...(5.29) ...(5.30)
DIving Eqs. (5.28), (5.29) and (5.30), we get W w-
<".
(PA)dyn[cos(a +8) - sin (a + 8). tan ~b] (l:tay)tan~b-ah
'utting (P A)dyn = ~ yH2 (KA)dyn and ah
1
= (1 :t:.a) tan 'V, the eq. (5.31) can be written as 2
Ww
= 2" yH
CIE
=
...(5.32)
(KA)dyn' CIE
cos (a + 8) - sin (a + 8). tan ~b "here
(l:tay)(tan~b-tan
\jI)
~orstatic condition, the weight of wall W is given by : W=- 1 y H 2 K .C 2 A I where Cl
rherefore, >ubstituting
.e.
- sin (a
(KA)dyn .
...W
'.
KA
...(5.33)
...(5.34) + 8) . tan
tan J..'l'b
W ~=
~b
tan ~b (l:tay)(tan
~b -tan \jI)
...(5.35) ...(5.36)
FT = Ratio of earth pressure coefficients in dynamic and static cases (KA )dyn FT
=
md FI n Eq. (5.36),
cos (a + 8)
=
...(5.31 )
W ~=F W
=:=
KA
. . Wall InertIa factor T
F=F I w
...(5.37)
=
tan
...(5.38) ...(5.39)
y,
~. i
204
Soil Dynamics & Machine Foundations~
;
,
1
F w is factor of safety applied to the weight of the wall to take into account the effect of soil pressure -~ and wall inertia. Figure 5.17 shows a plot of Fp F: and F w for v~ous values of ah. From this figure for t.' , F( = 1.0 and Fw =1.5, the value ah works out to be 0.18. However, if the wall inertial factor is considered, ;~$i the critical horizontal acceleration corresponding to F w = 1.5 is equal to 0.105. Thert:fore, if a wall is :' I designed such that W w = 1.5 W, the wall will start to move laterally at a ~alue of ah
= 0.105.
Hence for
no lateral movement, the weight of the wall has to be increased by a considerable amount over the static condition, which may prove to be uneconomical. Keeping this in view, the actual design is carried for some lateral displacement of wall.
i . j
I II
,. ...
I
14 fw 12 '" 10
8 3'
u.;
..
u:..
u:
6
/
4
L"/ 2
FT = 1.5
':.--- ==.
oD
0.105 I I
0.1
~.
- -
I .10.18
0.2
-=-
..::;:::-
y
./ ññóä T
~ .
0.6
"
~
Fig. 5.17: Variation of Fr F. and Fw with ab (Richards and Elms, 197?)
., , ;1 i ~ . ,, ~ ~.~
205
zm;c Eartll Pressure
Richards and Elms (1969) have given a design procedure based on a limited allowable wall movet, rather than on the assumption that the wall will not move at alL Such procedure is as follows: (i) Decide upon' an acceptable maxilllum displacement, d. .
.
.
(ii) Determine the design value of ahd from Eq. (5.40) [Franklind and chang, 19771
5ah *
-
...(5.40)
- ah ( d )
ahd
where ah = Acceleration coefficient from earthquake record d = Maximum displacement in mm (Hi) Using ahd' determine the required wall weight, Ww by substituting it in Eq. (5.31). The value ahd of avd may be taken as 2'
/ (iv) Apply a suitable safe~y factor, say 1.5, to Ww' There are three limitations to Richard-Elms analysis (Prakash, 1981): ~hese are: 1~ The soil is assumed to be a rigid plastic material. The walls do undergo reasonable displacements before the limiting equilibrium conditions (active) develop and experience very large displacements before the passive conditions develop. 2. The physical properties of the system and its geometry (particularly its natural period) are not considered. 3. Walls may undergo displacements by either sliding or tilting or both. This method does not apparently consider this difference in their physical behaviour, although it is logical to conclude that displacements computed by this method are in sliding only. .2. Solution in Pure Translation. A method for computation of displacement in translation only, of d retaining wall under dynamic loads had been developed by Nandakumaran (1973).
t
Earth
pressure (p)
--+ BF
Active C
Di sptaceme ".
Fig.S.tS:
nt
(a) (a) Earth pressure (P) versus displacement of wall
206 tJ
G
(Rep + Pp!r~
I
x. I
,I . :
I
Base friction
{B.FJ RBP--,';
H
(R8A- PA) J
RSA
oX
I
Displace m ent
Displacement / (b)
x ,°2
(c)
z
T
X,Ot
B "2 tan
-11..
.~B (d)
~
(e) Fig. 5.18: (b) Base-Friction (B.F.) versus displacement (c) Resultant of'P' and B.F. versus displacement (d) Simplified bilinear forces-displacement diagr.am (e) Computation of base resistance
The force-displacement relationships considered in this analysis are shown in Fig. 5.18. Fig. 5.18a shows the variation of earth pressure with displacement. In Fig 5.18b, variation of base resistance with
displacementis given. The net force away from the fill is the differenceof active earth pressurePA and the base resistance, RBA(Fig. 5.18c). The net force towards the wall is the sum of the passive earth pressure, Pp and the base resistance, RBP(Fig. 5.18c). The resulting bi1ine~rforce-displacement relationship is shown in Fig. 5.18d and is characterized by the following parameters:
t I .t I
I
namic Earth Pressure
.
207
(i) Slope of force displacement relationship on the active and passive sides as Kl and K2 respectively, where K2 = n . Kt.
(ii) Yield displacement, Zy For the resistance of the base, it is assumed that a column of soil of height (B/2) tan $ provides all the resistance in a passive case (Fig. 5. 18e), B being the width of the wall at its base. The mathematical model is shown in Fig. 5.19. The parameters that are needed to define the system r displacement analysis are: 1) the mass of system, rn, 2) period of the wall-soils system, 3) yield splacement, 4) damping in the system, and 5) parameters of ground motion. ºÖ´ñ×ùºþþùºþòòô
K
m .,
c y = Y sin GJt
z
=
.
x-y
x
Fig. 5.19: Mathematical model considered for the analysis (Nandkumaran, 1973)
The vibrating mass of the system consists of the mass of the wall and that of the soil vibrating witL e wall. Nandakumaran (1973) conducted vibratory tests on .translating walls and found that for the lIposes of matching the computed frequency of the wall with the measured natural frequency, the soil ass participating in the vibrations is 0.8 times the mass of soil on the Ranking failure wedge. Yield displacement for a given wall can be determined by considering the force-displacement relaonships. The ground motion is considered to be a sinusoidal motion of definite magnitude and period. The equation of motion can be written in the following form (Fig. 5.19) : ...(5.41 a) rnx + C (x - y) + K (x - y) = 0 rnl + Ci + Kz = - my ...(5.41 b) Z + 2 11~i 4-112z = - y ...(5.41 c) .
.
where z = (x - y), 112= Kfm where K has been defined as the stiffnesson the tension side and .
~ = Damping
C ratio = 2~Km
For ease in computations, all the three equations obtained by linear acceleration method (Biggs, ~63) to be satisfied at each instant of time or at the end of each time interval selected, c~n be divided by y, the relative displacement on the tension side at which the resistance becomes constant (yield displacelent) to obtain the following relations: 2
'l'n+_1= 'l'n+ 'i'n: t + t6 (\jJn-1+2\j1n)
..:(5.42)
~~ ~. ~-«(Pn+1+2\Vn)
...(5.43)
~n+1 =
.:
"
JI.~ . j j
:~.;t
I
,
\J. j\
208
Soil Dynamics & Machine Foundations
\Jin+\ =-YIl-21l~ Z
where
\Jin+\-1l2.K.Zy('I'n+
.
'1'=-
!
,
...(5.44)
I)
'1'=-
'1'=-
; !
...(5.45)
Zy Z
...(5.46)
Zy Z
...(5.47) Zy With these relationships, the analysis is performed for the range of variables listed in Table 5.2 Table 5. 2 : Range of Variables Considered in DisplacementAnalysis in Translation Variable
Range of values
Ground acceleration amplitude (a) gals Period of ground motion (T)s Damping (~) %
100, 200 and 300 0.5, .3, 0.2, and 0.1 5, 10, 15 1.0,0.5,0.3 and 0.2
Natural period (TII)see.
1.0, 2.0, 3.0, 5.9, aJ1d10.0
Yield displacement (Zy) mm
To study the response characteristics of the system, two cases were considered; one in which plastic deformation does not take place and the other in which it does. Figure 5.20 shows the response of the elastic system. It is evident from this figure that steady state conditions are attained in about 6 cycles and also that displacements on the tension side are larger than those on compression side. The response of the system wherein slips take place has been plotted in Fig. 5.21. This shows that even when plastic deformations occur, a sort of steady state is achieved in the sense that slip per cycle becomes a constant after about 6 cycles. 12
8 E E
4-
+' C ~
0 Tim
E
~ u
(5)
.E -4a. III 0
-8
-12
t
I
A =300 ga[5 T = 0.3 5 Tn = 0.3 s
Zy = 200 mm Tt. = 2.0 Damping = 10°/0
0.3 s
L.;
; f
Fig. 5.20 : Response of an elastic system with different stiffnesses of tension and compression sides (Nandkumaran,1973)
I 4:
'namic Earth Pressure
-0
,-...
c 0 '" c
12
A = 300 gals .T = 0 3 5 Tn = 0.3 5
8
Zy= 10 mm n = 2.0
'-E "0
I C
0 c
---
209
Damping
=10%
4-
+C E u
-0 a.
.-\11 Q
-
0 Tima (5)
-4
i 0.3s
,, Fi~. 5.21: Displacement versus time (Nankumaran, 1973) -.-.
80 Zy
= Smm
Zy =10 mm
~ =10°,. n =2.0
; =10°'0 n
70
=2.0
60 A 1 T 0.39.0.55
E E ~
u > u
SO
A 0.39
~ 4.0
..c:
toI
E
~
30
0
alii
0
20
I
10
0'29,0.3)-
0.1 , Q.3 -
0.4.
0'.6
0.2
0.8 Natural
0.4.
0.6
ptlriod.S
Fig. 5.22: Natural period versus slip per cycle (Nandkumaran, 1973)
IT . 0.5 r
..J
!10
Soil Dynamics & Machine Foundations
Fig. 5.22 shows typical set of results in the form of slips per cycle versus the natural period of the NaIl in seconds for the yield displacement Z= y 5.0 mm. and 10:0 .mm, ~= 10% and n = 2 for different ~round motions. The ground motion is considered to be an equivalent motion of uniform peak accelera:ion of well derIDedcycles. . Any problem can be solved with the following stepes : 1. Determine the natural period of the wall using the following equation: T=21t~ where K = stiffness on the tension side and In = mass of soil and the wall.
...(5.48)
2. Determine the yield displacement. 3. Determine the slip per cycle from Fig. 5.22 or similar other plots corresponding to the yield displacement, the natural period of the wall and the ground motion considered. 4. Compute the total slip during the ground motion. This method of analysis is better than the one proposed by Richards and Elms (1979) in that (i) definite procedure for determining the natural period of the soil-wall system in translation has been formulated, and (ii) physical behaviour of the retaining wall is considered in developing the force-displacement relationships. The method, however, suffers from the-fact that the tilting of the wall has not been considered. 5.3.3. Solution in Pure Rotation. A method of analysis for computing the, rotational displacement of rigid retaining walls under dynamic loads has been presented by Prakash etal (1981) and it is based on the following assumptions: (i) Rocking vibrations are independent of sliding vibrations and the rocking stiffness is not affected by sliding of the wall. (ii) The earthquake motion may be considered as an equivalent sinusoidal motion having constant peak acceleration. (Hi) Wall may be assumed to rotate about the heel. (iv) Soil stiffness for rotational displacement of wall away from the backfill may be computed corre~ sponding to average displacement for development of fully active conditions. (v) Soil stiffness for rotational displacement of the wall towards the backfill may be computed corresponding to average displacement for development of fully passive conditions. (vi) The stiffness values computed in (iv) and (v) remain unchanged during phases of wall rotation towards and away from backfill respectively. (vii) Soil participating in vibrations may be neglected. The mathematical model base upon these simplifying assumptions is shown in Fig. 5.23a. Figure. 5.23 b shows the scheme for calculation of side resistance corresponding to active and passive conditions. If fully active conditions are assumed to develop at a displacement of 0.25% of height of wall, then soil stiffness Kt in active state is giv~n by ,
=
K I
KpyH2 - KoyH2 2 2 Po-PA = 2.5 H average displacement ( 100 ) .
...(5.49)
.~ 1
t 'f Ii 1, ,1 1 » .'1"""'
'namic Earth Pressure
r-,
b t --t .t-=a~
.
211
B
ti'.x
~
, I I
'
L.. :::J IJ' Ut
+' C C:>I
~
.-
L..
I
I
U
a.:;:
~
(kA or kP) H
I J
kp
.t:
C:>I
... u0
+'
1
\
//R
Backfill
kA
rotation
Lf-"[ . b
~
-1
Q.2SH
Z.5H
F,
lOO."t-
100
Di splacement (a)
Fig. 5.23:
~
(b)
(a) Mathematical
model for rotation of rigid walls,
(b) Scheme for computation
of spring stiffnesses (After Prakash
et al., 1981)
Similar if fully passive conditions are assumed to develop at 2.5 % of wall height, soil stiffness Kz 1passive state may be computed as : Z z Kp rH - Ko rH 2 2 Pp - Po K ...(5.50) 2.SH , z - averagedisplacement ( 100 ) where:: PA Pp
= Active earth pressure = Passive earth pressure
= Earth pressure at rest ,KA = Coefficient of active earth pressure Kp = Coefficient of passive earth pressure Ko = Coefficient of earth pressure at rest Po
The rotation resistances of the base, in active and passive states (MRAand MRP)may be given by M~A=.C,.I'CPA
=
c,. I .
,',
...(5.51
a)
...(5.51 b) n which C, is coefficient of elastic.non-uniform compression, I is moI1!e~t'ofin~rtia ~fthe base about an lxis through. the heel of.the!walland.perpendicularto the plane ofvibra.tiQns;,an~~A and CPB itre angles - . MRP
)frotation'away
and
towards!he~backfill.
'1'-;,"
CP~
U'
,):~
"!';
'i'n!>d':~1;
..'
'-;'
0:
.,.
212
SoU Dynamics & Machine Fou1tdtztio1$ !$ ,. .,
The equations of motion for rotation of wall away and towards the backfill are respectively: .
Mmo ~A +
.
(
C.I-
13
)
4>A= M (t)
...(5.52 a)
4>p= M (t)
...(5.52 b)
-
i.
Z
..
and Mmo Ij>p+
-"
K H2
KzH
( C.
I-
3
J
Since the stiffnesses Kl and Kz are different, the period of the wall for the two conditions i.e. towards the backfill and away from backfill would be different. This would result in different values of 4> A and 4>p for each half cycle of motion and net rotational displacement of (4> A - 4>p)for one cycle of ground motion. The maximum displacement of wall for any number of cycles may be computed as : 4>T =
n
(4>A
-
4>p)
...(5.53)
.H
where n = Number of equivalent uniform cycles of ground motion H = Height of Wall Based on the above, a parametric study was made considering the range of variables listed in Table 5.3 It was observed that the contribution of rotational displacement may be significant. The contribution of rotational displacement using the above approach was compared with the sliding displacement for a 3 m high wall with backfill having angle of internal friction, 4>,equal to 36°, period of ground notion of 0.3 s, Table: 5.3: Range of Variables considered in Displacement Analysis in Rotation Variable
Range a/values
Height of wall (m) Angle of internal friction for backfill (degrees) Period of ground motion (5) Damping (~) C$ base kN/m3 Base width/Height of wall
3.0, 5.0, 7.5 and 10.0 30, 33, 36 0.3 0, 5, 10, 15 4 3, 4, 5, 6 and 8 (x 10 ) 1/3
horizontal seismic coefficient CJ.h equal to 0.25 and C$ equal to 3 x 104kN/m3. The total slip in 15 cycles due to sliding was 213 mm. Displacement of top of wall due to rotation found by this analysis was 147 mm. This illiustrates that the rotational displacement may not be negligible and an attempt should be made to account for it. The displacement analysis for rotational displacement is highly simplified. Nevertheless it shows explicitly that in some cases neglecting rotational displacement may seriously underestimate the total displacement. In actual practice it may be essential to account for combined effects of rocking and sliding that will affect the overall response of the system. 1j
-
5.3.4. Nadim- whitman Analysis. The Richard Elms model assumes a constant value of wall acceleration (C1.h. g) when slippage is taking place. But once the backfil1 beings to slip, compatability of movement requires the backfill to have a vertical acceleration, thus causing change in wall acceleration. .
~~
? ~ ~ I
213
ynamic Earl/. Pres~ure
Zarrab~(1979) considered the equilibrium of the.wall and the backftll wedge separately and.satisfied le continuity requirements at failure surfaces as shown in Fig. 5.24a. An iterative procedure was devel?ed for computing the instantaneous values of the inclination of failure plane, the dynamic active earth :essure and the acceleration of the wall, given the input of horizontal and vertical ground accelerations. he horizontal acceleration of the wall and the inclination of failure plane in the backfill are not constant I
Zarrabi's model. 7 6 S
I W.kn
\
:r I
R
w.~ -w
WJ
1
A= 0.2
w.k~
~
:1
I
N = 0.112 Uniform
G
.2 1
Ww Rw tan ~b Rw
3
f 1f1 (a)
(c)
Contact element Slip element
Slip
element
with Cn=lE8kN/mlm C5=0 with Cn=lEBkN/mlm,C5=12SkN/m/m
with,
Rigid
boundary
Cn = 1E 12 kN I m/m
'4m sea le ~
Cs = 1E S kN/m Im Cn = Normal stiffness C~ = Shear stiffness
,
of slip elements
of slip
elements (b)
Fig. 5.24: (a) Force resolution of wall and soil wedge in Zarrabi's model (b) Retaining wall and its finite clement idealization (c) Effect of ground motion amplification on permanent wall displacement (Zarrabi, ,1979) .
Generally, displace:nents computed with Zarrabi's model. are slightly lower than those computed lith the Richard- Elms model. Dynamic tests on model retaining walls performed by Lai (1979) sh~w lat Zarrabi's model predicts the movement of the wall more accurately than Richard- Elms model. Lai, Iso, obserVed, a single rupture plane in the backfill in contrast to Zarrabi prediction. Later pn;-Zarrabi
110delhas been modified to have a coiistanHncIlnationof failure plane 'in the b~ckfilt> .
'1:
<~
':i...;:'!'>'
":",
',..
~
'
,
""
'i
214
Soil Dynamics & Machine Foundations The Richard,
- Elms,model
and Zarrabi's model assume a rigid
- plastic
behaviour of the backfill
material. Hence the input ground acceleration is constant throughout the backfill. But due to more-orless elastic behaviour of soil at stress level below failure, the input acceleration is not constant. Hence amplification' of motion cannot be taken into account in these models. Nadim and Whitrnan (1983) used a two -dimensional plane - strain finite element model for computing permanent displacements taking into account the ground motion amplification. The slip element at the ,base of the wall has been assigned a very large v
- plastic
(Richard
- Elms
or Zarrabi's) model.
I, is the fundamental
frequency of wall and
I is
the
frequency of 'ground motion. It can be seen that effect of amplification of motion on displacement is greater when I I is greater than 0.3 The FE model predicts zero permanent displacement ,at high frequency, because in the analysis only three cycles of base motion are considered during which steady -state conditions can not be achieved. However, it can be said that large values off If, are not of great p'ractical interest because displacements are very small.
I,
Nadim and Whitman (1983) suggested the following simple procedure for taking into account the effects of ground motion amplification in the seismic design:
'
(i) Evaluate the fundamental frequency I, of the backfill fQf the d~sign earthquake using one-dimensional amplification theory by using the following equation and estimate the ground motion frequency, f
...(5.54)
I, = VI/4 H where H = Height at retaining wall in m VI
= Peak velocityof earthquakein m/s
(ii) If I Ifl is less than 0.25, neglect the amplification of ground motion. If I II, is in the vicinity of 0.5, increase the peak acceleration, A and the peak velocity, V of the desjgn earthquake by 2530%. Ifll I, is between 0.7 and 1, increase A and V by 50%. Obtain ah as A/g. (iii) Use the value of ah from the previous step in the Eq. (5.40) given by Richard
- Elms model for
getting ahd for known value of the displacement. (iv) The value of ahd estimatedin step (iii) is used as the value of horizontalseismiccoefficientin the Mononobe
- Okabe analysis to calculate the lateral thrust for which the wall is designed. The
value of vertical seismic coefficient may be taken as a~d
.
.
5.3.5.Saran, Reddy and viladkar Model. Saran et al (1985) have chosen the mathematical model in such way that it results translation and rotation simultaneously and therefore it has two degrees of freedom. '11 'I, "
'
,
,f
,
In practice, cro ~~ ':' section pf rigid ret~iiningwall vari ~s'to cl great exte!l.~' Areas
,
,
,
,
,b~ckfin soil is replaced by closely spaced independent elastic springs shown in Fig. 5.25. .;
,
,
". ",
. 215
nic, Earth PresslIre
t--°'--t
-
K, hl
Displaced position I
I I
K..,
hl
K...
I
:IT
~ I
h,
Retaining
wall
I
-- - - - - - - F=F
¸
ï
0
j -
sin
I
wt
Kn-"
--.
. . I
Inltla
position
I I I
ill
X
J I I I
.
I
I
¢
¾
¢
.
