotation at the center of crasity. Eqs. [22] rectangul r fo tings is rather uncertain. Some cive the m0tions of footind edces. indications of the possibic difTerences are Very often only the amplitud'es at the first menti ned in the nest paragraph. resonance with tw need to be found because these resonant sibrations cause freq uen t ' Comparison with Esperiments diflicul tics. (The amplitudes at ci, are slightly Relatiscly few experiments base been con-smaller then the maumum). ducted with embedded feelings subjated to Exaniptes of response curs es computed with herieontal excita; ion despne the fact that thi.. constant p a r.: m e t e r s are show n in Fig. 9. is the ca3e of major practical importance. The agreer"ent with tiie reschs ot,tcined with To proside more data. a series of field tests variable parameters a sausfactory for em. has been carried out at 1 he U nis ersit:. of 7" [). 'iit O / J v, d. a -. I ( d BEREDUGO AND NOVAK: E3IBEDDED FOO*lNGS 489 t e as. 3 -. w :ag ennaugtges L "*'* Co4sTapet papaugttps e gg
- r. . e n y . s .,
5.s . bos ,., d I k a \' t5 5 "' i . L. , i , 5 o o os so is oNEN*OLESS FMAENev o. . 8 Fro. 9. Companson of response curves compute' with unabic and constant parameters (n = 0, , -,1). 800 - 80-j ,. 60-
- *50' THEORY 9
(' E y, 4 o +-+- FIELD ' T ESTS ( SCU ARE FCO ilNG)
- 30- g g* aso 7mE0Ryi F,Jpg d 8 Ze "e
- 70 * '5 i5 $ 20- 2, /r, 1.90 1.90
,' g M/r, 3.17 3 s7 g se w 0.0 0 39 j D, 13.7I 15.48 f *- l e, 3.0 3.09 g 10 ' ,C,lj /qg op i7.07 is.s o d38 d - E8 ,; li) , f\ Es 5 do E 7' To \ 'g f1 m 6-I \ fa
- \
d y 5' j \ g 4-t .. 4 I \ , 4 of ir 3 5 E 3' l l i \' \A l'P >\ / N \ l e a 2- , j N , %s'\ ,% _ . e / -a.--- - - ,.- me - --e - - N $/ g t* 7% =a. , j ,. o -7 I } / 0.8 3.2 0.3 0. 4 0 *J 06 07 08 0. 9 1.0 1.1 0 OIMENSIONLES S FREQUENCY e , o w r, j
- - Fic. 10. Comparison of theor:tical and rneasured respense cunes for horizontal trans!ation in coupled rnotion
,. 5 ' of square footing Nndisturbed soil rr, = 1.71 lb-in.). g l (. !, I (3 7 /l, G r ..I G '"V) h M i 1 i j 490 cANAotAN GEoTECHNICAL JOURNAL. VoL. 9, 3972 7 ( j Western Ontario with two concrete blocks, linearities. This ef'-et is accepted as a scatter one having a square base, the other featuring in resonant frequencies and amplitudes in , a rectangular base with a side ratio of 2fl. this paper. (Some other ways of dealing with ; l The basearea was 5 ft (0.465 m2)in both eases. nonlinearities observed in experiments are 2 i The blocks were cast directly into neatly cut described in Novak (1971). Another diffi- , excavations. The embedment depth was chang- culty is to choose an equivalent radius ry. - ed by removing the soil in several steps. The for rectangular embedded footings vibrating" effect of embedment into backfill was investi- in a coupled mode. gated by stepwise backfilling of the soil and Figures 10-15 indicate the suitability of l by tamping to two different densities. The the theory and the differences in response of subsoil was composed of about 5 ft (1.5 m) the recta:. gular footings in the two major. of brown, sitty clay underlain by a glacial till directions. of considerable thickness. Shear modulus Figure 10 shows the comparison of theoreti- i of undisturbed soil was found to be 6.6 x cat and measured response curves of a 3 10' lb ft2 (320 kg em2) and Poisson's square footing. The equivalent radius r., - ratio was 0.38. Further details can be found 1.26 ft (0.38 m) was derived from the equality in Beredugo (1971) and Novak and Bere'dugo of footing bases. In Figs. I' and 12 the first , (1971). resonant amplitudes and frequencies are : The comparison of the theory wi;h the compared. It can be seen that the resonant experiments is complicated by distinct non- amplitudes are predicted much better for 110 9a 80 FrrtS TrPs
- y
- l. 0
,E g mg . 3,19o lh i, Ze / r,
- 4.71,Z, /r,= L90 do g ,, , 3 , ,7 30 r, = f.2 6 f t
- O eg *0 85 I w 8 ,,
e er .e.ra e o I s' . I l y IO f t e i g; 6 mom a Z! - xt 4 m S e H O ( l J,zl1 Y tc L i , 9 4 y 3 e d r. 5 E0 s 2 >.'O E O l.0 20 25
- eMBEDVENr Rario 8 = 1/t, Fac. II. Comparison of tt eortitcal and measured resonant amplitudes for horizontal translation in coupled motion (undisturbed soil, square footing, , = 0.38 b, = 15 . 4 8, b4 - 18.50L 6, ,h f), ,' r t !a
491 REILEDtlGO AND NOVAK: EMBEDDED FOOTING 5 er
- o. e ' ,
in th re fe o.1 ' e 5 s of I I . o .e ' * ,o h g. (i. 0 ' a 5 o os t..1
- se E
re nt ' $o* o
- or
@ F'Eto 'ESTS $ 03 g o eg,* 0 85 's- da ' {, E e es,
- i : s 10 - sn 5
2, /r,* e * , 2, / r,
- s. DO g
o.2 n n/r,
- L 17, r, e s.26 't
- 011. 0,
- i5 a 6,3,*'8 S o
- 0. 9 10 15 to 2S 0 0. S Ev er0ptNT Rario 8 *ter,
- 1. 0. . Fac. 12. Comparison cf theorencal and mcasured Tescc2. t frequencies in coupled motion Dr square footing, e = 0. 33 b, = ( 5.48, bg <* 13.50). the embe'dded foetings than for the surface cive some idea about the differences in re-ones. This important obsersation probably sponse in both major directions. Smaller can be attributed to the oserwhelming erTect amplitudes in the direction of the longer axis of geometric damping with embedded foot- can be recognized despite the scatter due to ings. The first resonant amplitudes of sur- nonlinearity (Fig. 14), while a somewhat face footirigs seem consid:rably oserestimated smaller relative increase in stitTness due to 'because the ceometric damping is sery small embedment can be seen in the same direction and the hysteretic damping is omitted in ( Fig. 15 ). 'the theory. The prediction of resonant fre- Until better guidelines are found it seems quencies appears quite reasonable too. Rec- that the equivalent ratios for rectangular tengu/ar footinp also show a better agree- footings can be derived from equality of base ment in resonant ampiitudes in the case of areas for translation constants k,, and c,, embedment (Fig.13). The .hoice of equisa!cnt in Eqs. [12] and [13), and from equality of radii for embedded rectangul.tr footings is, base moments of inertia for the other con. g of course, questionable. Figures 14 and 15 stants. Useful data on rectangular surface
- G
" r > ~1 (J ' r i jdj
- c ~~
I , 492 CANADIAN GEOTECHNICAL JoURNA1.Vol. 9.1972 l k 10 0-SO' APPROEt8dATE THEORY ., e- o- FIELO TESTS tRECTANGuLAR FOOTING ) -e0- a I g/r, TMEORY 1.70 Mkf.e b a { Zvre f. 9 0 iSS 7 30 \ H #'* 3'7 s g y I'. 00I 7 0.38 4 D, \ <.57 IS de p 9, f.O 3 09 z20 D 569 9.31 \ [,w 2. e 3 2.16 4 \ N \ 3' e i e' %/\ j 5' . o I* b N I y On of*
- l 1
a v 06 af / .o/'e \ j/ g / g N g */ p ~/ K.
- On .
l,Y , ep/ &l f \ O3 N , 0 gf i .' 7 N N y02- ! \ e / , c- /g - s s / //c N ( / / / / A l \ N Oe' / ! O. 2 03 04 05 06 07 08 09 10 el OwtNs.ONLEs1 FREQuCNCY e,* arr, & I Fic. 13. Comparison of theoretical and measured respom cur es for rocking in coupled motion of rectangular footing tindisturbed soil, rg,, 1.71 lb.in.). footines were derised by Kobori er al. (1971) this approach are its simplicity, the case with and may be helpful tacether with the intui- which parameters can be chanced and the i tion of the designer. Equisalent base radii ability to conuder layering and to introduce for surface footines can be found in Richart the soil reactions into the solution of any et al. (1970). structure. Field experiments were carried out with Summary and' Conclusions c nerete embedded footings subjected to horizontal exc:tation. The theoretical and Coupled forced sibration in horizontal experimental results were compared. , translation and rockinc of partially embedded The major findings can be summarized rigid footines was insesrigated both theoreti- as follows: cally and experimentally. (1) The response is usually dominated by An approximate analyt;4! solution was the first resonant peak and the second reso-
- used to derise directly usable formu!as and nant peak is entirely suppressed. Despite this, graphs, information about embedment into the omissior, of coupling leads to consider-backfill and relations ben e n uncoupled and able errors in both resonant frequencies and coupled motions. The nujar advantages of amplitudes.