-Dynamic"'" , --.QC tlve "
Dynamic pas slve ,
Fig. 5.25: Mathematical model for displacement analysis under dynamic condition
To determine the spring constants soil modulus values have oeen used" The s?~l modplus depends on type of soil. It varies linearly with depth in sands and n?rmally consolidated clays, but remains lstantwith depth in case of over consolidatedclays. For linearform of variation- k =:,11It;' h, where 11ft he constant of horizontal subgrade reaction and h is the depth below ground surface. Value of 11ftalso )ends on the type of movement namely (i) wall moving away from backfill (active) an (ii)'wall moving lards backfill (passive.). P,rpbable range of 11ft.in cohesionles .soils is given in Table, 504 ' Table 5.4: Rangle of,values of Modulus of Subgrade Reactions 1h' . TIlt
KNlm3
, ,
.-
-
Soil 'Active Loose 'sand
'200-300
passive ;
'400-60d'
Medium dense sand
400-600
800-1200
Dense sand
800-1200
1600~2400, ,
.
In case of soil modulus linearly-varying with depth, the soil-reaction -is assumed'to 'act as a loading tensity. Treating this load to be acting on a-beam of length, equal to the 'height 'of retaining w~ll,-the actions-at different points are evaluated treatIng this'beam'to
be siniply' suppo'ited:at tIi{spring points, Jr the retaining wall of height H~d'ivided into ~ t6nvenie-nf~~mber ot ~qual ~egrI{erits'of,height -~Ji,the 'actions hence'the.' spring constants vallies' at"~a:fiou~'ciivi'si~h p'oTnts\vould'oe
a~'~n~~r..." " .'
,
,~ #' 16
Soil Dynamic.s & Machine Foundations Dj'
k - 1 1 - 6 l1h(1:1hi
...(5.55 a)
k2
= l1h
k3
= 2 l1h (1:1h)2
...(5.55 c)
ki
= (i - 1) l1h (1:1h)2
...(5.55 d)
kn
=
(1:1h)2
1 "'6
lar en,. da: W2
...(5.55 b)
2
de ha se dr
...(5.55e)
(3 n - 4) T\h(1:1h)
IS
where kl and kn are the spring constants at the top most an bottom most points, ki, the spring constant at my division point 'f.
de S)
In case of soil modulus constant with depth, the soil reaction is assumed to act as uniformly distributed loading intensity. Treating this uniformly distributed load to be acting on a beam of length, equal to the height of retaining wall, the reactions at different points are evaluated treating th,isbeam to be simply supported at these points. The spring cons'tants would be as under,
5. u
.
-21 k (1:1h)
For the top most spring,
k =
For any intermediate spring,
k; = k (1:1h)
..(5.56 b)
For the bottom most spring,
1 kn =. -2 k(l:1h)
...(5.56 c)
1
...(5.56 a)
The method is based on the following assumptions: 1. The earthquake motion may be considered as an equivalent sinusoidal motion with uniform peak acceleration and the total displacement is equal to residual displacement per cycle multiplied by number of cycles. 2. Soil stiffnesses (or spring constants) for displacement of wall towards the backfill and away from the backfill are different. 3. Soil participating in vibration, damping of soil and base friction are neglected. Assumpations 1 and 2 are usually made in such as analysis while assumption 3 needs justification. It is difficult to determine analytically the soil mass that would participate in vibrations along with wall when it undergoes translational and rotational motions simultaneously. Neglecting this mass, the method gives higher displacements and the solution is conservative. However, the mass of vibrating soil can be found out by carefully conducted experim ~nts. For the case of pure translation, Nandakumaran (1973) has conducted experiments to determine the vibrating soil-mass and concluded that itcan be taken equal to 0.8 times the mass of Rankine's wedge. By adopting similar technique, the soil mass vibrating along with rigid retaining wall under combined rotational and translational motion~cim be found out. Then it is added to the mass of the wall to lump at centre of gravity'and the analysis can be carried out' without any changes. .'. '.-
,
.,
iI
~; )"~J I
-
11& III
it
0'
Earth Pressure
217
oils, it is customary to consider valUes of damping such' as 15% 'or 20% of critiCal in view of ergy absorption compared to other engineering structural materials. In the present analysis however, bsorption in the form of plastic displacement of the wall has been considered. Therefore smaller ; values would be appropriate. Neglecting even this smaller damping, the displacement of the this method will be more than the actual displacement. ~ displacementof ;
retainingwall is greatly influencedby base friction. In case of walls in alluvial and' at the waterfront, translation'al'motion,is predominant.In some other cases, the walls may
edominant rotational motion. But in general for any type of foundation soil, retaining wall pos'ansl..tional and rotational motions simultaneously. For'rigid retaining walls, the stability is mainly ts gravity, hence base friction, the analysis will lead to an overestimation of the displacement. wever, refinment of the model by including vibrating soil,mass, damping of soil and base friction ~d so that the analysis can predict displacement close to the actual displacements, study the response characteristics of the 'system, two casoesare considered, one in which plastic ations do not occur (elasti~ system) and the other in which plastic deformations do occur (plastic ). , Analysis of an elastic system active condition. The equations of motion of the retaining wall )' Alemberts principle can be written in general terms as .follows; ;=\ :'(' + Lk;[x+
{H -h)-(i-I)Md
9]
= Fa sin
...(5.57)
(J)t
11 ;=\
+ Lk;[x+{H-h)-(i-I)LlhJ
9) (H-h)-(i-l)
LllzJ]=0
...(5.58)
11
here M = Mass of reitaining wall J = Polar mass moment of inertia of the wall about the axis of rotation (J) = Frequency of the excitation force H = Height of retaining wall h = Height of centre of gravity of wall from its base x = Translatory displ~cement e = Rotational displacement ,etting : ;=\ "f.k; n
"1T F ...JL
M
;=\
,=
a
...(5.59) 4
'..
=a
...(5.60)
=b
...(5.61)
a
,:;
I,kj {(H-h)-(i-l)L\h} n
M ;=\ "f.
n
~
-
kj {(H - h ) - (i
J
'2
- 1)6.h}, '!
t ' , ~\ - - '. - , ,)
=c
'\
...(5.62)
-::r I
218
Soil Dynamics & Machine Foun~
The equations of motion of the rigid retaining wall cai1thu~ be written,as~ , , ,
:
",
be+
x' + ax ~ ,
,
'A" sih ro l' ".
,
0.,
'
"
,','
,
..:.(S.6?).
,
"" ,
,
'
"
b'
e + ca where
J
'"
( "7 )
=,
x';
, ...{S.64)
", ,
,
,
,
= Mr2, r being the radius gyration and 'b',can be called as coupling coeffici~ntbecause if b.=,o,
the two equationsbe~ome,independentof each other.
,.
"
The solutions of Eqs. (5.63) and' (S.64) can be written as X
,,'
, where X and
= X 'sinrot
~ sin
e = ~ are arbitrary constants.
.':.(5.6S)
(0 1.
"
...(S~66)
"',,
,Substituting Eqs.'(5.65):and(S.66) in Eqs. (S.63) and (S.64) , we get (- (02 + a) X
(ch
~
c)
= b~ + ao'
"',(5:67) ".
(:,)x
~
, ...(S'.6'8)
Solving Eqs. (S.67) and (5.68), we get "
ao
X =
2
(a
~= , ,
- 00
...(S.69)
b2
)
-,.2 ( c -
2 00
)
ao '
2
.'
2
...(S.70)
,.2,
(a-m Hc-oo )---b b u
Hence the solution becomes
'
"
X
=
ao (a-002)-
...(S.7!) 2 b2
,sinro(
,. (c-002)
a0
e=
..,(S.72)
1'2 (a-002) ( c- 002 )-T-b
sin rot
Therefore, the displacement of the top of rigid retaining wall is given by Xtop
or
...(S.73 a)
= X + (H - h ) e 2
x top =
2
.<0 I
-
r (c-ro) +btH-h) . a srn . ro( 2 2 2 2 0 { ( a-ro ) (c-ro ) r- b }
""
(S,73b)
! I
t,.
.ii
!icEarth Pressure
219
? Natural frequencies. Under free vibration condition, the equations of motion are: .
'x'
+ ax
:dba b
..
9 +
ca = ( ;
..
-
.-- . -;~;~,-
...(5.74) ...(5.75)
J x
ubstituting the solution: x = A sin ron t
...(5.76)
a = B sin ron t
...(5.77)
A and B are arbitrary constants ~quations 5.74 and 5.75 become:
~
(-00; + a) A (-
...(5.78)
= b .B
~
(:,)A
...(5.79)
b
From these we get,
A =B
...(5.80)
2
a -OOn 2
and'
...(5.81)
~
~
b
Equating,
(::) c-oo
(\r ~ )
a-oo; 4
2
b
OOn-(a+c)OOn+QC-; ( and solving we get,
2
2
...(5.82)
) =0
00;\ = 2
; (a+c)+J( c;a r +(~ r I
oon2 = 2(a+c)-J
c-a
2
b
...(5.83)
2
(-y- ) +()r
...(5.84)
.5.3. Passive condition. The ratio of stiffnesses on the compression and tension sides is denoted by n. nce in the passive condition, the values of a, band c change and these can be given by : a = n (a)a
...(5.85)
b = n (~)a
...(5.86)
C = n (c)aThe solution for this condition is similar to active condition described above.
...(5.87)
~ 220
Soil Dynamic'S & Mad
I
5.3.5.4. Analysis of a plastic system-active conditio,n. Assume that Zy and 9yare the yield displaceme~~\: " occurring simultaneously in all springs: the equations of motion can be written as:
...(5.88)
x. + a Zy = b 9y + ao sin Cl)t .. b e + ce = - Z
( r2 )
y
...(5.89)
y
Integrating the above equations twice, we get t2
x = (b ey - a Z) b
e = (?
T -
ao sin Cl)t 002 + Cl t + C2
...(5.90)
t2
Zy -coy
) "2
+ C3 t + C4
...(5.91)
Let 'fe' be the time after which displacement of top of wall (yto~ becomes greater than yield displacement (Yd) and plastic system starts. Let xe' xc' ee' 8e be the values corresponding to time te and can be calculated by using the equations developed for elastic system. The following boundary conditions can be applied to evaluate the constants of integration: (i) t = re' X = xe
...(5.92 a)
(ii) t = re' X = xe
...(5.92 b)
(iii) t = re' 8 = 8e
...(5.92 c)
(iv) t = re' e = ec Therefore, we have
...(5.92 d) ...(5.93) ...(5.94)
Zy = xe ey = ee C = x - (be - a Z ) t + I e y y e
a cos Cl)t
. C2
Cl) 2 te
e
_
C4 = 9 ere
e
...(5.95)
. ao sm 00te
= xe - xete + (bey - a Z) T+
C =8 3
0
002
leao cosoote ...(5.96) 00
te ( rb2Zy-Cey )
...(5.97)
b t; - 8 t ZZy-Cey _ e ( ) 2
...(5.98)
Displacementof the top of rigid retaining wall is given ~y x top
= x - (H - h) e
...(5.99)
5.3.5.5. Passive condition. The ratio of stiffnesses on the compression and tension sides is denoted by n Hence in the passive condition, the values of a, band c, change and these can be given by: a = 11(a)a
...(5.100 a)
b = 11(b)a
...(5.100 b)
c = 11(c)a
...(5.100 c) l' I.
nic Earth Pressure
221
'he solutiOltissimilar to the above procedure for active condition except the values of Zy and ay"In ression side (passive condition), the displacements for achieving yield condition are very large, ~in most of the cases plastic system for l"assive case is not considered.
lILL USTRA TIVE EXAMPLE~
mple 5.1
.
0 m high -retainin~ wall with back face inclined 20° with vertical retains cohesionless backfill 33°, 1/ = 1~_KN/m and 8 = 20°), The backfill surface is sloping at an angle 10° to the horizontal. (a) Determine the total active earthpressure using Coulomb's theory and Culmann's graphical construction. (b) If the retaining wall is located in a seismic region (ah = 0.1), determine total active earth pres-
sure using Mononobe's equation and- modified Culmann's graphical construction. ltion : .
-,. .
Static active earth pressure
,.-.,,0
2
PA-
1 H 2 cos2cosa = 2"1
I
)
«I> -(8a+ a) cos
.
-
{ I + [ cos (a
2
=
1/2 2
sin ( + 8). sin ( - i)
cos2 (33-20) ~ x 18 x 6.0 x cos220 cos (20+20)
- i) cos (8 + a) ]
}
1
x
I + sin (33 + 20) sin (33 -10) [ cos (20 -10) cos (20 + 20) ] {
1/2 2 }
= 168.42 kN/m Refer Fig. ".26 for Culmann's graphic,alconstructionfor getting static activepressure. Ds Es gives ~ total active earth pressure.
'
.
.
PA = 17 x 10 = 170 KN/m (b) Dynamic active earth pressure (P)d A yn
--
-1 1 H 2 col «I>-2 'V - a)( I:!: ay)
2
cos'I' cos a cos(8 + a + 'I')
'
x
,1 sin ( 4>+ 0) sin ( 4>- i - \jI )
{ 1+ [ cos (a - i) cos ( 0 + a + \jI) ]
Assuming a
v
'V
ah 0.1 = - = - = 0.05 2 2 .
= tan
-I
- ah
l~av'
= tan
-I
1:!:0.05
= 5.44° with ;:. ay and = 6.0° with'\- ay .
0.1
-
.
2
1I2
}
222
Soil Dynamics & . Machine. .Foundatioi$' . : i i, '0
6.0m
Linear scale - 1 : 100 F'orctZ Pr~ ssurtZ
scaltZ, -1mm
= 10KN
I i n~
Fig. 4.26: Culmann's graphical construction
Value of (f A)dyn with (+) ay 2 = .!." 18 x 60 2 cos (33-5.44-20)(1+0.05)
2
A.
1
2
cos 5.44cos 20 cos (20+ 20+ 5.44)
X
'
1I2 2
I + sin (33+ 20) sin (33-10-5.44) {
[ cos (20-10)
cos(20+20+5.44
]
}
= 214.26 kN/m Value of (P A)dynwith (-) ay 2 - 1 18 x 6 02 cos (33-6-20)(1-0.05) --x
2
.
= 198.05 kN/m
cos6cos220cos(20+20+6)
x
1 1+ ~in(33+20)Sin(33-10-6) [ cos (20-10) cos (20+ 20+ 6)] {
1/2 2 }
.
Therefore (+) ay case governs the value of dynamic active earth pressure. Hence, (PA)dyn= 214.26 kN/m .:. Refer Fig. 5.27 for modified Culmann's graphical constructi9n dynamic active earth pres, " for getting . '. . sure
'w,
2'3"
",ic Earth Pressure
m
Lin~ar, scalCl -1 :100 Forc~
scalCl -1mm:10KN
Fig. 5.27 : Culmann's graphical construction
DsEs gives the total dynamic activ~,earth pressure. (PA)dyn= 21 x 10 = 210 kN/m ,ample 5.2 retaining wall 8.0 m high is inclined 200 to the vertical and retains horizontal backfill with following aperties : y/= 18 kN/m3, ~ =300 and c = 6.0 kN/m2 There is a superimposed load of intensity 15 kN/m2 on the backfill. The wall is located in seismic gion having horizontal seismic coefficient of 0.1. Compute the dynamic active earth pressure and decmine the percentage increase in pressure over the static earth pressure. Solution: .~ 2c
(i)
ho =
=
1- sin 30
1
where KA = l+sin30
Y ~KA 2 x186.0 x..[j = 1.15 m .
1 = 3'
224
Soil Dynamics & Machine FoundatiiiJl;
H = 8.0 - 1.15 = 6.85 m ho 1.16 n=1[=-=0.167 .. 6.85. (ii)
For
= 30°, a. = - 20°and n = 0.167
Figs 5.7 to 5.23 give: = 0.33, (Naqm)stat= 0.512 and (Nacm)stat= 1.2
(Naym)stat
2
(PA)stat.= 18 x 6.85 x 0.33 + 15 x 6.85 x 0.512 - 6 x 6.85 x 1.2
= 282.0 kN/m (i ii)
For
= A.
Therefore,
30°, a.,= :- 20°, dh = 0.1, Fig. 5.14 gives
= 1.18
(Naqm)dyn= 1.18 x 0.512 = 0.609 and (Naym)dyn= 1.18 x 0.33 = 0.393 (PA)dyn= 18 x 8.852 x 0.393 - 15 x 6.85 x 0.609 - 6 x 6.85 x 1.2 = 345.18 kN/m
(iv) Percentage increase over static pressure 282.00 x 100 = 22.4 % = 345.10282.01 Example 5.3 A 5.0 m high retaining wall with backface inclined 20° with vertical retains cohesionless backfill «I> =
30°,Yr= 18 kN/m3and 0 = 20°). The backfill surface is sloping at an angle 15°to the horizontal.
Determine the weight of the retaining wall. «(1)For static condition, (b) For zero displacement condition under earthquake loading (c) For a displacement of 50 mm under earthquake loading. Solution: (a) ,For static condition .
.
. cos2 (cp-
KA
a)
1
= cos2acos(o+a)'
{ 1+ [ cos(a cos(30-20)
.'"
.
= 0.6225
.
-0
cos (8 +a.)
. '.
= cos2 20 cos (20+ 20) .
.
l/2 2
Sin(cp+O)Sin(cp-O
]
.
,}
1
sin (30+ 20) sin (3P-15) {
1+ [ cos(30-20)cos(20+20)
.
',' .
I/2 2
] }
!amic Earth Pressure
225-
From Eq. (5.35) Cl
=
Cos (a +0) -Sin (a +0) tan ~b tan ~b
= Cos (20 + 20) -Sin (20 + 20) tan 30 tan 30
= 0.6840 Thus from Eq. (5.34) 1 W
2
= "2y H KACl = .!. 2 x 1800 x 52 x 0.6225 x 0.6840 = 9765 kg/m
(b) For zero displacement condition (KA)dyn =
cos2(~cos \If cos
2
\If -
a)
1
x
1/2 2
~
a cos ( u + a + \If ) -
sin ( ~ + 0) sin ( ~ - i - \If) {
1+ [ cos (a-i)
cos(o+a+\jI)
]
}
Assuming ay = a2h = 021 = 0.05 -I ah \jI = tan l:tay
0.1 = 1:t0.05
= 5.440 with + ay and
= 6.00with -
--
ay
With + ay cos2 (30- 5.44 - 20)(1 + 0.05)
(K)Adyn =
1 x
cos5.44cos220cos(20+20+5.44)
{ 1+ [ cos(20-15)cos(20+20+5-44)
]
= 0.8311 With - ay: cos2 (30-6-20)(1-0.05) (KA)dyn = cos6cos2 20cos(20+20+6) x .
1
From Eq. (5.36) With + ay :
W
0.8311
tan30
-
W = 0.6225 x (1+ 0.05)( tan 30 - tanSAM
= 1.524
.
~in(30+20)Sin(30-15-6) [ 1cos(20~15) cos(20+20+6) ] {
= 0.7727
1/2
sin(30+20)Sin(30"""15-5.44)
2
1/2
}
2
}-
~.
226.
Soil Dynamics & Machine Foundations
With - av : W = 0.7727 x tan 30 ---!!. W 0.6225 (1- 0.05)( tan 30- tan 6) = 1.596 Therefore Ww = 1.596 x 9765 = 15585 kg/m (c) For displacement condition, d = 50 mm From Eq. (5.40) 5ah
a"d = ah [ -cl ] 5 x 0.1 - 0.1[ 50 ] = 0.03162
1/4
Assuming avd
= ahd = 0.3162 = 0.1581 2 2
With (+) avd 0.03162 = Tan [ l+ahd ] = Tan [ 1'+0.01581] -I
'"
ahd
-I
= 0.780 cos2 (30-1.78 - 20)(1 + 0.01581) (K)A dyn -- cos 1.78 cos2 20 cos (20+ 20 + 1.78) x
1 sin (30+ 20) sin (30 -15-1. 78)
1/2 2
{
1+ [ cos (20-15)
{
1 1/2 sin-(30+20)sin(30-15-1.84 1+ [ cos (20-15) cos (20+ 20+ 1.84)J
cos (20+ 20+ 1.78) J
}
= 0.684 With (-) a,'d -I
'"
0.03162
= Tan [ 1-0.01581] = 1.84
0
It gives = cos2(30-1.84-20)(1-0.01581) (KA)t(,'n
x
-
cos1.84cos220cos"(20+20+1.84)
2
}
= 0.672 Therefore, for (+) ve a,'d Ww - 0.684 Tan30 W - 0.6225' (1+0.01581) (tan 30-tan 1.78)
= 1.143 4
,
.'
:am;c Earth-Pressure""
227
For (- )ve avd Ww -
0.672 . Tan 30 W - 0.6225 (1-0.01581) (tan30-tan 1.84) = 1.162
"
Hence
ww =
= 11347
1.162 x 9765
kg/m
"'
If a factor of safety of 1.5 is used, then "
-.
W = 9765 x 1.5 = 14647 kg (Static condition) ~ = 15585 x 1.5 = 23377 kg (Earthquake condition - zero displacement)
Ww
Ww = 11347x 1.5 = 17020kg/m (Earthquakecondition - 50 mm displacement)
"
It may be noted that the weight of wall gets re~uced significantly if the wall is designed for some splacement. "
xample 5.4 ompute the displacement of a vertical retaining wall baving section and.~ackfill properties as shown in ig. 5.28. The characteristics of the ground motion are: Period = 0.50 s Average ground acceleration
= 0.2 g
Number of significant cycles
= 10
Yield displacement
= 5 mm; n= 2 and
"r
~ =10%
-tl.0m ~ ",
,I
,
I , "
",
" 1'Ranki'ne's
6.0 m
"
I
f
3.0.m
..-~-t
,,
wedge"
'tt ="18 k N I in3
,,,
I
~
"K.4s-
0/2
= 30°
"
Fig. 5.28 :'Section or retaining wall"
. .