w: mn . e . 'e - hm. , _ I o e P 493 BEREDt;GO AND NoVAC EMBEDDED FOOTINGS 12.0 4 t. - g \ 4 0.0 < g xs . o .... ..T, i.... . .....O..l.. ( x x 3,(v-v ) DEMOTE Arts OF ExctTAriON g g = .g e4 9.0 - \ tje a \ Y . \ g i s < E 8' x- - x \ g E \ * $ 7.0 h ' ( i \\ \ EU o \ ' \ \ S.O a \ \'t \ , \s $40 k ., e . \+'4 a N* G9 o % *+,\#\
- .o Yg i
8 y30 < 4 N'*, \ I 5 2
- +. \
(* 2.0 > \ \ -' \ 4.0 - 2.0 3.0 O ..Q EMilEpvENT RATIO 8
- 2 / r.
amphtude with embedment for rectangular footing in h Fic. 14. Variatien or measured horaontal resonant - coupled motion (undisturbed sod, ce,, - 0.85 and 1.71 lb.in.) e a' < ' , ,"y '"-'t 'h- (5) Equis alent frequency independent y (2) E rn be' ;' i- ,<~ '"- ~a-'nt damping and stifTnesses were derised for re<conse in th -~wn,nr desien purposes. They facilitate the prediction frequencies and d"--< - h of resonant frequencies and amplitudes from o p,lgdes. This etTect is much more pronounced l in coupled motion than in sertical transla- closed form formulas. d ! tion. Acknowledgments d (3) Backfill reduces the effect of embed. The study was carried out as a part of a ment. This can be accounted for in the broader research procram at The Unisersity y theory. '- (1) The approximate theory is better able of Western Ontario and supported by a to predict the coupled response of embedded crant-in-aid of research from the National - footines than of surface footines. The theory Research Council of Canada to the senior i of surface footines considerably'oserestimates author and by the award of a Commonwealth ' the resonant amplitudes. Scholarship from the Canadian Federal Go- , an m ,3 I w - / 4 494 CANADIAN GEOTECMNICAL jot!RNA! VOL 9. atn 4 l 1, x -' , t . -u ! 3
- 9 l y Y -Y X-I Y 3 8 4*
e a e g,
- 0.S S 80- M. L I
y o a es,. , .7 s en- m. w j 4 , (1L 8 e s t' i ^ 5
- . O a
i ,j ' O O.S tO
- i.S 2.0 EM9EDMENT RATIO 8 o f/ r, Fac.11. Variat.on of measured resonant frequency ratio with embedment for rectangular footing in coupled motion (e = 1.0).
I sernment to the junior author. The assistance Lysuta. J., and KtHLturytsi. R. L.1969. Finite dynamie of K. Sachs is gratefully acknowledged, model for innnite rnedia. J. Eng. Meth. Div.. I roc. (' A.S.C.E. 95, No. EM4. pp. 859-877 1971. Finite dynamic model for infinite media: Closure to discussions. J. Eng. Mech. Div., Proc. BARANov. V. A.1967. On the calcu!ation of excited A.S C.E. 97. No. EM I, pp. 129-131. vibrations of an embedded foundation. firi Russiani. Novae. M. I971. Data reduction from nonhnear response Vorrosy Dy na miki i Prochnocti, Poly technical ', institute of Riga. No.14. pp.195-209 curves. J. Eng Mech. Div., Proc. A.S.C.E. 97 No. EM4. po 110-1204. - Bratouco. Y.1971. Vibrahon of embedded symmetric Nos AK, M.. and Utarocco. Y. i97t. The effect of ern. fooungs. Ph.D. thcsn. Faculty of Engineering Scien. bedment on foonng ubrations. Proc. First Can. cc. The Unnersit> of Western Ontario. London. Conf. Earthquake Er g Res., May, Unnersit) of ' Canada. British Columbia. Va couser. B C., pp. 111-125. BvcacrT. G. N.1956. Forced ubrations of a rigid cir. I cular plate on a semi.ia6nsie clasoc half space and 1972. Vertical ubration of embedded foonngs.J. Soil Mech. Found. Div., Proc. A.S.C.E. Dec.1972. ' on an claine stratum. Philot Tra na.. Royal Soc.. , Lond. Ser. A. 243. No. 948 pp. 327-36R RaTAv. R. T.1971. Shding. rocking ubration of body ' Kosont, T., Misat. R.. and SuztKr. T.1971. The dyna. on clastic medium. J. Soil Mech. Found. D m. I Proc. A.S C.E. 97 No. SMI. rp. 177-192. mical ground compibnce of a rectangular founda. RicHami, F. E., H au J. R., and Wooos. R. D 1970. l tion on a viscoc!asne stratum. Buit. Disaster Prev. i Res. Inst., Kyoto Univ. 20. Part 4. No.183 pp. Vibrations of soils and foundations. Prentice-Hall s 289-329. Inc., Engiewood Clifk. U.S. A. l, Leco, Y. E., and Wastuaw R. A.1971. Dynamic Tuiut. H.1969. Dynamic anaI> sis of a structure em. ! i response of circutar footings. J. Eng. Mech. Div.. bedded in an clasne stratum. Proc. 4th World I Proc. AS C.E. 97. No. E MS, pp. 1381-1395. Conf. Earthquake Eng. Santiago, Chde. Sess:en A-6. pp. 53-69 Appendix 1. - StifTness and Damping Parameters C and S To facilitate the computation, the stiffness and damping functions C and S were appros-imated by the expression, given below. Their accuracy is sufEcient for practical applications. The clastic half space formu'as for C were computed with Bycroft's (1956) functions fu Functions S for the layer were calculated from ceneral equations derived by Baranov (1967). r s.u 1,
- w. (
. _1__ 495 statocco Axo Novanz zarsenoto rootixcs In all formulas dimensionless frequency a, = e;r,VpIG. Constaat parameters C and S can be I I used in frequency range 0 < u, < 2. The parameter 5,i exhibits pronounced variations with a, for Poisson's ratio r y 0.43. 4 Tasu I - i Validity Constant range parameters r Half space functions Mori:ontal translation 89.09a" -= C.: 4.30 C i = 4.571 - 4.653a, - a, + 19.14 , 0 $a,$ 2.0 0.0 -
- 0.1345a, C.2 = 2.70 C.2 = 2.536a, - , g ,93
~ 10.39a. ~ C.:= 5.10 - ' C.: = 5.333 - 1.584a, - , ,,333 0 $o,$ 2.0 v 8 0.5 0.174 ? L C.: - 3.15 C. = 2.923a, - _ g ,97 1 Rotation about hori:cntal a t os irocking) C Cg , = 2.654 - 0.196k - 1.729a2 , - i .485aj - 0.4881a' - 0.03498a?, 0$a,$1.5 .-h " 2.50 ,g 0.0 2 C_J 2 = 0.4 3 Cg, - 0.008025a, - 0.01583a , - 0.2035a! - 1.202aj - 1.448aj 4- 0.449f a3
- C TASM 2 Validity Ccnstant 1,a : ' , range parameters _
y Side layer functions oc. Hori:ontal translation nse 3.6090., 0.6,9 J 97 , S.: - 0.2328a, - a - 0.06159 -_ m- S.: - 3.60 an. 0 $a,$ 0.2 0.0 S.i = 150.* , - 3630ai - 3948a?, - 1934a.4, - 3488aj ' of 25. 0.8652a, 0 $a,$ 1. 5 f.:- 820 J. S*J 7 33J3 - a,-0.00$h 9
- 72. 0.2$a,$2.0 5.i - 2.474 - 4.119a., - 4.320a . --2 2.057a?, - 0. 362a.4 D S.: - 4.00 L 0 $ a,$ 0. 2 0.25 S. -1.468 v'a$ - 5.6628a', I~
- 70. 41. 5 9a,. 0 $a,$ 1.5 S.2 = 9.10 au S.i = 0.83a, 2 3.90 e o t t
in. . . . _ _ . . _ _ _ . . . _ _ _____ 0.2$a,'$2.0 S.i - 2.824 -- 4.776o, - 5.539a ,2 - 2.445a?, - 0.394aj Md S. = 4.10 $ on - 1.796 V'a, 6. 539 Ma., 0 $a,$ 0.2
- 0.4 S.