228
Soil Dynamics & Machine Foundations
Solution: (i) Refer Fig. 5.26 1 Weight of wall = 2" (1.0 + 3.0) x 6.0 x 24 = 288 kN Weight of soil vibrating along with wall 30 1 .
(
= 0.8 x 2" x 6.0 Tan 45-T ) Total weight = 288 + 149.6 = 437.6 KN
x 6.0 x 18
= 149.6 kN .
3
m = 437.6xl0 9.81 Let the coefficient of base friction = 0.6 .
= 44.61x 103 kg 1
.
288xO.6--xI8x-x62 K = 2 1 0.005
1 3
(Fig. 5.18d)
= 12960 kN/m = 1290 x 103 N/m T=21t~ n
VK;. 3
44.61 x 10
T = 2 1t n
V
12960 x 103
= 0.368 s (ii) From Fig. 5.22 (a), for Zy = 5 mm, TII= 0.368 s; n = 2, T = 0.5s, A = 0.2 g Slip per cycle = 24 mm Total slip = 24 x 10 = 240 mm Example 5.5 Determine the displacement of a model wall shown in Fig. 5.29 retaining medium dense sand ($ = 36°, Y= 18 kN/m2, and T)h= 520 kN/m\ The wall is subjected to following dynamic conditions: Yield displacement: = 6mm Ground acceleration: =.0.25 g Time period: = 0.3 s Solution: 1. The wall is divided into foul equal number of segments with 11ftequal to 0.75 m and the backfill soil is idealized by using springs as shown in Fig. 5.30. The mass of retaining wall is assumed to be lumped at its e.g. which is at a distance of 1.23 m above the base (Fig. 5.30). 2. Consider the backfill characteristics, the spring constants in active and passive cases determined using Eqs. 5.55 and are given below in Table 5.4. The ratio of stiffnesses in passive and active states is taken as 2.0.
.
~
..-~..'
-,"
--~~if'.t?~f~
229
nic, Earth Pressure
y I I
.
:-i°.3m~
A,
.
re
:
I I I I I I
,
,I ,
0
=
36
I
'(
=
18 kN 1m3
I
3.0m
I I I
7th: 520 I
,
'"
Retaining
0
I I
- - - ~ .?ll- -;-I - +- - - X , I I I I
,
0
f4
1.0 m
c --i
v Fig. 5.29 : Section of retaining wall
H = 3.0 m
M ~
a
io
=
a 0 sin wt
c I ......
b
=
1.0 m
Fig. 5.30 : Mathematical model adopted for solution
- '0
.," .'":':
::~;::~'~Yf~'~t*~11~{!',~~:r~i;:';'c::?0t~~,1;\q~i01;-J;{:;:':;~
0---
"ii14:Wf'.o~';.:'.~':,,'~.-'>c',"',",,',,';'.:'-', ,.','!,m,'.-'t:~'.:';,-~:",'-::'1'r'.:~\r-{<;'i-l;;:.:,';-Ji-;'-::,;~'rt','!;,<'-"':';:i',~"-;:~';',.'i-H~';''-,i~ !t=~"/i.'."-~t,~-,-'.,~'..~:,':t'.~..,-'::;~.-,(;',,'.-."i"""'~i,.:!tP1(F't-..':~'r;-;:;.;'.--' ,-, ~".~-.'i~'~-,,'~-',~""; ¢
.~ .".
"
230
Soil Dynamics
&' Machine
Founilations
Table 5.4 : Values of Spring Constants Spring -
Spring location
, Spring
w,r, to c,g"hj (m)
f
,Spring
constant
constant
passive case, 'kj (KN/m)
active case, kj (KN/m)
1.77
48.8
77.6
K2
1.02
292.5
585.0
K3
0.27
585.0
1170,0
K4
- 0.48
877.5
1755,0
Ks
-1.23
536.3
1072.6
KJ
L k,I
L kj = 2340.1
= 4680.2
3. Equation of Motion The quantities required in the analysis are calculated as shown below (Fig. 5.29) , 2 x 0.3 + 1.0 3 1.6 2 DIstance of e.g. from CD = 0.3+ 1.0 x'3 = 1.3 = 1. 3 m . ,
3xO.3xO.15+0.5x3xO,7
Distance of e.g. from BC =
= 0.13;;5°.56 Ivv
,~
=
0.3 x 3
.- .+0.3x3.0(1.5-1.23)+
12'
0,7
0.3+3
1+0.3 . 2 x 3.0
3
(
)
= 0.356 0.7 x 3
.
36
3
I
+-xO.7x3x(1.23-1)
2
2
= 0.675 + 0.06561 + 0.525 + 0.0555 = 1.321155 m4 3
I
YJ
=
.
'
.
0.3 x3+0.3X3.0(0.356-0.15)2+0.7
12
3
2
X3+.!.XO.7X3X
36
2
0.7 -0.056 ( 3
)
= 0.00675 + 0.0381924 + 0.285833 + 0.0330194 = 0.01065451 m4 Ixy
= 1.321155 + 0.1065451 = 1.4277002 m4
A = 1+2°.3x 3.0 = 1.95 m2 r = ~ 1.~~~7 = 0.8556 m '"
M = (°:3; 1.°) x 3.6 x 23.~~~03.~ 4.58, x 103'k~ - . . J = Mr2 = 3.35 x 103 kg-m2
l,
,'.1 ,
f
"'r:i;'j.:'::,'C":':,i',iL;;;";;[!';;;;:',',,;;,..,.j'..,,",
I11III
.'<'
..
lmic Earth Pressure
231
The values of a, band c in the active and passive cases can be detennined as below: Active case Passive case I.k. a = - 2.0 x 510.94 = - 1021.88 a = ---L = 51094 M . I. k. h. b =-2.0 x 117.51 =-235.02 b = I I = 117.51 M c = - 2.0 x 451.76 = - 903.52 c = I.k. I h~ I = 1513.393 = 451.76 J 3.35 The natural frequencies of the wall by considering the tension side (i.e. active case) are given by : '
1
2
c-a
oonl,2=2(a+C)I~ ( 2
2
b
2
()
) +-;:
Puttmg the values of a = 510.14, b = 115.51 and c = 451.76, we get 00/11= 24.94 rad/s oon2 = 18.46 rand/s
The natural time periods are therefore, Tnl = 0.25 s Tn2 = 0.34 s Earthquake motions are erratic and no two accelerograms are similar. The two main parameters of lY ground motion are the amplitude of acceleration and the number of zero crossings in unit time. A ~ry simple and convenient form of ground motion including the above two parameters, is a sinusoidal otion. Moreover, while proposing a method for analysing the liquefaction potential of sand deposits, eed and ldriss (1967) contended that any given accelerogram can be considered equivalent to some efinite number of cycles of loading of equal magnitude. Such idealization have the advantage that after tudying the effect of two parameters, the effect of a probable earthquake motion at any site, can be nalysed. Because of the above advantages, sinusoidal ground motions are utilised in the present study. Given
ao
= 0.25 g = 2.45 mIs2
21t T p = 0.3 s ; ro= 0.3 = 20.94 rad/s
a = 2.45 Sin (20.94 t) Predictioll of Displacements ill Elastic System The displacements in passive case (t = 0 to tpl2) can be cal~ulated as shown below:
x = X Sin Wt
e = ~ Sin Wt where
X
=
ao (a-ro2)-
2 b ,2 (c - ro2 )
=
2.45
.(-1021.88- 20.94')j.
.
2 (-235.02)'
0.8556 (-903.52-20.942)
=-
1.7448 x 10-3 m
H'
""""'i'-;:,:;"",:~',J,.""~',~,,J.':';',,
'""~';;,~,::,,,)':',.,:,;,:";'<
':":""":,/","",\i,,
"""
232
.,C-.;--,",:-.c
,',
'"
','
'.'
"""..'..n'-"':{!\;;{k,',',;.",'A"
i
Soil Dynamics & Machille Foundations Î ó ¬óù ó
ù
¿±
î
®î
î
ø¿ó³ ÷¨ø½ó³ ÷óó¾ ¾
ã
óì
îòìë
î
ãóìòïéìï ¨ ïð
®¿¼
øóïðîïòèèóîðòçìî÷¨ øóçðíòëîóîðòçìî÷ðòèëëêòóøóîíëòðî÷ óîíëòðî
Hence, we have x = - 1.745 x to-3 Sin (20.94 »
ã ó ìòïéì
¨ ïðóì Sin (20.94
¬÷
t)
The computed values of displacements from time 0 to 0.15 s are given in Table 5.5 Table 5.5 : Values of Displacement in Passive State (Elastic-Condition) Time
Translational
Rotation
t(s}
displacement x (mm)
e (rad)
0
0
IJisp. of wall at top due to rotation xe (mm)
0
Total disp. at top xro/mm)
0
0
0.0375
- 1.23
- 2.9510 x 10-4
- 0.52
- 1.75
0.0750
- \.74
- 4.1740 x 10-4
- 0.74
-2.48
0.1125
- 1.23
- 2.9510 x 10-4
- 0.52
- 1.75 .
0.15
̸»
0
0
f)
0
displacement in active case (t = t; to tp) can be calculated as shown below x = X Sin rot
~ Sin rot
e= where
-
x=
ao b2
/
(a-m2)-\
,.2 c-m2
=
2.45 (510,94-20,94')-
~= =
,
117,25' 0.85562 (451.76-20.942)
~-1.817
x \0-3 m
ao 2
., ., r (a -ro-).(c-ro-) --b b 2.45
ãóîòïç騴ðóîra ¼
2
. (~10.94 - 20.942).( 451.76- 20.942) °ig~561
117.51
Hence, we have x =-1.817 x Sin (20.94 t)
. :.:
e = -2.197 x Sin (20.94 ¬÷
\i
-- -
-
233
'namic Earth Pressure
The values of displacements in active state considering elastic condition are given in Table 5.6. Table 5.6: Value of Displacement in ActiveCondition (Elastic-State) Time t(s)
Translational displacement x (mm)
0.15 0.1875 0.2250 0.2625 0.3
0 1.28 1.82 1.29 0
Rotation
Total disp. at top xla/mm)
Disp. of wall at top due to rotation xB (mm)
e (rad) 0 0.02197 0.01552 0.02197 0
0 27.48 mm 38.89 mm 27.48 mm 0
0 28.76 40.71 28.76 0
The dynamic response of the retaining wall under elastic system is shown in Fig. 5.31 which indi:ates that the slip (permanent displacement) after one cycle of ground motion is zero. It can be concluded :hat in elastic system, after any number of cycles, the residual displacement would be zero. 70
~ 60
50 E E
."
0
~
40
.... 0
-c: ti
E to> u 0 a.
30
III
0
20
10
~-=Ji.!!'..!!'- - - - -- - --
- --
0
-
5-
0
0.0375
0.0750
0.1125
1 0.1500 Time.
1 0.1875
1
0.2250
s
Fig. 5.31 : Dynamic response of retaining wall
1 0.2625
1 0.3000
234 -
Soil Dynamics & Machitie Foundtitions
Check for Plastic Conditions
-
When the displacement ()fwall (Xtop)is greater than yield displacement (Yd)' the system would be plastic. Therefore the equations of plastic system should be used. In passive case, yield displacement (Yd)is so large that plastic conditions do not arise and elastic system is considered. In' active case, to identify the time 'le' afterwl}ich plastic conditions- exist, a line has been drawn cof!"esponding to Yd = 6mm (Fig. 5.31). It gives, te.=0.1631
sec
'
xe = - 1.817 x 10-3 sin (20.94 le)
= 4.9116
x 10-4 m. '
ae = - 2.197 x 10-2 sin (20.94 le)
= 5.9388
x 10-3 rad.
Based on the analysis,
- -
Z\. =x e =4.9116 -, x--10--4 , m
ae= 5.9388x
ay
"-
xe
= 0.3663 m/s
1O-3rad.
ae = 0.4429 rad/sec Cl = 0.3663-(117.51 x 5.9388x 10-3-510.94 x 4.9116 x 10-4) 0.1631+ 2.45 cos (20.94x 0.1631) = - 0.1489' . 20.94 C2 = 4.9116 x 10-4 -0.3663 x 0.1631+(117.51 x 5.9388x 10-3 -510.94 x 4.9116 x 10-4) x 0.16312+245 sin (20.94 x 20.1631) 0.1631 x 2.45 cos (20.94 x 0.1631) -?' 20.94 20.94
= - 0.0173 C3 = 0.4429C4
= 5.9388
117.512 x 4.9116 x 10-4 -451.76 x 5.9388 x 10-3 x 0.1631= 0.8676
[ 0.8556
x 10
-3
+ (0.4247)
]
x
0.1631
= - O.1009
2
- 0.4429
x 0.1631
Hence the governing equations for displacement become,
x = 0.2234 [2 - 5.587 x 10-3sin(20.94 t) - 0.1489 t - 0.0173 a = - 1.302 t2 + 0.8676 [- 0.1009 From time 0.1631 s to 0.2963.5, the computations of displacement are given in the following Table. Table 5.7: Values of Displacements in Active Condition (Plastic-State) Time
Translational
Rotatiol!
t(s)
displacement x (mm)
e (rad)
0.1531
0.47
0.1875
1.18
0.2250
0.69
0.2625
- 2.44
37.129 x 10-3
- 6.77
, 41.8624 x 10--
0.2963
Disp. of wall at top due to rotation Xo (mm)
5.97026 x 10-3 16.0015 x 10-3 8.396
.
10-3 "
Total disp. at top x/(JP(mm)
10.57
11.04
28.32
29.50
50.26
51.01
65.72 "
63.28
74.10
67.33
f
,;,)
-,t
)ynamic Eartl, Pressure
235
,.From t =e;2963 s to t = 03 s;the- displacements 'are computed using the expressions obtained by olving the equations of motion under elastic condition taking boundary conditions satisfying the previmsly computed values t = 0.2963 s. The complete solution can be expressed as : I.
l
t.
l
Xe
= A I sm rot + A 2 cos rot + A 3 sm rot + A 4 cos rot
ee
= BI sin ro t + B2 cos ro t + B3 sin ro t + B4 cos ro t
I
'11
I
Superscripts of A and B indicate the mode of vibration. Therefore Constants AI' A2' BI and B2 ::orrespond to the mode when system is vibrating with wnl' and A3, A4' B3 and B4 for the second mode. xe
= AI
e
m
sinrot+A2 cosrot+A3 sinrot+A4 cosrot A A A A e = -1.. sin rot + --1.. cos rot + J sin rot + -1. cos rot
m
m
m
Putting boundary conditions, we get - 0.0067669 0.0418624
= - 0.0786 AI - 0.9969 A2 + 0.0786 A3 + 0.9969 A4 = 0.0763 AI - 0.9679 A2 - 0.1139A3
+ 1.445 A4
- 0.133143 = 20.875 AI + 1.6459 A2 + 20.075 A3 + 1.6459 A4
0.09603 = AI
-
20.267 AI + 1.5979 A2 + 29.094 A3 + 2.385 A4
= - 0.00376
A2 = - 0.02171 A3 = - 0.00187 .
A4 = 0.0144~
,'"
Therefore for range of tp from 0.2963 to 0.3 s the equations of displacements will be : xe = - 0.00376 sin ro t - 0.02171 cos rot - 0.00187 sin ro t + 0.01448 cos ro t
ee = 0.003'65 sin rot + 0.02107 cos ro t - 0.002710 sinro t + 0.02099 cos ro t ,
Values of displacements in ~lastic condition from time ,0.2963 s to 0.3 s are given in Table 5.8 Table 5.8 : Values of Displacements in Elastic State
Time
TraJ;slational
Rotation
Disp. of wall at top
t(s)
displacement x (mm)
e (rad)
due to rotation xe (mm)
Total disp. at top XI (mm) op
0.2963
- 6.77
41. 8624 x 10-3
74.10
67.33
0.2975
- 6.85
41.916
67.34
0.298
- 6.98
41.98359 x 10-3
74.191. , 74.3109
0.299
- 7.10
42.0320 x
74.39670 ,,'
67.29
0.3
- 7.23
42.062 x 10-3
74.4498
67.21
x 10-3 10-3
67.33
,
.
Hence it is fo~rid that after 'one cycle, the disphi~ement ~sequal to 67.21 ,mm and it can be called as slip. The total displacement after n cycles is e'qual'to n times the slip. The 'final translational displacement and r~tation of the retaining wall are therefore known. The displacement curve in palstic state is also shown in Fig. 5.31.
.:.'..'
;'r:;,;"i:;":.~'d~~.';
.:;..,.-/:{;i;'
..,.,..,rtif~;~;8t. "'
236
Soil Dynamics & Machine
Foundations
ÎÛÚÛÎÛÒÝÛÍ
Biggs, J. M. (1963), "Introduction to structural dynamics", McGraw Hill Book Co., New York. Coulomb, C. A. (1776), "Essai sur une application des regles des maximis et minimis a quelque problems de statique relalifs a I'architecture", Mem. acad. roy. press. disversavants, vol 7, Paris. Culmann, K. , (1866), "Die graphische statik", Zurich. IS 1893 : I. P. (1962), "Earthquake resistant design of retaining walls", Proc. Symposium in Earthquake Engineering, University of Roorkee, Roorkee. Lai, C. S. (1979), "Behaviour of retaining walls under seismic loading", M. E. Report, University of Canterbury, New Zealand. Mononobe, N. (1929), "Earthquake proof construction of masonry dam", Proceedings. World Engineering Congress, vol. 9, p. 275. Nadim, F., and Whitman R. V. (1983), "Seismically induced movement of retaining walls", Jour. of Geot. Engg. DiYn.,ASCE, Vol. 109, No. 7, pp. 915-931. Nandakumaran, P. (1973), "Behaviour of retaining walls under dynamic loads", Ph.D. Thesis, Universityof Roorkee, Roorkee. Newmark, N.M. (1965), "Effect of earthquakes on dams and embankments", Geotechnique, Vol. 15, No. 2, pp. 129160. Ohde, S. 91926), "General theory of earth pressures", Journal, Japanese Society of Civil Engineers, Tokyo, Japan, Vol. 12, No. I. Prakash, S., and Saran, S. (1966), "Static and dynamic earth pressures behind retaining walls," Proc. 3rd Symposium 011Earthquake Engineering, University of Roorkee, Roorkee, Vol. 1, pp. 277-288. Prakash, S., Puri, V. K. and Khandoker J. U. (1981), "Rocking displacements of retainingwalls during earthquakes", Int. Conf. on Recent Advances in Geotcchanical Earthquilke Engineering and Soil Dynamics, Vol. 3, St. Louis, U.S.A. Prakash S. (1981), "Analysis of rigid retaining walls during earthquakes", Int. Conf. on RecentAdvances in Geotech. Earthquake Engg. and Soil Dynamics, Vol. 3, St. Louis U.S.A. Reddy, R. K., Saran.,S., and Viladkar, M.N. (1985), "Prediction ofdisplacements of retaining walls under dynamic conditiOns", Bull. of Indian Soc. Earth. Tech., Paper No.-239, vol. 22, No. 3. Richard, R. k, and Elms, D. G. (1979), "Seismic behaviour of gravity retaining walls", Journ. Geotech. Engg. Divn., ASCE, Vol. 105, No. GT4, pp. 449-464. Saran, S., and Prakash, S. (1968), "Dimensionless parameters for static and dynamic earth pressures behind retaining walls", Jour. Indian National Society of Soil Mech. and Found. Engg., July pp. 295-310. Seed, H. B., and Whitman, R. V. (1970), "Design of earth retaining structures tor dynamic loads", ASCE Speciality conference on Lateral Stresses in Ground and Design of Earth Retaining Structures, pp. 103-147, Ithaca, New York.
Zarrabi, k. (1979), "Sliding of gravity retaining wall during earthquakes considering vertical acceleration and changing inclination of failure furface", M. S. Thesis, MIT, USA.
< "., :. ~ .j J 4
;;:\r£;;:;i,~;~~. ;[;:J~1~;t!fi ,;~:~~-~ <:v~?';;:;~\.;':',
i~'w'~~f~il!¥~~
j'
,~
.: . . .
>.-"
237
Dynamic Eartlt Pressure
PRACTICE PROBLEMS 5.1 Explain with neat sketches the following: (a) Mononobe- Okabe's approach, and (b) Modified Culmann's graphical construction for getting dynamic active earth pressure. 5.2 How is the effect of partly submerged backfill considered in computing dynamic earth pressure? . 5.3 Explain the salient features of the following: (a) Richard-Elms model (b) Nadim-Whitman model (c) Reddy-Saran-viladkar model for getting displacement of rigid retaining wall. 5.4 A vertical retaining wall is 8m high and retains noncohesive backfill with y = 18 kN/m3, 4> = 30° 8 = 20° .The backfill is inclined to the horizontal by 15°.The wall is located in a seismic area where the design seismic coefficients are
ah = 0.10 ; al, = 0.05 Compute the static and dynamic earth pressure on the wall using both modified Coulomb's approach, and Culmann's graphical construction. 5.5 If the retaining wall (Problem 504)is to incline at lO° with the vertical, would you recommend its inclination towards or away from the fill. Justify your answer fully. 5.6 The backfill of retaining wall (Problem 5.4) is carrying a surcharge of 50 kN/rn2. Estimate the increase in static and dyn~mic earth pressures. 5.7 The backfill of retaining wall (Problem 5.4) is submerged upto 4.0 m from the base of wall. Estimate the total pressure on wall both in static and dynamic cases. 5.8 Compute the displacement of the wall (problem SA) f~r the following condition: Period of wall = 0.25 s
= 5.0 mm Period of ground motion = 0040 s Zy
Equivalent number of cycles in an earthquake o~ magnitude 7.0 will not exceed 15. DD
t
DYNAMIC BEARING CAPACITY OF SHALLOW FOUNDATIONS .