- 56. 55 a., 0$a,$1.5 S.: = f0.60 ,
S.2 = 0.96a, -- _ b
- Rotation about a hors:ontal a vis (rockine p g* 2 7.165aj - 4.667a ,i-- 1.093a?. Sh = 2.50 '
Sh " 3.142 - 0.4215u., - 4.209o , 0<a,$l.5
- "I ~
2 "I"# 0 Ol44a., 5.263a.i - 4.177aj i .M3a*. - 0.2542a! 542 - 1.80
- 9. 5,4 I
I )M Y J i h 4 / ^ 496 CANADIAN GEoTECHNICAL JoURNAt Vot 9,1972 Appendix II.-Notation m~ A, - m,e u, - dimensionless amplitude of horizontal vibration m A,. - u,, - dimensionless resonant (tr.uirnum) amplitude of horizontal vibration m,e i Ay - y, - dimensionless angular amplitude of rocking i Av., - m,e .v. - dimensionless resonant (masimum) amplitude of rocking
- o, - wr,@ - dimensionless excitation frequer'zy a,,, -
dimensionless resonant frequency (at maximum amplitude) B, - b,(7-8v) 32(1-v) - modified mass ratio for horizontal vibration B; - 3bg(1-v)l8 - modified mass ratio for rocking motion b, - m/pr/ - mass ratio for horizontal vibration. , bg - I/prf - mass ratio for rocking motion C. ,, C,,, - clastic half space stifTness and damping parameters for horizontal translation C,i , C.2 - frequency independent ha!f space stifTness and dampiag parameters for horizontal translation Cei,Cg, - clastic half space stitiness and damping parameters for rocking Cgi,Cg, - frequency independent half space stitTness and dampine parameter; for rocking c,, - equivalent damping constant for horizontal component of coupled motion c,4 - equivalent " crass" damping constant for coupled motion cgg - equivalent damping constant for rocking in coupled motion C , e - eccentrioty of rotatinc mass I f,,,f,, - compor'znts of Reissner's displacement fonction for horirontal vibration fgi.f;, - components of Reissner's displacement function for rocking i G - shear nodulus of clastic half space; shear modulus of undisturbed soil beneath j footing G, - shear modulus of side layers; shear modulus of backfill g - acceleration of grasity g,, - m,; - weicht of rotating mass
- - heicht of footinc Hl2' - Ha'nkel function'of the second kind of order n - J, - ir,,
h - thickness of clastic stratum / - mass moment of inertia about horizontal axis passing through center of gravity i - v' I J o.1,1, 3 Bessel functions of first *:ind of orders 0.1. and 2 respectisely A,, - equisalent spring constant for horizontal component of coupled motion A ,4 - equivalent " cross" spring constants for coupled motion , kg; - equivalent spring constant for rocking component of coupled motion L - lencth cf footinc ~ / - depth of embedr'nent of footing M - amplitude of excitation moment MD) - excitation moment about horizontal asis m - mass of footing; mass of footing and oscillator m, - unbalanced rotating mass N,(t) - horizontal side reaction due to embedment in coupled motion ,m 6 n, .c h <
- J
-p-l _ 497 sensot;co axo sova.2 assascoro rooriscs ( l Ngt t) - reactive torque due to embedment in coupled motion Q, - amplitude of horizontal excitation force . Q(i) .. horizontal escitation force ' y - (1 - 2r) 2(I - r) - function of Poisson's ratio R, - resonant amplitude ratio (relatise herease in resonant amplitude) 3 Rf = resonant frequency ratio (relative increase in resonant frequency) j 3 R,(t) - horizontal reaction in the footing base . Rg t t) - half space reactise torque in coupled motion
- r. - radius of cylindrical footing: equivalent radius of rectangular footing j S,Sg - Baranov's ricidity parameters S,p S., - side layer stiffness and damping parameters for horizontal translat:on S;,,Sg, - side layer stiffness and damping parame:ers for rocking 5p5,,, - frequency independent side layer stiffness and damping parameters for translation ,
SgeSg, - frequency independent side layer stiffness ;rnd damping paran eters for rocking . t - time t u, - u, +iu, - comptes implitude of horizontal displacement I u,, - resonant (maximum) amplitude of horizontal displacement u, - re.?! amplitude of horizontal displacement . u u, - real and imacinary parts of complex amplitude u, ', r u(t) - horizontal displacement ,n x., - n,4 parameter d:pending on e; and r 3g
- . - heicht of horizontal excitation force abcse base of footing
- , - height of center of grasity above footing base Y,, - Bessel functicns of the secc nd kind of order n
'E :, - height of horizontal ewitir e force abmc center of gravity J - / r, - embedment ratio n - p,rp density ratio C r - Poisson's ratio ' es - circular excitation frequency ei,,, - frequency at maximum amplitude c;j - joi undamped natural frequency - first and second undamped natural frequency e>a
- ,.o; - phase angles
- v, - vi +iv, - comples amphtude of angu!ar (rocking) displacement . v., - reasonant amplitude of angular displacement vn - real amplitude of angular displacement voy, - real and imaginary parts of v, v(t) - angular displacement p a mass density of clastic half space: mass density of undisturbed soil j ,y p, - mass density of side layer; mass density of backfill , I I 5 I ^wm ges .