6.1 GENERAL Foundations may be subjected to dynamic loads due to earthquakes, bomb blasts and operations of machines. The dynamic loads due to nuclear blasts are mainly vertical. Horizontal dynamic loads on foundations are mostly due to earthquakes. Basically there are two types of approaches namely (i) pseudo-static analysis and (ii) dynamic analysis for getting the solution. In this chapter, pseudo-static analysis is first presented and it is followed by dynamic analysis. Design of foundations of different types of machines have been given in detail in chapters 8 to 10. 6.2 PSEUDO-STATIC
Pseudo-static analysis Adopting appropriate can be conveniently subjected to eccentric
ANALYSIS
is more commonly used for designing foundations subjected to earthquake forces. values of horizontal and vertical seismic coefficients, equivalent seismic forces evaluated. These forces in combination of static forces make the foundation inclined load. In Secs. 6.3 and 6.4, the procedure of detennining bearing capacity.
s.:ttlement, tilt and horizontal displacement of shallow foundations subjected to eccentric
- inclined
loads
ha\'e been prL'scnted. It is preceded by brief description on fundamental concepts involved in bearing capacity analysis. 6.3
BEARING
CAPACITY
OF FOOTINGS
6.3.1. Modes of Shear Failure. The maximum load per unit area that can be imposed on a footing without causing rupture of soil is its bearing capacity (some times termed critical or ultimate bearing capacity). It is usually denoted by quoThis load may be obtained by carrying out a load test on the footing which will give a curve between average load per unit area and settlement of the footing. Based on pressure-settlement characteristics of a footing and pattern of shearing zones, three modes of shear failure have been identified as (i) general shear failure, (ii) punching shear failure and (iii) local shear failure (Caquot, 1934; Terzaghi, 1943; DeBeer and Vesic, 1958; Vesic, 1973). In general shear failure, well defined slip lines extend from the edge of the footing to the adjacent ground. Abrupt failure is indicated by the pressure-settlement curve (Fig. 6.1a). Usually in this type, failure is sudden and catastrophic and bulging of adjacent ground occurs. This type of failure occurs in soils having brittle type stress-strain behaviour (e.g. dense sand and stiff clays).
,..,:",,;,,"d:"~'tX,~,'.","",r,d.:"Y""::'-"
"J:'.~'f.,,'"~~:"~:"
w,::":'~.'
",r;1',;;~"f"':"",
"",'"
"'"
"",.0,
':"':'<"",,"
,>,'[:";':""'0,
..,.""...;~.;t~A",.
,,';"~',:':},,~~,
239
Dynamic Bearillg Capacity of Shallow Foulldatiolls
In punching shear failure, there is vertical shear around the footing perimeter and compression of soil immediately under the footing, with soil on the sides of the footing remaining practically uninvolved, The pressure-settlement curve indicates a continuous increase in settlement with increasing load (Fig, 6,1 b). .
Load
-
+-
c
<::.I
(a)
(b)
E
.
-+-
.-: I
<::.I
+<::.I Load c
"
<::.I.
E
,"
'<::.I
++<::.I lI'I
I
(c)
---
",
,,-...."
/
.... +C
.,'
Load
<::.I
E
<::.I +<::.I If)
-
Fig. 6.1 : Typical modes of failure (a) General shear, (b) Punching shear and (c) Local shear
The local shear failure is an intermediate failure mode and has some of the characteristics of both the general shear and punching shear failure modes. Well defined slip lines immediately below the footing extend only a short distance into the soil mass. The pressure-settlement curve does not indicate the bearing capacity clearly (Fig. 6.1 c). This type of failure occurs in soils having plastic stress-stram characteristics (e.g. loose sand and soft clay). In Fig. 6.2, types of' failure modes that can be expected for a footing in ~ particular type of sand is illustrated (Vesic, 1973). This figure indicates that the type of failure depends on the relative density
and depth-width ratio (D/B) of the footing. There is a critical value of (D/B) ratio below which only punching shear failure occurs.
-
, 240
Soil Dynamics
0
0
Relative density, 0.2 0-4 0.6
& Machine
Foundations
Or 0.8
1.0
Gen eral sh ear failur( zont
elcc ..
..a. .J::.
5
'"0
-
.->
Pun ching shear fai lu re zone
0
!'t:
10 Fig. 6.2 : Region for three different modes of failure The criteria
given in Table 6.1 may also be followed
for identification
of type of failure:
Table 6.1 : Identification of Type of Failure Type of failure
Relative density Dr (%)
(Deg)
Void ratio e
70
36°
S 0.55'
S 20
S 29°
0.75
1.
General shear failure
2.
Local shear failure or punching shear failure
6.3.2. Generalized Bearing Capacity Equation. In the design of foundation usually net bearing capacity is computed' and used. It is defined as the maximum net intensity of loading at the base of the foundation that the soil can support before failing in shear. It is denoted by,"qnu'Therefore qnu=qu-YIDf where, q u = Ultimate bearing capacity
...(6.1)
The equation of net bearing capacity developed for strip footing considering general shear failure (Terzaghi, 1943; Meyerhof, 1951) is extended to consider variations from the basic assumptions by applying modification factors that account for the effect of each variation (Hansen, 1970). It may be written as : q
nu
=cN.S.d'i.b c
c
1"
c
eel
+ y .D . (N -1 ) óÍò¼ù·ò¾ò® +-'Y 2 'B.N y .S y .d y .i y .b y .r' w f q q q q q w 2 ...(6.2)
241
Dynamic Bearing Capacity of Shallow FoundatioJIs
where qllu
= Net ultimate bearing capacity
c = Undrained cohension of soil B = Width of footing Df = Depth of foundation below ground surface Nc- Nq, Ny = Bearing capacity factors Sc' Sq, Sy = Shape factors for square, rectangular
and circular foundations
dc' dq, dy = Depth factors ie' iq, iy = Inclination
factors
bc' bq' by = Ground inclination factors rw' r:v = Ground water table factors 6.3.2.1. Bearing capacity factors. Nc' Nq and Ny are non-dimensional factors which depend on angle of shearing resistance of soil (Terzaghi, 1943; Terzaghi and Peck, 1967). Their values may be obtained
from Table 6.2.
"
Table 6.2 : Bearing cp
Ne
0
5.14
5 10
6.49 8.35
Capacity
Factors Hq
Hy
\.00
0.00
1.57 2.47
0.45 1.22
Deg
.; .'
15
10.98
3.94
2.65
20
14.83
6.40
5.39
25
20.72
10.66
10.88
30
30.14
18.40
22.40
35
46.12
33.30
48.03
40
75.31
64.20
109.41
45
138.88
134.88
27 I. 76
50
266.89
319.07
762.89
6.3.2.2. Shape factors. Approximate values of shape factors which are sufficiently accurate for most practical purposes are given in Table 6.3. Table 6.3 : Shape Factors S.No. (i) (ii)
Shape of footing Continuous strip Rectangle
Se
Sq
1.00
1.00
1+ 0.2 B/L
1+ 0.2 B/L
Sy 1.00 1-0.4 B/L
(Hi)
Square
1.3
1.2
0.8
(iv)
Circle
1.3
1.2
0.6
(B = diameter)
242
Soil Dynamics- & Machine Foundations
6.3.2.3. Depth factors. The bearing capacity factors given in Table 6.2 does not consider the shearing resistance of the failure plane passing through the soil zone above the level of the foundation base. If this upper 'soil zone possess significant shearing strength, the ultimate value of bearing capacity would be increased (Meyerhof, 1951). For this case, depth factors are applied, whereby D d = 1 + 0.4 --L
...(6.3)
B
C
2
d = 1 + 2 tan <1>(1 - sin <1» . q
,
DJ
(6.4)
B
dy = 1 ...(6.5) The use of depth factors is conditional upon the soil above foundation level being not significantly inferior in shear strength characteristics to that below this level. 6.3.2.4 Factors for eccentric-inclined loads. The effect of eccentricity can be conveniently and conservatively considered as follows: One way eccentricity (Fig. 6.3 a) - If the load has an eccentricity e, with respect to the centroid of the foundation in only one direction, then the dimension of the footing in the direction of eccentricity shall be reduced by a length equal to 2e. The modified dimension shall be used in the bearing capacity equation and in determining the effective area of the footing in resisting the load. Two way ecc?ntricity (Fig. 6.3 b) - If the load has double eccentricity (eL and eB) with respect to the centroid of the footing then the effective dimensions of the footing to be used in determining the bearing capacity as~ell as in computing the effective area of the footing in resisting the load shall be determined as given below: -
2 eL
...(6.6)
B' = B-2
eB
...( 6.7)
L' = L A' = Lt
X
B'
...(6.8)
~
Qv G.
e
I
I
,
,t"'.
Qh
"
""':"""-"""';""" ." ".'. "
,\7 '«'\~1\Z2?r"
.,,:~..,.,. J..
I
t
.., : ," :":';:'::"""':""~
B or L
.
, ::.::~
~
Fig. 6.3 : Eccentrically-Obliquely loaded footings (a) One way eccentricity
.W'«)
a-
243
Dynamic Bearing Capacity of Shallow Foundations
~
.... ....
Fig. 6.3 : Eccentrically-Obliquely
for B- axis
loaded footings (b) Two way eccentricity
In computing the shape- and depth factors for eccentrically-obliquely
loaded footings, effective width
(B') and e_ffe~tive length__(~? will be used in place of total width (B) and total length (L). For a design,;'eccOentricity' should be limited to one-sixth of the foundation dimension to prevent the condition of uplift occurring under part of the foundation, Inclination factors -
Following inclination factors may be adopted in design. m+l ,
-
ly -
[
1
Qh"
...(6.9)
- Q + BLc cot ]
Q
. Iq
= [ 1-
V
iq
'
m
...(6.10)
Q + BLch cot ]
- [N:~~$] (when p
...(6.11)
O.
= 1 mQh - cNcBL (when = 0°)
...(6.12)
where Qh is the horizontal component of the load Q acting on the foundation at inclination i with vertical. Values of m are taken as given below: (i) If the angle of inclination i is in the plane of L-axis
111=(2+~)
..(6.13)
(1+~)
(ii) If the angle of inclination is in the plane of the B-axis
.
~, y m'= -0" ~',./. ,'"
-
(.'
2+oB 1+B L) ( L)
...(6,14)
,
"
,.'
-" As P~!)~,':640].:)981,:tlfernclination "...' .- - ------ -,-- - ,--,~.- -,-'factors -- are give,n"by-~~:- ,.--.~:-,:::--".-' .)
'q~cq:
~'
. '; ..'
:;!:f
'..~
244
Soil Dynamics & Machine Foundations 0 ID
-
..0 '-"
0 Lt)
0
-
~
00 11
0 ~
1J '-" 0 ~
.-
Col V
0
0 M
Z
0
11
N
0 N
0 ~
.-
0
0
.-
0 N
0
0
tII V
Z
0
0 ~
.-
0
t
0
0 ~
(6~p)q,
0 N
(6~p)
0
0
cp
-&-
." ." ... '" " 'E 7 ""': -.::; ;1 .....
0 0
-
.-
-
0 ~
u
d
'-" 0 co
0 M
0
°0
0 N
0 ID
11
11
Col V Z
0 N
0 ~
0
0 N
0 ~
'0 ~
0 0 N
( 6~p)
0
0
0
t
0
0 N
0 ~
(6~p)
0
et>
0
t>I V Z
.. Dynamic Bearing Capacity of Shallow
I
~.
245
Foundations 0 1.0 '"
.0 .........
0 \J')
-
00
1
tI
I
i
'" "0
.........
0 tIJ C') eT Z
0
1
00 C')
tI
,-
1\
N
0
1
to... '\.u.
18
r. 0
0
..t
0....
N
(60p)
II
"0
0
C')
0
0
0
0
M
10
0
0
""
N
(6<>p)
°g
Z
er '" :: '"
''"' ,.
cp
7.
..
'.
Ir:
-.::> O£
'. '"
1
0
I
l
'"
u
.........
i
00
1
11
0 0 N
.0
11
\
tIJ eT
Z
,-
10N
0
0 ~
- o.
et)
~~
-0
0 N
,'. (6<>p)
-0
0
0 ~,
o.
et)
0 0 N
(6<>p)
0
et>
'0
eT tIJ
Z
"}:Jl
..."".
246
Soil Dynamics & Machine
Foundations
0 N
0 -..t
".......
.c
_°
,
"tJ 0 00
°0
, 0 M
° .0
JI
M
"
0
C:.I
,-
"N
')0 Z
C:.I )0
Z
0
-.t 0
I.t'I M
0 t
0 M
I.t'I N
0 N
0
0
0 M
I.t'I M
0
0
...t
(6ap )
(6ap)cp
0
et>
~ '" :::I ~ ... ;> .~ 7.
0
'c: ~ ~ '.
U)
"....... 0
'-"
0 N
°0 JI C:.I )0
"....... u
z
0 U)
,
°0
0 co
N JI
0 ...:t
0
...t
0 ::>
.t
0 M
0 N
øê¿°÷ et>
0
0
0 N
b -..t
0 M
'1f) M
(6ap)
1f) N
et>
00 N
ID
J)ymimii: 'Bearing 'CapJCity"of'Sh~llow
,<
.<'
-,'
,,' ,,(i'
'f"
"247 ---
Foundations ,
ic
"
,
2
= iq = ( 1 ~ ;0
)
::':
'(
,t,
, ":..(6.15) ...(6.16)
~J
iy= (1-
';-
For more accurate estimation of bearing capacity of a eccentrically-obliquely loaded footing, bealing capacity fa~tor:; as shown ,in Figs. 6.4 to 6.6 developed by Saran and Agarwal (1991) may ,b~,u$ed. These factors have been obtflined by carrying out a theoretical analysis based on limit equilibrium and limit analysis approaches'.
~":":
As evident from these figures, bearing capacity factors (Nyei'Nqei and Ncei) depend on
~,
qnu
= cNeei .
-.
1,
Se . de + YiD! (Nqei -1)
...(6.17)
Sq . dq . 'rw + 2"'Y2 . B '. ~'Yei. S1' dy' r'w
Use of Eqs. 6.6 to 6.16 make more'cons~rv~tiveqestimate of~he bearing capacity. , 6.3.2.5. Base inclination factors. If the base of foundation is inclined from the horizontal and an applied load acts nonnal to the base (Fig. 6.7), the pattern of rupture surface beneath the foundation will be different from the pattern that develop beneath the level footing carrying a vertical load (Meyerhof, 1953). For this condition base inclination factors as given'" below may be used: . , ,
(l-b)
,
...(6.18)
b =' b ,q fer tP > 0°, c q N e tantP '
= 1-
0.0067 a
for 4>= 0°
b = b =
...(6,19)
2
,
(1 - ~
tan'"
"
:;'
"
.q 1 57.3 If ) , .'[' the angle of the base inclination in degree with horiz~ilta1.. ' ,
where a
represents
~
" _'0
rJ
,.,c
Fi~.6.7 : Inclined, footing
',," ",~s
"
...(6.20)
,.;.r"','.,, 171.
,: 248
Soil Dynamics & Machine Foundatitnu ,:..".
"
~
6.3.2.6. Water table factors. (Fig.
Correction factors r w and r'w may be computed using following Eqs.
6.~).
' da
rw
Df
= 1.0 - 0.5
(For da ~ DJ Y
...(6.21):'
.
"
d'
~
(For, db ,~ B) . ...(6.22} , where do and db represent the position of water table with respect to the base of the footing as shown: in fig. 6.8. For the position of water level 'below the base of footing, do = 0 i.e. rw = 1; and for the r~ = 0.5 + 0.5
position of water level at depth more than B, db = B i.e. r'll' = 1.
'
"'"
B Footi ng
vwot.rzr Of
t
'
Irzvel
do
-t-
I
I water
B
I
I
I (rzv
I
L L______J
'
\!
db
L
(0)
1.0
L.~ 1.0
,~L.
L." 0.9 ..0 u
.:: 0.9 0 .. u
0 Of-
.B 0.8
c 0 0.7
~
?
0.8
c:
.~ ..u 0.7
.. u ::J '0 0.6
::J
'0 tI
tI Cl:
j-. "". .f
0.6
a:
0.5 0 o. Z O. 4 0.6
0.8
do/Of
..
0.4
0.6
0.8
1.0
~, " 'f
¼¾ñÞ ..
(b) Fig. 6.8 : Correction
(c) factors 'cor position of water table
,J..I ,.
,
.-,,"~
;'~,,; '~,0 ";;. ~', :'
~
~-_.-
J,~II1I~-.,
--
ID
~,,'
249
Iynamic Bearing Capacity of Shallow Foundations
).3.3. Local and Punching Shear Failure. The assumption that the soil behaves as a rigid material s satisfied for the case of general shear but is not appropriate for punching and local shear. Comparison )f the relative pressure-settlement curves (Fig. 6.1) indicates that, for punching and local shear failure ;ases, the ultimate pressure is less and the settlement is greater than for the condition of general shear :'ailure.For design purposes, the general shear, local shear and punching shear failures can be identified 1Sper the criterion given in Table 6.1. Terzaghi (1943) proposed empirical adjustments to shear strength parameters c and <{I to cover the case of local and punching shear failure. Shear strength parameters cm and <{Im should be used in the bearing capacity equation and the bearing capacity factors are obtained on'the basis of
2 = - C
c
<{Im
...( 6.23 a)
3
III
= tan
-1
...(6.23
(~ tan ~)
h)
If the failure lies between general shear and local shear failure, then linear interpolation is done to evaluate
the value of bearing
capacity
factors.
(N )
'Y 4>=36°
For example,
of Ny for "
(N ym ) CP=29°
36° - 29°
(NY)4> = 34°= (Nym )$ = 29° +
value
~=
34° will be
,
...(6.24 )
x (36° - 34° )
where (Nyrn)$ = 29° value of Ny factor for <{I::;:29° considering local shear" failure condition. Therefore its value will be obtained using Table 6.2 for <1>m = tan -1 [(2/3) tan 29°] = 20.29°. , .
6.3.4. Factor of Safety. The net bearing capacity of the soil is divided by a safety factor to obtain the net safe bearing capacity. It is denoted by qnF' '
.
A factor of safety is used as a safeguard against (i) natural variations (ii) assumptions (hi) inaccuracies
in shear strength of soil,
made in theoretical
methods,
of empirical methods and
(iv) excessive settlement of footings near shear failure. A factor of safety of 2.5 to 3.0 is generally used to cover the variation or uncertainties listed above. Therefore
...(6.25)
qnF = qnu F 6.4 SETTLEMENT,
TILT AND HORIZONTAL
DISPLACEMENT
An eccentrically-obliquely loaded rigid footing settles as shown in Fig. 6.9 in which Se and Sm represent respectively the settlement of the point under load and edge of the footing. If' t' is the tilt of the footing, then Srn is given by :
'
Srn
= Se + (BI2 - e ) sin t
...(6.26)
In Fig. 6.9, Ho represents the horizontal displacement of the footing. '
<;~"",.;.
---....----
"
,,250
Soil Dynamics &, Machine {1:
~ ,
.'
,
" " ,~
I
I
'
I
Se
5,0
..--
I ~ l_.l.
i
(e/2-~)
.
e!--I
-.","'IQd(Zi,
"
I.,
Foundations
'
e/2
-1
-E
"
Srnl , Final position'
l-:HD~
'
"
- --~)
I
E
~
~~el1.Fig,6.9: Settlement,
tilt and horizontal displacement
of eccentrically-obliquely
loaded footing
Agarwal (1986) carried out modeltests on eccentrically-obliquely loaded footings resting on sand Footings of different widths and shapes were used. In each test, for a pressure increment observations were taken to record Se' t and Ho' Effect of relative density of sand was also studied. In addition to these tests, pressure.settlement and pressure-tilt characteristics of eccentrically-obliquely loaded footings resting on clay and Sand beds were also obtained using non-linear constitutive laws of soils (Saran & Agarwal, 1989). From the model test data and results of analysis based on constitutive laws, plots of SeI So versus e. I Band Sm IS o versus elB were prepared for different load inclinations (Figs. 6.10 & 6.11). So represents the settlement of the footing subjected to central vertical load (i.e. elB = 0 = i) and obtained corresponding to the pressure intensity giving the same factor of safely at which Se and Sill values are taken. These plots were found independent to the type of soil, factor of safety, size and shape of footing The average relationships can be represented by the following expressions:
~
1 - 0.56
,
~:=Bo + BI (~)
-
where.A"
...(6.27
Ao + AI (~)+A2(~r
~e 0 =
...(6.28
,..(6,29
(~ )-0,82 (~)'
...(6.3(
A, ~.. 3.51 + 147 (~ )+5,67( ~)'
A,
~
(~ )-lZ,4S(
4,74 - \.38
,
,
',-
I
) Bo = I - 0.48 ( ~,...~.~;. ,
,/
""
2
~
( q, )",...' '
r
..:(6.31 -" . ,..(63
i_h,"""'"
ilntlmic Bearing Capacity of Shallow Foundations
B\
~ 1.8~ + 0,94
~
251
W- L63( ~)'
...(6.33)
0.8
i ::150
. .
0.6 Se So
. 0-4
0.2
0
~-
0
0.1
0.2
0.3
'.,
ù´×Þ Fig. 6.10 : 8/80 versus e/B for j
= 15°
1.2 ,
I =
0
15
0.8
-Srn
So 0.4
0
0 ,
.
0.1
0.2
0.3
e/B ',;
" < :., ,{,;;-~Fig.6.11
: Sm/SOversus elB for i-15°
'Co):, ..
252
SoU Dynamics & Machine Foundations
Values of So can be obtained using the data of plate load test or standard penetration test in cohesionless soils, and consolidation test data in clays in conventional manner. Similarly a unique correlation was obtained between Ho / B and i/~. Ho
B
i i 2 i 3 i 4 =0.121 (~)-0.682(~) +1.99(~) +2.01(~)
...(6.34)
The above correlation is also found independent to the type of soil, factor of safety, size and shape of the footing. The effect of eIB was found small and the displacement value decreased little with the increase in eccentricity: This effect is neglected considering the results slightly on the safe side. 6.5 DYNAMIC ANALYSIS .
The dynamic bearing capacity problem attracted attention of the investigators in 1960 when the performance of foundations under transient loads became of concern to the engineering profession (Wallace, 1961; Cunny and Sloan, 1961; Fisher, 1962; Johnson and Ireland, 1963; Mckee and shenkman. 1962: White, 1964; Chummar, 1965; Triandafilidis, 1965). All analytical approaches are based on the assumption that soil rupture under transient loads occurs along a static rupture surface. In this section the sailent features of th~ analysis developed by Triandafilidis (1965) and Wallace (!961) for transient vertical load; and by Chummar (1965) for transient horizontal load have been presented.
~
8
qu
(enter
~
I
-T
0.43 8
I
ZPrandtl's
~
\ ~
B
q
u
.
-
:
-r,
of rotation
(Fczllllnius)
---r
rupture
C Ilntllr of rotation (Fllllllnius)
I', r =2.20SB/rr I
, --- - --7 , - -L-
w
surfacll
-
WCO5~
Fig. 6.12 : Illustrations of mode of failure, and dynamic equilibrium of moving soil mass
-",.-""
')ynamic Bearing Capacity of Shallow
253
Foundations
5.5.1. TriandafiIidis's Solution. Triand~rllidis (1965) has presented a solution for dynamic response of continuous surface footing supporting by saturated cohesive soil (I\>= 0 condition) and subjec,ted to vertical transient load. The analysis is based on the following assumptions: (i) The failure surface of soil is cylinderical for evaluation of bearing capacity under static condition (Fig. 6.12) .
(ii) The saturated cohesive soil (I\>= 0) behaves as a rtid
plastic material (Fig. -6.13).
(iii) The forcing function is assumed to be an exponentially dec~ying pulse (Fig. 6. 14) (iv) The influence of strain rate on the shear strength is neglected. (v) The dead weight of the foundation is neglected.
"0
er U\ U\ C:II ~
CJ\ CJ\ I. C:II ...
..U\
..
v
t/)
E 0
c >-
0
'.
,
,
strain
rima Fig. 6.14: Transient
Fig. 6.13 : Assumed stress-strain relationship
vertical load
Analysis
Let the transient stress pulse be expressed in the form qdoli= q e-~ I = A. q e-~ I
...(6.35)
where, qd = Stress at time t ~ = Decaying function q u = Static bearing capacity of continuous footing qo = Instantaneous peak intensity of the stress pulse A.
= Over load factor = qo .
.
.
qu
".
The rupture surface is shown in Fig. 6.12 with centre of rotation at point 0 loc~ted at a height of 0.43 B above the ground surface. The equation of motion is written by equating the moment of the disturbing and restoring forces taken about the point O. The only disturbing and restoring force is an externally applied dynamic pulse. The restoring forces consist of shearing resistance along the rupture surface, the inertia of the soil :rp~sspa~icipating in motion and the resistance caused by the ~isplacement
111&1::,
254 r.
Soil Dynamics
,
~
Machine Foundations
of c~ntre of gravity of soil mass. Driving moment MdP due to applied dynamic pulse is I 2 .
.
= "2qd B
Mdp
...(6.36)
where, B= Width of footing The static bearing capacity of a continuous footing along the failure surface (Fellenius, 1948) is qu = 5.54 Cu where,
...(6.37 a)
Cu= Undrained shear strength
Resisting moment Mrs due to shear strength is Mrs
=
; qu B2
...(6.37 b)
An applied pulse imparts an acceleration to the soil mass. The resisting moment Mri due to the rigid body motion of the failed soil mass is
= Joe Jo = .Polarmass moment of inertia
Mri
where,
-- WB2 1.36 g W = Weight of the cylinderical soil mass = 0.31 Y1tB2
...(6.38)
...(6.39)
...(6.40)
Y = Unit weight of soil WB2 .. Therefore,
Mri
...(6.41)
= 1.36,ge
The displaced position of the soil mass generates a restoring moment Mrw ' which may be expressed as
Mrw = W r sin e Mrw =Wre
For small rotations,
...(6.42 a) ...(6.42 b)
2.205 B
r =-
where,
...(6.43)
1t
By equating the moments of driving forces to those of the restoring forces, the following equation of motion is obtained. Mdp = Mrs + Mri + Mrw Sub'5tituting
for moments
..
and rearranging,
3g
we get
[,
O.68g
e +-e= 1tB
...(6.44)
-~t,
q ""e - 1] ...(6.45) [ W ] u Equation (6.45) is a second order, nonhomogeneous, linear differential equation with constant coefficients. The natural frequency and the time period of the system are given by co.
T
~
~
...(6.46)
~;~ 2~ ~ ~
:
.,.(6.47)
..
~.JI,,~'{!:
255
DynU'tJic Bear;n~ C9pacity of S~lllloHJ.Fpundatiolls
Solution of Eq. (6.45) gives the following relation W
'T2
(8) ==
0.68gqu'
1-).+-
[{
41t2+~2T2
~2T2
}
- .41t2
cos
21tt
~).T .
2~t\
( )+-sm (-
T
21t'
J
T
+).e
-f}t
,~2T2 óóóï
41i2
-,
]
...(6.48)
The ab
~(e)
~
0.68gqu
4~2+~2r
21tt
21tt
+ ~ATcos ~2T2 Sin - ~)'T e-J3t { T } 21t { T } 21t 41t2,} ]
A-I-
21tT [{
For obtaining the critical time t = te which corresponds Eq. (6.49)is equated to zero. Since 2 1tT/ (41t2 + ~2 T2) :t=0
[
.
]
~1tT -21tt +-cos
/\,- 1-- ~2T2 SIn
'\
[ ]
41t2
T
T
By using small increments t = le can 'then (W/0.68
~).T -J3t ~-
21t t
[ ].--e
.
-
T
in to Eq.
side of
...(6.50)
,'h
,
of time t in Eq. (6.50),
be substituted
to 8 = 8max' the right-hand
0
21t
.
...(6.49)
(6.48)
the value of le can be obtained.
with
known
values
of ~, ). and
This value of B' to obtain
= K, dynamic load factor. Figures 6.15, 6.16 a~~ 6.i7 give the values of K (s2)
g qll )8max
for B = 0.6, 1.5 and 3.0m, respectively, with), = 1-5 and 10
-
N
+0 u
-2
10
.
0 ....
/ f
'0 0 0
-3 10
u
-4
I,
0
/
/
'/
~
/'
"
"
/"
/<'
1 ......
B
0
+0
.'
-" "
, ---,.,
/",
.'
=
O.Gm
0.0
0.5 -------1.0 ---->.-en 2.0 ----
/
"
"u-
oT
OQ1. 5,0 -.-.u 10.0 ~.,
.. .
..
'
"/"'"
,f
-6 10
g
.f
2.0
1.0 .) "0'- "
-
---,--. .-'-' --''-
/'
/'
I
,
I
-5
10
,/
-
,,/",""""'" ~~ .- -,,
1.// .
'10
-7'
,
/
f,' /" , , /
-E 10
0 c ,>.
'
;
..
I0
s-1
------- - -...----...- -::::.---'"---
10
~
0-50
~-------
-I
V\
~=
Ov(Z,rload
---,-""
..
;...
50.0
~
3.0
4,0
ratio»"
5.0
F' _.
"
Fig. 6.15 : Relationship between overload ratio and dynamic load factor for continuous footings 0.6 m wide ,. , " 1..1 .,,;,i
---
'..u.
-
~~,~-~~
256
Soil Dynamics & Machine Foundations'
10
10
-,'
-I
10
N
","'"
III
..., ~ ...
.. 0
t
-2
10
-1
2
10
0 0
-10
-0 -u e 0
10
/
'
'I
I
{/
/
" ' -------. -' -' ,
,---'-,,'..-" -" , """
;'
/' /'
/" """,
'/
I,' '/
/
,
/
,/
-7
10 1.0
0.0
050.0-"'-
., -
.. ,'."'" -'
~
"/- /-
i
102
'0
10
-3
.~ 10 E 0 c
l; 105
,,'/ .,'
;"/'/'
/./.
I .'// ;/ " ./
-c
10
f/ '
..-'
/,/'
" ---",
../.....-
,,/ ....... ..- /r
i.'
"'-"
B
+~
= 3.0m
0.0 0.5--00-1J.O - ----
>o..!!!. 2,0 --=--0 CQ,5.0 - ---
I 0~
/
-~_.-,....-
10.0
50.0
107 1.0
2.0 Qverlood
Fig. 6.16 : Relationship between overload ratio and dynamic load factor for continuous footings
-- - - ;:;.::;-:..-;::.::;=~-
- ,-"" ../-: ------------
-I
10
-/0.
u CQ, 10,0
/
III
0 0
S'=1.Sm t 0.5--- ---2:- 1.0---->0.1111 2.0---0"" 5.0-,-,0
-, ./
10
-
./
--"
,/." "
,... N
u 0 00-
/../"
f:
~ 105
0
I
I
'" ..," "",.'' /,'
---;::.::::'::===:. -" - -------
Fig. 6,17 : Relationship between overload ratio and dynamic load factor for continuous
1.5 m wide
footings 3.0 m wide
Analysis presented by Trianadafilidis (1965) is based on rotational mode of failure. However, it is possible that a foundation may fail by vertically punching into the soil mass due to the application of vertical transient load. Wallace (1961) presented a procedure for the estimation of the vertical displacement of continuous footing considering punching mode of failure. The analysis is based on the following assumptions:
6,5,2, Wallace's
Solution,
(i) The failure surface in the soil mass is assumed to be of similar type as suggested by Terzaghi (1943) for the evaluation of static bearing capacity of strip footings. This is shown in Fig. 6.18.
~B
.
Fig, 6,18 : Failure surface
9 ton r: ro t
~ <'
257
Dynamic Bearing Capacity of Shallow Foundations
(ii) The soil behaves as a rigid plastic material (Fig. 6. 13) (iii) The ultimate shear strength is given by ...(6.51)
s=c+crtan
strength
= Normal stress
(j
peak triangular force pulse (Fig. 6.19).
(v) The footing is assumed to be weightless and to impart uniform load to the soil surface.
-Peak
intensity,
q
.->-
"t:! ..... c: 0
'" c:
..-
(1 ....
(y
..c:
:J
-0
CT\
-0
,-c:
(1 0 ..J
"U (1 0 ..J
'.,
Tim e Fig. 6.19 : Loading function
Analysis The applied load is assumed to be an initial-peak triangular force which decays to zero at time td (Fig. 6.19). The peak load q is expressed in pressure units. Since the function is discontinuous at time td' two equations are necessary
For 0
,;
t
,;
td' Loading function
~
qB (1-
...(6.52)
~)
...(6.53) For t ~ td' Loading function = 0 In Fig. 6.18, BD is an arc of a logarithmic spiral with its centre at O. It is defined by the Eq. (6.54). r
...(6.54)
= ro ea tan'
= Distance
OB (Fig. 6.18)
where, "0
..
.!
-.-..-.......
....
--_Uu_--~---
---
_';::'77~--
I
---
IT' <
258 -
Soil Dy~ic$.
& ¥lIchine Foundations
-
The static bearing capacity - qu for such a' failure surface is -given by 1
...(6.55)
¯« ã ½Ò½õ¯Ò¯õóþøÞÒ§ 2 -
where,
c = Cohesion ...(6.56)
q = "(DJ "( = Density of soil DJ = Depth of footing
Ne, Nq, and Ny = Bearing capacity factors The bearing capacity factors depend on and K, K being 2 (Distance OA)/H, Fig. 6.18. The value of K locates the centre of the spiral which is the centre of rotation. Obviously the correct value of K is that which yields the minimum value of the bearing capacity. It is obtained by trial and error for each set of problem parameters. The values of N"{'Ne and Nq for various values of - and K are given in columns 3, 4 and 5 of Table 6.4. Table 6.4 : Bearing Capacity Factors (N'Y'Ne, Nq' NI' NR) K
(deg)
(1) 0
(2)
- 0.05
10
(3)
-
Nq
(4)
(5)
NI'
NR
(6)
(7)
NI (8)
0.0000
5.7277
1.0
0.0633
2.0125
5.6366
0.0000
5,7124
1.0
0.0631
1.9723
5,5887
+ 0.05
0,0000
5.7258
1.0
0.0633
1.9433
5.5394
- 0.65
0.,1454
79.6255
7.9664
0.3755
8.9076
4.8709
- 0.60
0,1445
29.8163
3.6086
0.2280
6.4362
5.3126
- 0.55
0,1481
18.9958
2,6619
0.1579
5.0332
5.6460
- 0.50
0.1553
14.3469
2.2552
0.12\3
4.1699
5.8636
- 0.45
0.1655
11.8179
2.0339
0.1011
3.6088
5.9750
- 0.40
0,1786
10.2699
1.8985
0.0897
3.2299
6.0020
- 0.35
0.1945
9.2580
1.8100
0.0833
2.9674
5.9698
- 0.30 - 0.25
0.2\31
8.5723
1.7500
0.0799
0.2344
8.1007
1.7087
0.0786
2.7828 2.6523
5.9005 5.8108
- 0.20
0.2585
7.7778
1.6805
0.0785
2.5604
5.7116
- 0.15
0.2855
7.5629
1.6617
0.0793
2.4969
5.6099
- O.IQ - 0.05
0.3154
7.4291
1.6500
0.0809
2.4547
5.5096
0.3483
7.3580
1.6437
0.0829
2.4288
5.4128
0.00
0.3843
7.3366
1.6419
0.0853
2.4155
5.3205
+ 0.05
0.4233
7.3553
1.6435
'0.0881
2.4122
5.2330
- 0.60
0.5700
53.9491
10.5127
0.1120
5.7922
7.1922
0.00 5
Ne
Ny
,-
Dynamic Bearing Capacity -OfShallow Foundations - 0.55
0.5588
28.9945
- 0.50 - 0.45.
0.5645 0.5832
20.5266 16.3539
0.40
0.6127
-:-0.35
' 0:0935
4.8411
7.1948
4.6194 3.8837
'0.0833 ',0.0779
4.2238 3.8095
7.1228 6.9932
13.9337
3.4569
0.0757
3.5264
6.8273
0.6521
12.4031
3.1870
0.0755
3.3323
6.6445
- 0.30
0.7008
11.3881
3.0080
0.0767
3.2008
6.4587
- 0.25
0.7586
10.7004
2.8868
0.0790
3.1147
6.2781
10.2345
2.8046
0.0821
3.0625
6.1071
2.7503
0.0858
3.0360
5.9474
0.0901
3.0294
5.7994
~
- 0.20
,0.8253
.:...0.15'
0.9012
' 9.9267
- 0.10
0.9863
'9.7361
1.0807
9.6352
3.0386
5.6676
9.6049
2.6936
0.0999
3.0604
5.5360
0.05
1.2986
9.6313
2.6983
0.1053
3.0923
3.4187
- 0.55
1.5462
46.5473
13.4724
0.0707
5.2677
8.6324
- 0.50
1.519830.2759
9.1124
0.0696
4.7177
8.2310
- 0.4'5
1.5342
23.2038
7.2175
0.0707
4.3564
7.8481
- 0.40
1.5806
19.3483
6.1844
0.0734
'4.1189
7.4903
- 0.35
1.6540
16.9964
5.5542
'0.0773
3.9669
7.1622
- 0.30
1.7520
15.4722
5.1458
0.0823
3.8766
6.8645
- 0.25
,1.8730
14.4550
4.8732
0.0881
3.8322
6.5961
- 0.20
2.0166
13.7730
4.6905
'0.0947
3.8232
6.3542
2.1825
13.3257
4.5706
0.1020
3.8418
6.1361
-0.10
2.3710
13.0501
4.4968
0.1101
3.8825
5.9388
-0.05
2.5823
12.9048
4.4579
0.1183
3.9413
5.7596
2.8168
12.8613
3.6745'
46.2884
- 0.45
3.6419
33.8986
- 0.40
3.6943
27.6099
'
.
',-
'
9.1768
13.3381
0.0728
5.3067
8.5380
, ,. 11.0492 "
0.0796
5.0886
7.9941
""",,',
20.0542
4.5161 '
19.0369\"7~9289
5.6804
"'.17:~542
"",:6.1717 .~~tt~:l...§r,'
.,' ,
5.6658
,
4.2298'
18.J742:
-
' ,,0.0673
. 8.&572
17.?678
' ';;
17.8477
. 9.7067
5.8533
0.50
5.4463
23.9713
. 5.2413;
'::-" , ,~"
5.5961
4.1008
2'i.5'S75
0.10
"',," :. . .,'tS"t,~p{~2.
4.0149
0.1383
3.8151
- 0.15
:. O~OO, '",~,',.,?'r"".,..,.
0.1282
,
.
-0.05
'4.4462 :'4.4563
3.9952"
,
0.25
- 0.20 ~
"
'12.8991
- 0.50
- 0.30
. '"
' 2.6990,0.0948
- 0.15
, -.0.35
'. "',
,2.7167
1.1848
0.00
25
;,
0.00
+ 0.053.0750 20
' 6.1125
- 0.05 +"
15
: 259
': "8.5665", "', ':,'~; '. "
, , ." -': , . '
i "
.4.9684
,
7.5267
4.9199
7.1214
4.9258
6.7672
'
0.1076 '
:4.9746
' 0.1194
6.4552
'. ' '7 :6f,77
,< 0.1325
5.0582
6.1783
'1.5-398
,', ,,0.1470
5.1704
5.9309
,-..:0.,1629
5.3068
5.7084
5.4638
5.5072 ,
;'
".7.4620 .,
7.4368
l:~...!l.
73.8778
0.0~77
., '0.0970
'8.2992'
:~~'.J7.6903
'
"35.4499
"",-.0.1802,
'..",,",
,-
'.\,.;,~~~l,989 ,,~5,~486 ',0.0732,
'
,
7.23469.9384'
,~:3243
,/~;:~{~~r~,'1frY~/*~;;:~:' '/;2d'i~:!,
; ,";"J'\
~
',,' ";;"':~~!.;'"
-
260
;'.l';'t,',,~:
~'i""\
.,;c,-'", ,
...'
~.,
Soil Dynamics & Machine Foundations
30
35
- 0.45
8.3599
51.2706
24.9079
0.0835
6.8363
9.0503
- 0.40
8.3728
40.7056
19.9814
0.0954
6.6214
8.3291
- 0.35
8.5541
34.7663
17.2119
0.1094
6.5339
7.7297
- 0.30
8.8760
31.1015
15.5029
0.1254
6.5404
7.2223
- 0.25
9.3230
28.7315
14.3977
0.1437
6.6199
6.7864
- 0.20
9.8871
27.1750
13.6720
0.1646
6.7584
6.4075
- 0.15
IQ.5646
26.1681
13.2024
0.1882'
6.9462
6.0748
- 0.10
11.3542
25.5533
12.9157
0.2148
7.1761
5.7803
- 0.05
12.2569
25.2309
12.7654
0.2445
7.4429
5.5178
0.00
13.2745
25.1345
12.7205
0.2775
7.7423
5.2825
+ 0.05
14.4095
25.2180
12.7594
0.3139
8.0710
5.0704
- 0.45
19.3095
80.8644
47.6872
0.1064
9.3123
- 0.40
19.1315
62.4470
37.0539
0.1267
9.0899
9.3540 , 8.4705
- 0.35
19.3718
52.5548
31.3426
0.1506
9.0494
7.7518
- 0.30
19.940
46.6067
27.9084
0.1787
9.1446
7.1533
- 0.25
20.1887
42.8208
25.7226
0.2116
9.3473
6.6458
- 0.20
21.9566
40.3597
24.3017
0.2500
9.6392
6.2095
- 0.15
23.3512
38.7778
23.3884
0.2944
10.0081
5.8303
- 0.10
24.9984
37.8159
22.8330
0.3456
10.4452
5.4979
- 0.05
26.8993
37.3127
22.5425
0.4041
10.9441
5.2044
0.00
29.0580
37. 1624
22.4558
0.470<:
11.4998
4.9436
+ 0.05
31.4810
37.2926
22.5309
0.5457
12.1084
4.7107
- 0.45
46.2942
134.3023
95.0397
0.1527
13.4981
9.4021
- 0.40
45.4427
100.6609
71.4837
0.1887
13.2639
8.3844 ,
- 0.35
45.6687
83.4477
59.4308
0.2323
13.3114
7.5703
- 0.30
4.6.7356
73.3676
52.3727
0.2849
13.5708
6.9017
- 0.25
48.5145
67.0529
47.9511
0.3481
14.0015
6.3419
50.9356
62.9887
45.1052
0.4237
14.5786
5.8661
- 0.15
53.9640
60.3926
43.2874
0.5133
15.2895
5.4569
- 0.10
57.8568
58.8199
42.1862
0.6191
16.1127
5.1018
- 0.05
61.805 I
57.9989
41.6113
0.7428
17.0515
4.7911
0.00
66.6296
57.7539
41.4398
0.8868
18.0970
4.5175
+ 0.05
72.0773
57,9662
41.5884
1.0529
19.2451
4.2753
- 0.40
115.7097
112.8231.
.146.0161
0.3229
20.8738
8.0404
- 0.20
'40
.
.
-
,
,
261
Dynamic"Bearing ,Capacity of Shallow Foundations
- 0.35
115.5504
141.1002
119.3973
0.4107
21.1138
7.1701
- 0:30
117.6386
123.0124
104.2199
0.5195
21.7125
6.4650
- 0.25
121.5875
111.8576
94.8599
0.6536
22.6077
5.8817
- 0.20
127.1879
104.7472
88.8935
0.8175
23.7619
5.3914
-0.15
134.3346
100.2323
85.1051
1.0168
25.1570
4.9741
'- 0.10
142.9868
97.5069
82.8181
1.2572
26.7775
4.6 I52
- 0.05
153.1451
96.0866
81.6263
1.5450
28.6173
4.3038
164.839
95.6630
81.2709
1.8870
30.6724
4.0317
+ 0.05
178.1176
96.0303
81.5791
2.2904
32.9409
3.7924
- 0.40
327.6781
322.2748
323.2752
0.6576
36.2961
7.4295
- 0.35
325.4943
259.1345
260.1349
0.8611
37.0113
6.5559
- 0.30
329.9752
224.0769
225.0772
1.1194
38.3965
5.8568
- 0.25
339.8627
202.7837
203.7840
1.4447
40.3468
5.2846
- 0.20
354.4804
189.3358
190.3361
1.8515
42.8070
4.8083
- 0.15
373.4971
180.8450
181.8452
2.3565
45.7496
4.4062
- 0.10
393.7473
175.7358
176.7361
2.9784
49.1634
4.0628
- 0.05
424.2605
173.0775
174.0778
3.7386
53.0475
3.7669
0.00
456.1177
172.2851
173.2853
4.6607"
57.4067
3.5096
+ 0.05
492.4763
172.9729
173.9732
5.7709
62.2499
3.2843
0.00 45
,
"
,
.
'
Any acceleration of the soil mass ACDBA due to the downward movement of the footing will cause inerti::1 forces which will resist jmch movement. The inertial forces are directly proportional to the acceleration of each individual soil mass and thereby dependent on displacements. The effective total inertial force is obtained by combining the inertial forces on ech separate mass using energy considerations. 2
The inertial
force
...(6.57)
is given by, If ==NI Y B ddt "
~ == Displacement
where,
NI
==
at any time t
Coefficient of dynamic inertial shear resistance
The coefficient NI. depends on cl>and K, and its values are listed in column no. 6 of Table 6.4. Displacement of the soil mass within the failure surface due to downward movement of the footing will increase the restoring moment about the point 0, and the increase in moment will be proportional to the displacement provided the rotation is not excessive. It is expressed as RF = NRB Y~ '.The
coefficeint NR also depends on .
...(6.58)
and K. Its values are listed in column no. 7 of ,. Table 6.4. The differenti~J" '; "'.." J ' . " equati~~s ""," are .'" setup",~Y,_~~uatingtheJour , .". . " - ~, 'vetiCalfo~ces \",.. to z~ro. .There ." ." m~st be qi
"
separ~te equation,s for be(qre!lP,~i~fteF >titpe id'._, sin~e the loading ~uncn~m is def11}edin !ha,t man!1er. , . ">,,,) ,..,.",.1"""," U,. ,.'~ , . 0" ~~"" ".h' 'J,,~".' ) ~
"ForO -
"
"',
,,'
"
ó
óÁò
ô
þþôù
ùóóóùó
Soil Dymimics &: MaclrineFouniJiitions
óîêî
ùî
ò ÿòô
¬
ôó
Ò×§Þ óóó®õÒΧެÿòõ¯«Þó¯Þ£óó ô
¼Ö
ó
ø
ùò
óô¬¿ ÷
¼î¬ÿò õ¢¬ÿò
or,
¼¬î Ò×Þ
...(6.59 a)
ãð
ã ó¯ó¯« ÒקÞ
¯
ô
Òקެ¼
¬
...(6.59 b)
For t ~ td 2 d2A
"
NI yB ---r+NR dt
...(6.59 c)
yBt!.+qu B =0 "
d2t!. NR t!. = -- qu -+...(6.59 d) d t2 NIB NI YB The solution of the differential equations will yield equations of footing displacement versus time. The forms of the particular solutions ofEq. 6.59 (b) and Eq. 6.59 (d) are found to be
or,
and,
t!.
= Cl
A
= C3 cos (K't) + C4 sin (K't)
cos(K't)+C2
Sin(K't)+
¯ó¯«
q
( NR"t, ] ( NRytd )
t ...(6.60 a) -
...(6.60 b)
!k-
,
respective! y, in which K' ~
-
NRy
~ ~~~ ;and ~.. C2 C3 and C4 are
coefficients
of integration,
The coefficients
Cl and C2 are evaluated by the initial conditions. The coeffIcients C3 and C4 are evaluated by the conditions of displacement and velocity at td as defined by Eq. 6.60 (a). Solution and substitution of
the coefficients yield nondimensional Eqs. 6.60.
.
For 0 ~ t ~ td
q NR "t t!. ( qu .)
= L-I ( qu ) [I-COS(Klt)]+ tdq~/[Sin(K1t)-:-(K1t)]
...(6.60 c)
For t ~ td
(N:, Y) ~
~
\(1- æ÷
ÖÕù Í·ÑøÕþ¼ÝÑÍøÕùId) + [JK' {1-COS(K"d)+in(K")-1
...(6.60 d)
The coefficientsNY'Nc' N q, NI and NR are dependent only on values of and . K. Using magnitudes of from 0° to 45° and ofK for the region where the ultimate static shear resistance could be a minimum, these coefficients were evaluat.ed. The' values obtai~ed are given ,in Table 6,.4for every fifth degree. The maximum displacement from Eq. 6.60 (a) and Eq. 6.60 (b) is the predicated permanent footing displacement,sitice-doWnward niotion ceases at the.time'ofma?ci~Um displacemetit'imd rebdund is not -
~onside~ed. The'se data aI6ng' ;ith th'etim~s'~;of~~rifu'fdl~pIac:ement
are giv'e~ ID'Fi~'-'6;20...
If' .~~
"amic Bearing Capacity of Shallow Foundations
(p\)
JlZ =i~ Pl-
UO!~OJ np
263
pOO1 IOUO!SU2>W!PUON
.-
0
00
..
U") 0
x C1
N
ci
E
..---. --; 0
:
:J
~ , er
U")
-I.
+E
E ::I E
c to'
to' U 0 N 0
..
c,; '" C.:!!
........
0
= '"
a.
U\ ."0
"0
... '"
:: '" .~ ... ~
E :J .-E x: 0 E
0' <'. ' ~ \\
0 c 0
'),
-d
.~
d
C to'
N 0 0
.
.0 0
-
6
\l'
N
-
U")
0 U") .'
0
( P~ ) lLZ =
>I
P ~- uoHoJnp
.
-:0 00
.
POOl )OUOISU2>WIPUON
E ."'0 C 0
Z
... 0 7.
c .....
..i
OD ~
,)~f
264
Soil Dynamics
& Machine
Foundations
6.5.3. Chummar's Solution. Chummar (1965) presented a solution for dynamic response of a strip footing supported by c - 4>soil and subjected to horizontal transient load. The analysis is based on the following assumptions: . (i) The failure of the footing occurs with the application of
~ horizontal dynamic load acting at
a certain height above the base of the footing.
.
(ii) The resulting motion in the footing is of a rotatory nature. The failure surface is a logarithmic spiral with its centre on the base corner of the footing, whic.h is also the centre of rotation (Fig. 6.21). (iii) The rotating soil mass is considered to be a rigid body rotating about a fixed axis. (iv) The soil exhibits. rigid plastic, stress - strain characteristics. Q ~
AQ
...
r
C2
Log spiral
r
=
ro eetan
~
/c
-
Re~u1tant
friction
Fig. 6.21 : Transient horizontal load on a continuous footing resting on ground surface.
Analysis The static bearing capacity of the footing is calculated by assuming that the footing fails whe' acted upon by a vertical static load, which causes rotation of the logarithmic spiral failure. The ultimat static
bearing capacity qu is given by
1 + -2 y BN y .
qU =cN where,
c
c = Cohesion B = Footing width and equal to the initial radious of spiral y = Unit weight of the soil
Ne and Ny
= Bearing capacity factors fOl:the assumed type of failure
...(6.61
..
I
(~:'
265
Clmic Bearing Capacity of Shallow Foundatiolls
Considering
moment of the forces about 0, the centre of rotation: 2
d Moment where
'
2 . cB 27ttan~ B ue to cohesIOn c, MRC = 2 tan cl>(e -1) = 'I' C ( 27ttan~ 1) e -
\jI=
,
...(6.62a)
...(6.62 b)
2 tan cl>
Moment due to weight W of soil wedge, M RW
= 'YB3 tan 4>(e31ttan
...(6.63 a)
tan 4>(e37t tan tjI+ 1)
where,
E = 4>
...(6.63 b)
2 9 tan 4>+ 1
= Angle of in~ernal friction B2
Moment
It gives,
of qu about
0
=
qu
2
=
...(6.64)
MRC + MRW
-
q -u
c
(e
- 1) + 2'YB tan 4>(e
37t tan
+ 1)
21t tan
'
...(6.65)
9 tan2 4>+1
tan 4>
Combining Eqs. 6.61 and 6.65, we get '.,
N = 4 tan 4>(eh tan~ + 1) y 9 tan2 4>+ 1 and,
N = c
...(6.66)
e27ttan~ - 1 tan cl>
...(6.67)
With a suitable factor of safety F, the static vertical force on the foundation per unit length can ~ gIven as
Q = ~(cNc+i'YBNy)
...(6.68)
The variation of dynamic force considered in the analysis is shown in Fig. 6.22. In tnIS Qd (max) =
AQ
...(6.69)
where Qd (max) = Maximumvalue of horizontaltransientload per unit length actingat height H above
base of the footing A = Over load factor
",':m
"-="~'.'~..~
~.."'~..'.~~~
166
,..
.
~
Soil Dynamics & Machine FolJndations
"U 0 0
.~ P E 0
Pmax
c >Cl
~
~
td Timq
Fig. 6.22 : Loading function
For considering of the dynamic equilibrium of the foundation with the horizontal transient load, the moment of each of the forces (per unit length) about the centre of the log spiral needs to be considered: 1. Moment due to the vertical force Q M = I
-1 Q B
...(6.70
2
2. Moment due to the horizontal' force Qd at any time t Qd(max) Ht M2
= QdH =
td
Md(max) td
t ...(6.71
where Md (max)= Qd (max)H 3. Moment due to the cohesive force acting
along the failure surface is given by 'Eq. 6.620
4. Moment due to weight of soil mass' in the failure wedge is' given by Eq. 6.630. 5. Moment of the force due to displacement Figure 6.21) from its initial position: M)
of the centre of gravity of th~ failure wedge (C'
= W d..X
...(6.,
where W is the weight of the failure wedg~, and given by
W=
'YB2 (i
It tan $
- 1)
(4 tan cp) /1 X = R cos (11- ex) - R cos 11
and R
= QCI
...(6....( 6.
(Fig. 6.21). When ex is small, Eq. (6.74) can be written as /1 X = (R sin 11)ex
...( 6
Ilj,~"r:~!)"
267
Oynamic Beating Capacity of Shallow Foundations
However, where,
R
=
X
=
...(6.76)
~(x)2 + ("2)2 -4 B tan 2 <1> ( e31t tan cj)+ 1)
(9 tan2 <1>+ l)(i1t
-
4Btan(e31ttan~+1)
z =
...(6.77)
tancj)-1)
...(6.78)
3( ~9tan2 <1>+l)(i1ttan~-l)
Combining Eqs. (6.72) - (6.78)
.
.
= PB
M3
where,
.
3
...(6.79)
(sin 11)a
p =(e31ttan~+1)
...(6.80)
)
3 ( ~9 tan2 <1> +1
6. Moment due to inertia force of soil wedge: d2a
M = -Z J 4 ( d[ ) where J is the mass moment of inertia of the soil wedge about the axis of rotation 1 B4
J =
[ 16gtan<1>]
(i
1t tan $
- 1)
...(6.81 )
...(6.82)
.
and g is the acceleration due to gravity. Substitution of Eq. (6.62) into Eq. (6.81) yields M = ~c 1 B4 .-d2a 4 g d [2 where
~c =
...(6.83 )
(e41ttancj)-I)
...(6.84)
16 tan ~
Moment due to the frictional resistance along the failure surface will be zero as its resultant will, pass through the ce'ntre' of log spiral. Now for the equation of motion, 'M\ + M2 = MRC+ MRW+ M3 + M4 .Substitution of the 'proper terms for the moments in Eq. (6.85) gives
...(6.85)
...(6.86)
(~'t~)+ K' a where
~
A [( Md;;")}
-
k - -
,
gp sin 11 ~c B
; QB-E] ...(6.87)
...(6.88)
A=~ (1 B4 ~c) E = \jIC132+ E 1 B3
...(6.89)
,.-fj:
III'II:~
268
Soil Dy"amics & Machine Fou"datio"s
Solution of the differential equation of motion [Eq. (6,86)] with proper boundary conditions yields the following results: For I ~ Id A
1 E--QB k2 2
(
a =-
)
cos(kl)--'
A Md(max) k3
td
, A Md(max).t 1 srn(kt)++-QB-E kZ ( Id 2
)
...(6.90)
For I > Id a = (
~
) [G 1 k
cos (k Id) - G 2 sin {k td) ] cos ( k t) +
()
[G 1 k sin ( kId)
~
...(6.91
-Gz COS(kld)]Sin(kt)+(~ )(~QB-E) A
h were, G I = 2" k and,
.
1
A
Md (max) .
( E - -2 Q B) cos (k Id) - -3" k
A 1 G2 = -k' E--QB ( 2
)
.
Id
A
sm(ktd)-Z'
srn ( k Id) +
Md(max)
~
k
A Md (max)
k
...(6.92
z
A Md(max) cos(ktd)+
k
2
...(6.93
~
The procedure of computations have been discuss~d in example 6.4.
!
ILL USTRA TIVE EXAMPLES I
Example 6.1 Proportion an isolated footing for a column of 500 mm x 500 mm size subjected to a vertical 10 of 2400 kN. The structure is located in seismic region. The earthquake force results a moment of 4 kN-m and shear load of 360 kN at the base of the footing. The soil properties are as follows: C
=6
. kN/m
z
, =
39° and y = 18 kN/m
3
A plate load test was performed at the anticipated depth of foundation on a plate of size 600 r x
600 mm and a pressure settlement record as given below was obtained. The permissible values
settlement, tilt and lateral displacement 2 240 Pressure (kN/m ) 0.0 Settlement (mm) Solution:
0.0
are 50 mm, 1 degree and 25 mm respectively. 480 720 1200 1440 960
2.0
5.0
7.5
12.0
16.0
(i) Safe bearing capacity Eccentricity of load,
e =
~
=
~OOOO
Inclination of load ~ i ~ tan-I ( ~:)
= 0.1667 m ~ tan- t U4~O ) = 8.5 3'
23.0
l( 2
~n 269
Dynamic Bearing Capacity of Shallow Foundations
Let the size of footing is 2.0 m x 2.0 m and is located at 1.0 m depth below ground surface. Hence, and
B' = B-2 e = 2 - 2 (0.1667) = 1.667 m L' = L = 2 m . qnu = [e Ne Se de ie + YIDlNq - 1) Sq dq iq rw + 1. 2 Y2B NySydy iy r'J
Since <1> = 39° it is the case of general shear Ny = 100.71 Depth factors are given as-
failure. For <1> = 39°, Ne = 70.79, Nq = 59.62 and
DJ ~ " de - 1+ 0.4 I3'"" - 1+ 0.4 1.667 - 1.24
=
dq
. 2 DJ 1+2 tan (I-sm°
= 1+ 2 tan 39 (J - sin 39)2 1.;67 = 1.13 dy = 1
and
Shape factors are calculated as : Br
Se = 1+0.2T 0"
1.667
= 1+0.2~
B'
Sq = 1+0.2T
1.667
= 1+0}-2 , ,
B' Sy = I-OAT Inclination and
factors:
= 1.167
1.667
= 1-0.4~
= 0.667
Q = ~Q~ + Q~ = J3602 + 24002 = 2426.85 kN m
=(2+~)(1+~)
= (2+1.~67)(1+1.~67)= 5.195
.
Therefore,
= 1.167
i-I q
m
Q Q + B' Le cot ]
[
360 = [ 1- 2426.85+ 1.667(2)( 6) cot 39 ] and
i =i - (l-iq) e
q
. 'y -
1-
[
50195
= 0.438- (1-0.438)
Ne tan
Q
[
= 0.428
70.79 tan 39 m+1
h Q + B Lc cot ] "
= 1
= 0.438
360
2426.85 + 1.667 (2) (6) cot 39 ]
6.195
= 0.374
".'
270
Soil Dynamics
& Machine Follndation~
Assuming water table to be below the ground surface at a depth greater than (Dj + B), hence r '" = r' '"= 1, and also sa y Y1 = Y2 = Y hence,
qllll = CNe Se de ie +YI Dj (Nq -1) Sq dq iq rw+~ Y2 B Ny Sy dy iy r~ qllll= 6 (70.79) (1.167) (1.24) (0.428) + 18 (1) (5~.62 - 1) (1.167) (1.13) (0.438) (1) + (0.5) (18) (1.667) (100.7) (0.6677) (1) (0.374) (1) = 263,06 + 609.45 + 376.92 = 1249.43 kN/m2
According to Meyerhof-
.
I
"
2
( ) (
le =Iq = 1-90
=
853
1- 90
2
)
= 0.82
. 2
. -
I
Iy - ( 1-~
8.53 2 = 0.61 ) = (1-39)
Therefore, ql1l1= 6 (70,79) (1.167) (1.24) (0.82) + 18 (1) (59.62 - 1) (1.167) (1.13) (0.84) +
I
(18)(1.667)(100.71)(0.667)(1)(0.61)(1)
= 503.99 + 1140.98 + 614.76
From charts -
kN/m2
(According to Saran & Agarwa1, 1991)
=0.0
For
= 2259.74
= 39° , ~ = 0.1g67 = 0.083, and i = 8.53°
(Fig. 6.4 to 6.6)
For
= 0.10
(Fig. 6.4 to 6.6)
i = 0°
i = 10°
i = 8.53°
; = 0°
; = 10°
i = 8.53"
Ny N
144
58
70.64
75
42
46.85
68
41
44.97
47
30
32.50
N .c
88
48
53.88
58
37
40.09
q
Hence for
~ = 0.083 and i = 8.53,
Ne = 50.89, Nq = 34.62 and Ny = 42.43 In this case - [As effect of eccentricity and inclination has considered already] ;=;=;=1 e q y B =2m
..
-~'~
.---
Dynamic Bearing Capacity of Shallow Foundations
271
D
Depth factors will be :
de
= 1+ 0.4
1
.
£ = 1+ 0.4"2 = 1.2
dq = 1+ 2tan 39 (1- sin 39)2 ~ = 1.11 dy = I and shape factor will be<
Thus
S e = 1.3, Sq = 1.2 and Sy qnu
=
0.8
(As footing is square)
= 6 (50.89) (1.3) (1.2) (1) + 18 (1) (34.62 - 1) (1.2) (1.11) (1) (1)
+ ~ (18) (2) (42.43) (0.8) (1) (1) (1) = 476.33 + 806.07 + 610.99
= 1893.39 kN/m2 Therefore, value of qnufrom charts lies between values of qnuobtained by Eq. 6.17 and Meyerhofs method. Hence
= qnu x Area of footing Qnu = 1893.39 x 2 x 2 = 7573.56 kN
Qnu
Factor of safety = 7;:t~6 = 3.16 > 3, Therefore foundation is safe against shear. (2) Settlement computationWhen footing is subjecting to a central vertical loa~.only, in that case e / B = 0 and i = 0 For '" 't' = 390'e N = 88'yN = 144 and Nq = 68 thus qnu =(6)(88)(1.33)(1.2)(1) + 18(1)(68-1)(1.2)(1.11)(1)(1) + ~ (18) (2) (144) (0.8) (1) (~) (I)
= 823.68 + 1606.39 + 2703.6 = 4503.67 kN/m2 Pressure on footing corresponding to a F.O.S. = 3.16,- will be = 453~~667= 1425.22 kN/m2 From plate load test data corresponding to pressure 1425.22 kN/m2, Sp = 22.2 mm 2
. Sf Now smce Sp
.
Bf (Bp+30)
= [ Bf
(B f + 30) ]
Thus settlement of footing when it is subjecting to central vertical load will be
200(60+30)
Now,
So
= Sf = [ 60 (20 + 30) ]
-!...
- 8.53,,;,0.2187
~ - 39
2
x 22.2 = 37.77 mm .
I I
-
I
.;'!
~
272
Soil Dynamics & Machine Foundations .
Hence AD
1-0.56W-O.82(~J
~
= 1 - 0.56 (0.2187) - 0.82 (0.2187)2 = 0.838 AI ~ -3.51+ 1.47
=-3.51 A2
(~ )-5.67 (~)'
+ 1.47(0.2187) + 5.67(0.2187)2 = -2.917
4.74-1.38(~)-12.45(~)'
~
= 4.47 -1.38 (0.2187) - 12.45 (0.2187)2 = 3.843 Se So = Ao+AI
e B +A2
e B)
2
() (
2
..
Se
= 0.838-2.917(°.1;67)+3.843(°.1;67) = 0.6216 = 0.6216 So ~ 0.6216 x 37.77 = 23.48 ³³
and ÞÜ
¢
ïóðòìèÉóÑèîø¢®
= 1 - 0.48 (0.2187) - 0.82 (0.2187)2 = 0.856 B, ~ -1.80+0.94(~)+1.63(~)'
= - 1.80 + 0.94 (0.2187) + 1.63 (0.2187)2 = - 1.516
i:
~
Bo+BI(~)=
0.856+(-1.516)(°.1;67)
= 0.7296
Srn = 0.7296 So = 0.7296 (37.77) = 27.56 mm < 50 mm (safe) .
sm t
= S m -S B
T-e
e
= "27.56- 23.48 = o 0049 2000 . ~-166.7
t = 0.28° < 1° (safe) and
H -1l
- 0.682 (0.2187)2 + 1.99 (0.2187)3- 2.01 (0.2187)4 B = 0.12 (0.2187) . = 0.0101
Ho = 0.0101 B = 0.0101(2000) = 21.21 mm"< 25 mm (safe) Example 6.2 A 1.5 m wide strip foundation is subjected to a'vertical transient stress pulse which can be given as qd = 650 e-IOt kN/m2. The soil supporting the foundation is saturated clay with Cu= 60 kN/m2. The unit weight of soil is 19 kN/m3. Determine the maximum angular rotation of the footing might undergo.
.
~~".<
273
Dynamic Bearing Capacity of ShaUow Foundations
Solution: 2
1.
qu =.5.54 Cu= 5.54 x 60 = 332.4 kN/m 650 A.= 332.4 = 1.955
2. Reffering Fig. 6.16, For A.= 1.955 and W
( 0.68 g qll )
8max
~ = 10s-1
= 0.00298 2
W=0.311tyB
=0.311t19 x (1.5)2= 41.61 kN Therefore.,
8max = (0.00298) (0.68 x 9.81 41.61x 332.4)
= 0.158 rad
= 9.1°
Example 6.3 A 2.5 m wide continuous footing located at 1.5 m below the ground surface is subjected to a vertical transient load (qd (max) = 3000 kN/m2, td = 0.3s). The properties of the soil are y = 18 kN/mJ, ~ = 30° and c = 50 kN/m2. Calculate the maximum vertical movement of the foundation. Solution: 1 1. qu = CNe + y DJ Nq + 2" y B Ny
'.
1 = 50 Ne + 18 x 1.5 Nq + 2" x 18 x 2.5 Ny
= 50 Nc + 27 Nq + 22.5 Ny The computations
of qu are done for 'I>= 30° and different values
factors from Table 6.4. These are given below in Table 6.5. Table 6.5: Computations k
- 0.45 - 0.40 - 0.35 - 0.30 - 0.25 - 0.20 - 0.15 . - 0.10 - 0.05 - 0.00 +0.05
of k by taking Nc' Nq and Ny
.
of qu for Different Values of k 2 qu (kN/m ) 5765 4553 3910 3533 3304 2512 2204 3070 3080 3118 3181
274
'. . Soil
The minimum value of qu is obtained at k
2. For
=-
Dynamici & Machine' Foundations
0.15 as 2204 kN/m2.
k = - 0.15 and = 30°, from Table 6.4.
{N;:
NR
= 10.0081 and .
V~ = 5.8303 .
NR Y = 10.0081 x 18 = 0 08174 qu
2204 k'
~
3.
-
~~~~
= 5.8303 qd (max)
3000
-qu
= 2204=
.
~~~ .
~
.
~ 2~5 = 5.22
x
1.36
td K' = 0.3 x 5.22 = 1.566
For qd (max) = 1.36 and td k' = 1.566, Fig. 6.20 gives qu Y NR -'Umax qu
= 0.034
A
t:. = 0.034 max 0.08174 =0.416m.
Therefore,
Example' 6.4 A 2.5 m wide continuous surface footing is subjected to a horizontal transient load of duration 0.4 s applied at a height of 4.0 m from the base of footing. The properties of the soil are y = 17 kN/m3, c = 30 kN/m2 and = 32°. Determine the value of the maximum horizontal load that can be applied
on the footing. Also compute ,the rotation at time equal to 0.6 s. Solution: (i) Determine Q using a suitable factor of safety (= 2.0). c = 30 kN/m2, = 32°, Y = 17 kN/m:
md B = 2.5 m
. .
.N =
e21t
c
-1=
tan,
~
tan
e21t tan 32° - 1 . = 79.4 tan 32° .
4 N = 4 tan~ (e31ttanljl + 1) = y 1 + 9 tan2 ~ Q
= !2 B =
;
x
(
c N
c
tan
31t tan 32°
(e
+
1)
= 200
1 + 9 tan2 32°
+! YB N 2 y
2.5(30 x 79.4;
32 °
)
;
x 17 x 2.5 x 200)= 8290 kN
275
Dynamic Bearing Capacity of Shallow' Foundations (ii) Determine
'11, E, /le'
~ and
sin 11
H = 4.0 m td=O.4s i1t
V= E=
-
- 1
tan'
79.4
2 tan ~
=2
.
= 39.7
tan ~(e31ttan, + 1)
(1 + 9 tan2 ~) = 200 = 50 4 e 41t tanl\» - 1 ~c =
16 'tan~
= 256
e31ttan, + 1
~=
=' 56.6 3[ ~9 tan2~+I]
"
,
'....
- = -4Bta~2~ (e31ttan'+I) (1+9 tan2~) (i1ttan, -1)
X
50 = (- 2 B) 39.7 ~,- 2.52 B z =
.' 4B tan (e31t tan cl»+ 1) 3[ ~9 tan2 + 1] (i1t ta~ cl»-1)
= (2B) . SIll 11
~~:~
Z,
=
= 2.85 B =
~x-2+z-2
2.85 B .
= 0.75
"
~(-2.52B)2+(2.85B)2
(iii) Determine k, A and E,
k ~ ~g
A = E
9.81 x 56.6 x 0.75
(256x 2.5)
P lleB = sin
1]
g = 9.81 'YB4 Il 17 x 2.~ x 256
= 'IIcB = 39.7
3
= 807 ,
"
= 0.0000577 .
3
+ E 'YB
x 30 x 2.52 + 50 x 17 x 2.53
= 20700 kN
Soil Dynamics & Machine Foundations
276 (iv) Determine Md (max)in terms of Md (max) =
A.
H.
Qd (max) = H .
= 4 x 8290 A. = (v) Determine A.crwhich corresponds to a.
A
a. = - 2
k (E
A.
=0
1
A Md(max).
A
2
k3
k2
óóÏÞ ) cos(kt)--.
td
.Q
A.
33160
sm(kt)+-
Md(maX)1
[
td
1
õóÏÞóÛ] 2
For t = td
)
a. = 0.00005277 20700-.!. x 8290 x 2.5 cos (0.8070 x 0.4)
0.807
(
2
0.00005377.33~~~A sin (0.8070 x 0.4) 0.807 .
+ 0.0000577 . 33160 A x 0.4 + 1. x 8290 x 2.5 - 20700 ] 0.8072 ( 0.40 2
= 0.9159 cos (0.3228) = 0.05 A. - 0.0474
- 9.10 A.sin (0.3228) + 2.94 A.+ 0.9181 - 1.834
For a. = 0, A = 0.948 = Aa (vi) Determine Md (max)for Md(max)=
33160Acr = 33160 x 0.948 = 31436kN-m
(vii) Determine Gland A
G1 = 2k
(
= A.cr
A.
G2
1
A Md(max) .
)
E-- 2 Q.B cos(ktd)--r
k
td
A
sm (k td ) +2 k
M
d(max)
= 0..8685 - 2.89 x 0.948 + 2.94 x 0.948 = 0.9159 1 G 2 -- --A E--Q.B k 2
(
)
.
A Md (max)
sm(ktd)--' k2
= - 0.9159 x 0.807 sin (0.3228) = - 0.2346 - 6.60 + 2.79
td
A Md (max)
cos(ktd)+-.k2
td
- 9.1 x 0.948 x 0.807 cos (0.3228) + 2.94 x 0.948
= - 4.05 (viii) Determine a. for t = 0.6 s
CL
~
G }GI k cos (kid) -G, sin (kid)] cos(k 'd )+G}GI +
E ~K2(.!.QB) 2
k sin (kid )-G,
cos(k 'd)] sin (k I)
;!&~t'
277
Oynamic Bearing Capacity of Shallow Foundations
1
"
= 0.807 [0.9159 x 0.807 cos (0.807 x 0.4) + 4:05 sin (0.807 x 0.4)] cos (0.807 x 0.6) + 0.;07
[0.9159 x 0.807 sin (0.807 x 0.4) - 4.05 cos (0.807 x 0.4)] sin (0.807 x 0.6)
+ 0.0000577
0.8072
1. x 8290 x 2.5-20700 [2 ] '
0
1 1 = 0.807 [0.701 + 1.285] x 884 + 0.807 [0.2346 - 3.841] x 0.466 - 0.9159 = 2.175 - 2.082 - 0.9159 = - 8229 º¿¼ '
'
'
,
ÎÛÚÛÎÛÒÝÛÍ Agarwal, R, K. (1986), "Behaviour of shallow foundations subjected to eccentric
University of Roorkee, India. Caquot, A. (1934), "Equilibre
"
- inclined loads". Ph,D. Thesis,
'
des massifs a frottement interene", Ganthiev-Villars,
Paris.
Chummar, A.V. (1965), "Dynamic bearing capacity of footings", Master of E'ngineering Dissertation, University of Roorkee,
India.
'
,
Cunny, R.W. and'Sloan R. C. (1961). "Dynamic loading machine and results of preliminary small-scale footing tests", A.S.T.M. Symposium on Soil Dynamics, Special Technical Publications. No. 3.5, pp. 65-77. De Beer, E. and Vesic, A. (1958), "Etude experimental de la capacite portante du sable sons des fondations directed etablies en surface", Annales des Travaux Public ,~e Delgigue, .59 (3), pp 5-58. Fellenius, W. (1948) "Erdstatische bcrchnungen", 4th ed., W. Ernst Und Sohn, Berlin. Fisher, W. E. (1962). "Experimental studies of dynamically loaded footings on sand", Report to U. S. Army Engineer Waterways Experiment Station, University of Illinois, Soil Mechanics Series No.. 6. Hansen, J. B. (1970), "A revised and extended formula for bearing capacity", Bull. No. 28, Danish Geotechnical Institute, Copenhegen. IS : 6403 (1981), "Code of practice for determination of bearing capacity of shallow foundations"., Johnson, T. D, and Ireland H. O. (1963), "Tests on clay subsoils beneath statically and dynamically loaded spread footings", Report to U. S, Army Engineer Waterways Experiment Station, University of Illinois, Soil Mechanics Series No. 7. M<:.Kee,K, E., and Shenkman S. (1962). "Design and analysis of foundations for protective structures", Final Report to Armour Research Foundation, Illinois Institute of Technology. Meyerhof, G. G. (1951), "The ultimate bearing capacity of foundations", Geotechnique, Vol. 2, No. 4, pp. 301331. Meyerhof, G. G. (1953), "The bearing capacity of footings under eccentric and inclined loads", Proc. Third Int. Conf. Soil Mech. Foun. Engg. , Zurich, vol. I, pp. 440-445. Saran, S. and Agarwal R. K. (1989), "Eccentrically -obliquely loaded footings", AS~E, Journal of Geot. Engs. , Vol. 115, No. 11, pp. 1673-1680. . Saran, S. and Agarwal R. K. (1991), "Bearing ~apacity of eccentrically
of Geot. Engs., Vol. 117, No. 11, pp. 1669-1690.
-obliquely
loaded footings", ASCE, Journal '
~~~';';'-;";;;>;".~~;:."-'
--
'
278
Soil Dynamics & Machine Foundations
Terzaghi, K. (1943), "Theoretical soil mechanics", John Wiley and Sons, New York. Terzaghi, K. and Peck, R. B. (1967), "Soil mechanics in engineering practice", 1st Ed. , John Wiley and Sons, New York. Triandafilidis, 'G. E.' (1961), "Analytic~l study of dynamic bearing capacity of foundations", Ph. D. Thesis, University of Illinois, Urbana, Illinois. Triandafilidis, G. E. (1965), "Dynamic response of continuous footings supported on cohesive soils", Proc. sixth Int. Conf. Soil Mech. Found. Engin., Montreal, Vol. 2, pp. 205 - 208. Vesic, A. S. (1973), "Analysis of ultimate loads of pp. 45-73.
shallow foundation", J SMFD, ASCE, Vol, 99, SMI, .
Wallace, W.L. (1961), "Displacement of long footings by dynamic loads", ASCE, Journal of the Soil Mechanics and Foundation Division, 87, SM5, pp. 45-68. White, C. R. (1964), "Static and dynamic plate bearing tests' on dry sand without overburden", Report R 277, U. S. Naval Civil Engineering Laboratory.
PRACTICE PROBLEMS 1. Describe stepwise pseudo-static analysis of designing footing subjected to earthquake loading. 2. Differentiate between Triandafilidis and Wallace analyses of dynamic bearing capacity of footing subjected to transient vertical load. Give the salient features of anyone. 3. Describe the method of obtaining the maximum horizontal dynamic load that can be applied on the footing. Give the expression of determining the rotation of the footing. 4. A 2.0 m wide strip footing is subjected
to a vertical
transient
pulse (qu
= 600 e-st). The soil
supporting the foundation is Clay with Cu = 50 kN/m2. The unit weight of soil is 18 kN/m3. Determine the maximum angular rotation of footing that it might undergo. 5. A 2.0 m wide footing located at 1.0 m below ground surface is subjected to a vertical transient load (qd(max) = 2000 kN/m2, td = 2 s). The properties of the soil are y = 17 kN/m3, = 32° and c = 30 kN/m2. Using Wallace's approach, determine maximum vertical movement of the foundation. 6. A 3.0 m wide surface footing is subjected to a horizontal dynamic load having duration 0.3 s. The properties of soil are y= 18 kN/m3, = 35 and c = 20 kN/m2. Using Chummar's approach, determine the value of maximum horizontal load that can be applied on the footing. Also determine the rotation of footing after 0.2 sand 0.4 s'
DD
.
LIQUEFACTION OF SOILS
7.1 GENERAL \1any failures of earth structures, slopes and foundations on saturated sands have been attributed in the literature to liquefaction of the sands. The best known cases of foundation failures due to liquefaction are [hose that occurred during the 1964 earthquake in Niigata, Japan (Kishida, 1966). Classical examples of 'iquefaction are the flow slides that have occurred in the province of Zealand in Holland (Geuze, 1948; Koppejan, et al. 1948) and in the point bar deposits along the Mississippi river (Waterways experiment 'itation, 1967). The failures of Fort Peck Dam in Montana in 1938 (Casagrande, 1965;Corps of Engineers, 1939; Middlebrooks, 1942), the Cal~veras Qam in California in 1920(Hazen, 1920) and the Lower LaB \Jorman Dam during the 1971 San Fernando Earthquake (Seed et al., 1975) in California provide typical ~xamples of liquefaction failures of hydraulic-fill dams, Liquefaction often appears in the form of s~nd fountains, and a large number of such fountains have Jeen observed during Dhubri Earthquake in Assam in 1930 and Bihar Earthquake in -1934(Housner, ,958; Dunn et al., 1939). When soil fails in this manner, a structure resting on it simply sinks into it. The nost recent Koyna earthquake of 1995 is an illustration of liquefaction phenomenon causingcatastrophic Jamages to structUres and resulting in loss of life a,nd property. 7.2 DEFINITIONS
7.2.1.Liquefaction. It denotesa conditionwherea soil willunde~gocontinueddeformationat clconstant ow residual stress or with no resIdual resistance, due to the build-up and maintenance of high pore water Jressure which reduces the effective confining pressure to a very low value; pore pressure build-up leadng to true liquefaction of this type may be due either to static or cyclic stress applications. 7.2.2. Initial Liquefaction. It denotes a coI1ditionwhere, during the course of cyclic stress applications, he residual pore water pressure on completion of any full stress -cycle becomes equal to the applied 'onfining pressure, the development of initialliquafaction has no implications concerning the magnitude )f the deformations which the soil might subsequently undergo; however, it defines a condition which is :
useful basis for assessingvariousposs,ibleforms of subsequentsoil behaviour.
'.2.3. Initial Liquefaction with Limited Strain Potential, Cyclic Mobility or Cyclic Liquefaction. It lenotes a condition _inwhich cyclic stress applications develop a condition of initial liquefaction and ubsequent cyclic ~tress applicati?ns c~u~e limi~ed strains to develop either because of the remaining esistance ~f the ,soil to-deformatio~ .or because the soil dilates, the pore pressure drops. and the soil tabilizes
under. the applied ,,'.- loads. I. ! J,
('i, . ~
j'.
c'
c,
;: :.'
-,' -i'
;'
,.
-
-- - ~-="-~~
11
..-.
.::...~.""~<.. <~--, ..~ ~~ ""'~
--"7
Ih,
280
Soil Dynamics & Machine Foundations
In laboratory undrained cyclic tests (triaxial, direct simple shear and gyratory shear) on saturated sands, cyclic mobility has been ob8erved to develop and to result in large strains (Lee and Seed, 1967; Seed and Lee, 1966). It is controversial whether cyclic mobility occurs in dilative sands in situ during earthquakes to the same extreme degree as has been observed in the laboratory. A simple means for understanding the difference between liquefaction and cyclic mobility as observed in the laboratory is through the use of the state diagram, which is shown in Fig. 7.1 (Castro and Poulos, 1976). The axes are void ratio and effective minor principal stress" The steady state line shown represen~sth~ 10c1,1s q[ states in which a soil can f1owOa(c~onstant effectiv.erilinor principal stre"ssa)'and constant shear stress. The void ratio at the steady state is the same a:>the critical void ratio. 'TI C d
\J)
.x. .-u :J
_0 Q 0-
-
Ftow at con stant volume
()I
"
dI- c()I
L i que fa c t i on
'TI '-E .u
..
>
()I Cl. ()I 0111\ 0 I- (\) (\).1:.
>
-
Contractive
soils (loose)
:
I J
B
- ~--
D I
I
Cyclic mobility
I
(
,C I I
0
Larg e strains and D.I LatI .Ive softening caused soiLs by cyclic loading, (dense) I state may reach B) I
0
-
C3t ( During) flow
. I I :
/.
Monotonlc Load ing
I
I I
-
I sta t e
J I
,
.
'ne
Cyclic or monotonic Loading {of dilative soil s1 arts here
.I
I
-
-
03c minor
principal
stress,
OJ
Fig. 7.1: Undrained tests ~n fully saturated sands depicted ODstate diagram (Castro and Poulos, 1976)
LIquefaction is the result of undrained failure of a fully saturated, highly contractive (loose) sand,' 1\"1 example starting at Point C and ending with steady state flow at constant volume and constant <1) a' Point A. During undrained flow, the soil remains at Point A in the state diagram. , The quicksand condition that is so familiar through the use of quicksand devices for instruction il soil mechanics is depicted by points on the zero effective stress axis at void ratios above Q. In this state sand has zero strength and is also neither diiative nor contractive. At void ratios above Q the sand grain are not in close contact at all times.
~'
IDi1!1iII
Liquefaction
281
,of SoUs
The mechanics of cyclic mobility may also be illustrated with the aid of Fig. 7.1. Consider first the behaviour in Fig. 7.r when a fully saturated dilative sand starting, for example, at Point D is loaded monotonically (statically) in the undrained condition. In that case the point on the state diagram may move slightly to the left of Point D but then it will move horizontally towards the steady state line as load is applied. If one now starts a new test at Point D, but this time applies cyclic loading, one can follow the behaviour by plotting the average void ratio and the effective stress each time the applied cyclic load passes through zero. In this case the state point moves horizontally to the left, because the average void ratio is held constant and the pore pressure rises due to cyclic loading. .
The magnitude of pore pressure build up in the cyclic test will depend on the magnitude of the cyclic load, the number of cycles, the type of test, and the soil type, to name a few variables. In particular, it has been observed in the laboratory that in-triaxial tests for which the hydrostatic stress condition is passed duvng' cycling, and if a large enough number of cycles of sufficient size are applied, the state point for t1w d'\f,erageconditions in the specimen eventually reaches zero effective stress at Point B each time the hydrostatic stress state is reached. Subsequent application of undrained monotonic loading moves the .
state point to the right toward the steady state line, and the resistanceof the specimenincreases. : During cycling in the test described above, strains develop and the specimen becomes softer. If these strains are large enough, one can say that the specimen has developed .~yclicmobility. Adequate evidence has been presented to show that most of the strains measured in cyclic load tests in the laboratory are due to internal redistribution of void ratio in the laboratory specimens. For example, at the completion of such tests the void ratio at the top of the specimen is much higher than at the bottom (Castro, 1969). Thus the horizontal line D-B in Fig. 7.1 is fictitious in the sense that it represents average conditions. Near the top of the specimen, the void ratio increases, and near the bottom the void ratio decreases. The pore pressures that build up and the strains measured in the lab~ratory are due to the formation of such loose
zones (Castro and Poulos; 1976).
'
In summary then, specimens that lie above the steady state line on Fig. 7.1 can liquefy if the load applied is large enough. Such liquefaction can be triggered by monotonic or cyclic undrained loading. The further to the right of the steady state line that the starting point is, the greater will be the deformation associated with the liquefaction. If the initial point is above Q, the strength after liquefaction will be. zero. If the starting point is below Q, the strength after liquefaction will be small but finite. Saturated sands starting at points on or below the steady state line, will be dilative during undrained monotonic loading in the triaxial cell and the state point will move to the right. If cyclically loaded the state points will shift to the left as strains occur and the specimen softens. If enough cycles are applied, if they are large enough, and if the hydrostatic stress condition is passed during each cycle, then the zero effective stress condition (i.e. initial liquefaction) can ultimately be reached in the laboratory. 7.3 MECHANISM OF LIQUEFACTION The strength of sand is primarily due to internal friction. In saturated state it may be expressed as (Fig. 7.2). ...(7.1) S = on tan cp where, S = Shear strength of sand on = Effective normal pressure on any plane xx at depth Z = Yhw + Ysub(Z - kw) cp = Angle of internal friction
.
Y = Unit weight ~f soil above water table Ysub= Submergedunit weight of soil
.
~..
0...
". ."0_"""",'.",,
.
- _. , ,., ...,
282
Soil Dynamics & Machine Fo!,ndations
Ù®±«²¼
Surface
t ---------
hw
.
-
~-[-
~
-1l..-
z '(sub
x
1
x
Fig. 7.2: Section of ground showing the position of water table #
If a saturated sand is subjected to ground vibrations, it tends to compact and decrease in volume, if drainage is restrained the tendency to decrease in volume results in an increase in pore pressure. The strength may now be expressed as, ...(7.2)
Sdyn. = (an - Udyn)tan $dyn
where, Sdyn= Shear strength of soil.under vibrations U~yn = Excesspore water pressuredue to groundvibrations
$dyn = Angleof internal frictionunder dynamicconditions It is seen that with development of additional positive pore pressure, the strength of sand is reduced. In sands, $dynis almost equal to $, i.e. angle of internal friction in static conditions. For complete loss of strength i.e. Sdynis zero. Thus, or
a" - Udyn= 0 (Jn = Udyn
or
Udyn = 1 an
...(7.3)
Expressing Udynin terms of rise in water head, hw and Y5ubas (G
written as :
Yw.hw
-
1/1 + e) Yw' the Eq. (7.3) can be
=1
G-l.'Yw'Z l+e h G-l -1f. =-=i Z l+e where, G = Sp~ific gravity of soil particles e = Void ratio
or
'-;0 ".
ier
.
= err Heal hydraulic gradient.
er
...(7.4)
283
iquefltct;oll of Soils
It is seen that, because of increase in pore water pressure the effective.stress reduces, resulting in loss f strength. Transfer of intergranular stress takes place from soil grains to pore water. Thus if this 'ansfer is complete, there is complete loss of strength, resulting in what is known as complete liquefac.on. However, if only partial transfer of stress from the grains to the pore water occurs, there is partial JSS of strength resulting in partial liquefaction. In case of complete liquefaction, the effective stress is lost and the sand-water mixture behaves as a ;iscous material and process of consolidation starts, followed by surface settlement, resulting in closer Jacking of sand grains. Thus the structures resting on such a material start sinking. The rate of sinking )f structures depends upon the time for which the sand remains in liquefied state. Liquefaction of sand may develop at any zone of a deposit, where the necessary combination of in,itu density, surcharge conditions and vibration characteristics occur. Such a zone may be at the surface ,w at some depth below the ground surface, depending only on the state of sand and the induced motion, However, liquefaction of the upper layers of a deposit may also occur, not as a direct result of the ~roltnd motion to which they are subjected, but because of the development of liquefaction in an underlying zone of the deposit. Once liquefaction develops at some depth in a mass of sand, the excess pore water pressure in the liquefied zone will dissipate by flow of water in an upward direction. If the hydraulic gradient becomes sufficiently large, the upward flow of water will induce a quick or liquefied condition 111the surface layers of the deposit. Thus, an important feature of the phenomenon of liquefaction is the fact that, its onset in one zone of deposit may lead to liquefaction of other z~nes, which would have remained stable otherwise.
7.4 LABORATORY STUDIES 7.4.1. Field Conditions
for Soil Liquefaction.
,.
An element of soil located at depth Z below the horizon-
tal ground surface will be subjected to vertical effective stress er, which is equal to
..,
. .
i}:;"",':>','" ".-
:.,\'~ ,.-
,//~,'
,;.,.. ""',',.,;;",",' "' -,"'-,..~,,",
'
',,'-"'.'" --".
',', _..
-
28~
Soil Dynamics & Marhine
f---
=+ --- --.2.--~
/~
H'
0-= OVi
~
'/"/'
.
OVi= ohw + lsub (z-hw)
..
z
'1
(a)
Ko ()
DA
'//1'/1'1'
1////1'
"'1'
No gr,ound
Ill'
'"
'YI
"
Stresses
on element
A
shaking
óóóóó.2-
--=+
"
""
~
0-= OVi
-- -
Jh;""
rh
z
'L
Ko 0-
OB
~h
"" , /" '" "'./ , / "" "" """""
+
(b)
Earthquake
~ ~. if .~i ~. ~, FOUlJdlltioll... .~ ;:,. .. ~
" '/' ,. ""./
Stresses
on element
8
loading
Fig. 7.3 : Stress conditions for soil element below horizontal ground in cyclic loading conditions
285
Liquefactiollof Soils /
() = Uvi
! ..:hi
.jor-
DA
0'"
" ""
/ /
r
" /
No ground
(0)
/
/
/
.I'"
I
".I"/
Stresses
shaking
KO
on element
A
cv= av i --L
=r=
I'
DB
-r;h Chi
1HDHPo-
~
'"
(b)
>"'+"O'
uu"
Earthquake
Fig.
7.4:
Stresses
loading
Stress
conditi ons
for
soil
ele me nt
below
sloping
ground
in
on element
cycliiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiii iiiiiiiiiiiiii
B
--
k -..,
.
.. 1
I I I
r
Soil Dynamics & Machine Foundations
286
. I
/I' /l1I/'"
I
\
"""V/I\I'
I
00
00
I J
---c=J--
I
,
K0
t
. t,
I I I
t
t
1I
I
"A' 11/'\\
---
--11 ---
0;
00 -C_t
-C
-+- \\ --- ~\--
1!-KoCTO
t
t
Initial
stresses
Ko 0-0
Cyclic toad
sequence
Fig. 7.5: Idealised field loading conditions
I
40
I
30
I .
1:max. Dept h:; 14.0m
,
20
N
E --
z -
I.
Gav
10 0
..... 1I1 11
JI
ill
ili li' ai ffi
- 30
óìðÔ 0
6
18
12 Ti.me
24
30
(5)
Fig. 7.6 : Sh~ar stress variation determined by response analysis
7.4.2. Different Laboratory Tests. Simulating cyclic shear stress conditions, following types of test procedures have been adopted for liquefaction studies: (1) Dynamic trhixial test (Seed and Lee, 1966; Lee and Seed, 1967). I
(2) Cyclic simple shear test (Peacock and Seed, 1968, Finn et aI., 1970, Seed and Peacock, 1971).
I
(3) Cyclic torsid~al shear teSt (Yoshimi and Oh-Oka, Ishibashi and Sherif, 1974). (4) Shaking. table test (frakash and Mathur, 1965; Yoshimi, 1967~Finn et aI., 1970). Typical studies 011above mentioned laboratory tests are described herein.
I I
r
I I
r
l
101,.
11
- ----
..
IIIm
I!Io
J
288
Soil Dynamics & Machine 'Foulldations
,
7.5 DYNAMIC TRIAXIAL TEST Seed and lee (1966) reported the first set of comprehensive data on liquefaction characteristics of sand studied by dynamic triaxial test. They used the concept of developing cyclic shear stress on section xx of the soil sample as shown by the stress conditions I and 11on the sample given in Col. 1 QfFig. 7.7. The stress condition I is achieved by increasing axial stress on the specimen by an amount ad' keeping lateral stress a3 constant, and simultaneously reducing the all round pressure on the specimen by an amount adl2 (Cols. 3 and 4 of Condition I). Similarly the desired stress condition 11can be induced by reducing the vertical stress by ad' and simultaneously applying an increase in all round pressure equal to ad 12 (Cols. 3 and 4 of Condition II). It may be noted that during testing the pore pressure should be corrected by reducing it by ad 12 in condition I, and increasing by ad 12 in condition H. Seed and Lee (1966) performed several undrained triaxial tests on Sacramento river sand (emax = 1.03, emin = 0.61). The grain size ranged between 0.149 mm and 0.297 mm. The results of a typical test
in loose sand (e = 0.87, DR = 38%) are shown in Fig. 7.8. In this testthe initial all-around pressure and the initial pore water pressure were 196 kN/m2 and 98 kN/m2 respectively, thus giving the value of effective confining pressure as 98 kN/m2. The cyclic deviator stress ad of magnitude 38.2 kN/m2 was applied with a frequency of 2 Cpg.The test data in Fig. 7.8 shows the variation of load, deformation and pore-water pressure with time. "0
~
8
DR~38°/o
01.S z
-
0"""
I/)
"3
Go
~
6
-
'15
E L.. .2E
1 eo=0'87,
100 0 100. 2St-
Time '" h 38.2
kNjm£
Compression Time
0
"0 E.
OJ=98,OkN/m2
0-
)(
'H::L
~
Extension
~
::J
I/)N L..
'-E
-
o..z ~x.
.L..
a. 0
100 '
0
. '-Time
I
- - .,' -,
f sec.t--
Fig. 7.8: Typical pulsating load test on loose saturated Sacramento river sand (Seed and Lee, 1966)
289
Juefactioll of Soils
---
:ompression 30 0
0
OR
-
201
.
-
~=. 98.0kNjm
10
o' L.. ...... \/)
38 °/0
.
..
'-c
=
at Od =(+)38-2 kNj m
2
at Od = 0
04
.i..#, 0
2
-10
)(
C:Xten sion
2
.
-20
«
- 30
2
1
4 Number
at Od =(-)38'2 kN/m I r 10 20 40 of cycles
100
(a) Axial strain -versus number. of cycles
L..~N
-
d~ E ~z L..~
0-
150 100
~ L..
~ ::J en\/) C III d ~ £. L..
50 0
ua.
,-atCJd,=O -. at-Od =+38,2 kN/m2
confining pressure
a.::J .S
- - -
InitiaL e tfe ctive
1
2.
4 Number
1°--' of cycLes
at Od :-38'2 20
kNjm
40
2
lOO
(b) Observed change in pore water pressure and number of cycles
150
-
L.. """N
-
d E
z
confiningeffective pre ssure 10°r-Initial '"
L..
0-
a. ::J .-c L.. ::J en\/) C
\/)
0 .I:. L.. ua.
50
,
,
- SO
2
.
4
10
20
40
Number of cycles .
(c)Changein porepressure and numberof cycles
Fig. 7.9 : Typical pulsating load test on loose Sacramento river sand (Seed and Lee, 1966)
100
290
Soil Dynamics & Machine Foundations
From this data, variation of axial strain amplitude, the observed changes in pore-water pressures, and the pore water pressure changes correct~d to mean extreme principal stress conditions with number of stress cycles have been plotted in Fig. 7..9.It was observed that, during the first eight cycles of stress application, th~ sample showed no noticeab!7.d~formati()n although pore-pressure increased gradually. The pore pressure became equal to O"}during the ninth cycle, indicating zero effective confining pressure. During the tenth cycle, the axial strain exceeded 20% and the soil liquefied. Similar tests as described above were performed by Seed and Lee (1966) for different values of 0"d' The relationship between O"dagainst the number of cycles of pulsating load applications is shown in Fig. 7.10. It is evident from this figure that number of cycles of pulsating load application increases with the decrease of the value of 0"d'
N
-
E 60
z~
OR ::. 38
I;'.. 50 ~ ~
-
eo
.
40
~
::.
0/0
0-87
03= 98,OkN/m2
III 0\
c:
-d
-
III :) Cl.
~
d ~ Q.
30 20 10 0
,
3
10
100
30
300
1000
:~ .
Number
.of cycles
to cause
faiLure
Fig. 7.10: Relationship between pu~s~tingdeviator stress and number of cycles required to cause ~ailure in Sacramel1to river sand (Seed and Lee. 1966)
Test data obtained on dense Sacramento river sand is shown in Fig. 7.11. It may be noted that the change in pore water pressure become equal to o"}after about 13 cycles, however the axial strain amplitude did not exceed 10% even after 30 cycles. This is due to the fact that in dense condition soil dilates, and the pore water pressure reduces which in turn stabilises the soil under load. As discussed earlier this corresponds to cyclic mobility (Seed, 1976; Castro, 1976).
291
.iquefaction of Soils 11 P re
sSlon 15
--0
OR =
0 ,-
.,0)=
'
78
0/0
",,'"
'
"
_9~'.0~N/m~r-
+68'7 kN/nf
c: 0
-
0
....
- 5
L-
at Od =
at Od
=
0
111
« d x
"
'..
-:-10
"
"at Od =
'
tension
-15
1
2
4 Number
10 of cycles
20
2
~,-68'7kN/m
100
40
(a) Axial strain versus number of cycles "
~ :::I 111 111 ~ L-
0..
150
100
L~ .....
d-
AA-- -a--
Initial effective confining pressure
-A~at Od =0
~N ~ E ~ ........ 50
u at Od =
0 Z o...x
- -\--:r..s-..t::~
c<:I) CJ) c d £. U
68.7 kNjm2 uat Od-
- 68'7 kNim2
0
- 50 1
2
10
4 Number
20
40
100
of cy cl e s
(b) Corrected change in pore water pressure Fig. 7.11 : Typical pulsating load test on dense Sacramento river sand (Seed and Lee, 1966)
Lee and Seed (1967) have extended this work for studying the various factors affecting liquefaction and identified the followings : Relative density, Figure 7.12 shows the plots 'of peak pulsating stress ( Le.. the stress causing liq'uefaction) against number of cycles of stress in loose and dense sands. The initial liquefaction corre-
spondsto the conditionwhen the pore waterpressurebecomesequal to the confIDingpressure 0')" Criterion for complete liquefaction is taken corresponding to 20% double amplitude $train. The figure indicates that in loose sand, initial liquefaction and failure occur simultaneously (Fig. 7. 12a). With the
-
" .1'"",,'
291
.iquefaction of Soils
TtpreSSlon 15
-0
10
OR = .78
0/0 '
. .","
0
-
.-c: 0
\..... 111 ....
0 -
.~,:03'= _98.0~~/m~r..
5 O
.-K""
at Od = 0 I
- 5
"
"
'..
)(
« -:-10 tension
.
-15
at Od =
+68'7 kN/m2
1
2
4 Number
10 of cycles
20
.
at Od =
2
~,-68.7kN/m
100
40
(a) Axial strain versus number of cycles "
~
150
::J
111 111 c:J 1... a.
100
Initial effective confining pressure
1...
c:J .....
0-
~N c:J E 1..."0 Z a..x
Od =0
u at Od =
50
-..:\--:r:s--t.:~68.7 kN/m2
c-
c:J 0'1 C 0 L:. U
- -a-- -6~at
AA-
-
0
- 50
1
2
10
4
Number
20
u .at Od 68.7 kN/-m 2
40
100
of cy cle S
(b) Corrected change in pore water pressure Fig. 7.11 : Typical pulsating load test on dense Sacramento river sand (Seed and Lee. 1966)
Lee and Seed (1967) have extended this work for studying the various factors affecting liquefaction and identified the followings : Relative density. Figure 7.12 shows the plots .of peak pulsating stress ( Le.. the stress causing liquefaction) against number of cycles of stress in loose and dense sands. The initial liquefaction corre-
sponds to the conditionwhen the pore waterpressurebecomesequal to the confming pressure 0"3'Criterion for complete liquefaction is taken corresponding to 20% double amplitude strain. The figure indicates that in loose sand, initial liquefaction and failure occur. simultaneously .. (Fig. 7. 12a). With the
-
" .#",'..'
0292
Soil Dynamics
"O""'-'O~~"-'-""-'o"""o~"'
"o
,
~ Machine
"""_0"-'
Foundations .
"':'~'"
-
incre~se in relative density, the difference betWeen the number of cycles to cause initialliquefacticin and'~ f?ilure increases. .N .
E
Z
'.:;100
'~.r '..';
11' "
~ tIJ
80
..
strain
L.
111
eo
=
OR
=
OJ =
0'\ C
0 -87 38
0/0
9 8.0 k N/ m~
-
0 VI ::J
a. x 0 ..~ .
0 0
100 1000 Number of cycles
10
010,000
100,000
(a) Loose sand
N
E
-
200
eo = 0.61 OR= 1000/0
..........
z x
'" VI
VI
-c0
160 \
120
-
...
\
...... .......
......
\I) ::J
a. x 0 a.
98'OkN/m2
, , Liquefaction " .... ......
80
=
Initial
\~
L. \I) 0'\
OJ
40
-- - --. .~...
0
1
10 100 1000 Number of cycles
10,000 100..000
(b) Dense sand
Fig. 7.12: Peak pulsating stress venus number of cycles (Lee and Seed, 1967) L \,
0 ._--. .-.......-.....
- _.0--
..
293
[que/action of Soils
Confining pressure. Figure 7.13 shows the influence of confIning pressure on initial liquefaction md failure conditions. At all relative densities for a given peak pulsating stress, the number of cycles to :ause initial-liquefaction tFigs:-7:B"u and b}-or-fai1ure-(i-.e:-26%strain, Figs:-7:i3 c and d) increased vith the increase in confIning pressure ,N
E
-~
180 'OR
...
eo
120
=7 8
0/0
= 0.71
L. ....
DJ (kNfm2) 1500
CTI
.-c:
-0
60
-'
:J'
~~
-
Co d
-It 0
a.
. -
'A
10
1
.
A
I
500 '.
I
100 1,000 Number -of cycles'
I
100
10~000
100,000
(a) For initial tiquefaction in medium dense sand
N
-
2000 ,
E
-
............
z
--
'500
OR =
100 0/0
eo
0.61
=
l5' " ....
1000
OJ (kNfm2) 1500
CTI
c: .-....
0
-' 500 :J
Co 0
a.
1-
&
0-
10 -
,
.
4 100 Number
500
-
.,1,060
OO ' 10.000
100,000
of cycles
(b) For initial liquefaction in dense sand Fig. 7.13: Peak pulsating stress versus number of cycles (Lee and Seed. 1967) (...Contd.)
...:t-.;",,!1;~,,!"J)'..
,f; <.,'~"";..!",,..~
-1.> 'b\':;:"""';?"
i';~k1t~;!;-';:".';';t;r;ft"';+,;:-;;~("i,,(,t~~l't.~;;,
, '.:..Y""f.'i.':"~,;;;".
.' '~ii-" ;;":8"';f
",;;.
~f~:ii:',~
294
SoU Dynamics & Machine Foundations
---.
,
.
",
,.,.
,..',
i~
,',-
N.
E ,Z..¥ 180
-
H'
"'
OR : 78°/0
VI t.I .. VI
- eo :,0,71
120
0'\
-:J
VI
.. ,-
~ (kN/m2)
c:
-CS
".
1500
60
a.
CS t.I a.
0
1
10
100 Number
500 10°1 10~000
1,000 of cycles
100,000
(c:)20% strain (i.e. failure) in medium-dense sand 2000 N
E
......
z-" ~
~
OR = 100
1500
eo
=
°'0
0.61
III III 1:>1
;:III 1000 C7I c:
0
III ..... :J Q
500 500
-" C
100
1:>1
'Q.
0
1
10
100 1,000 Number of cycles
L10.000
(d) 20% strain (i.e. failure) in dense sand
100,000 -,'
Fig. 7.13: Peak- pulsating stress versus, number ofc:yc:les(Lee and Seed,1967)
Peak pulsating stress. Figures 7.14 a and b show respectiyely the vari~t!onof peak pulsating stres~ Odwith confining pressure for initial liquefaction and 20% axial strain in 100 cycles. It may be noted tha for a giverirelative density and number of cycles ofload applicafion;the"odmcreases linearly with 03 fo initial liquefaction, while for 20% axial strain condition, similar linear trend exists only in loose sand5 ;," . "" " .. :. .";. I . . ~;'~;'~
Liquefaction
of Soils
2000 ........ N
E
z 1600 oX
\11
~ 1200 .L..
Ini.tiol void ratio.)e6 0 '61
\11
01 C
.-0
800
0.71
-
\11
:J a.
400
.x d
(:,) Cl.
-0 0
800
400 ...
3600
1200
2000
03 (kNfm2)
(a) Initial liquefaction in 100 cycles
2000 , .'
.........
N
E --Z 1600 oX
....... \11 \11 ~ ....
.-
Initial ratio.)
1200
\11 01 C
";::
d
800
0 -71
-'\11
ðóéè
:J a.
void eo 0.61
400
0'81
::s:
d ~ Cl.
0
0
400
800
1200
1600
2000
03 (kNjm2) (b) 20% strain in 100 c)'des Fig. 7.14 : Influence of pulsating stress on the liquefaction of Sacramento river sand (Lee anti Seed. 1967)
--
-..'."
'--".-'
_u_-,,--"-- -
---~
296
Soil Dynamics & Machine FOlllldations
Number of cycles of pulsating stress. From the Figs. 7.12 and 7~13,one can conclude that for a given p\llsating stress, number of cycles needed for causing initial liquefaction and failure increases with the increase in relative density"and confining pressure. 7.6 CYCLIC SIMPLE SHEAR TEST The cyclic simple shear test device is already described in C;:haptet 4:with a mention that it simulates earthquake. condition in a better way. Peacock and Seed (1968) reported the first set of comprehensive data on liquefaction studies by using this test. They performed tests on Monterey sand (SP, emax= 0.83, emin = 0.53, DIO = 0.54 mm). The sample size was 60mm square and 20mm thick. The samples were tested at relative densities (DR) of 50%, 80% and 90% giving the sand in loose, medium dense and dense states respectively. The oscillatory shear stress was applied at a frequency of 1 cps or 2 cps keeping the normal stress constant.
Initial
relative
Initial
void
density
ratio,
eo
°R=50olo =
0-68
Initial confining pressure,OV Frequency ã ï ئ
¢ Äïï
\11C>lN
;: E Äïïóòòòòòòòòò
1- Z d
C>I
.I:. if)
~ ---
40
I'
î0
I
0
f---' 20 -
=
500 kN/m2
I
1111 cum-
M
mum
40 (a) Applied cyclic shear stress
--
20
0
0 ---;- 1
I
c .0 1-
0
1-
10
-\11
j
1ftttt
d C>I .J::. If)
2u
124,eyc: le s (b) Shear strain response
Fig. 7.15 : Record of typical pulsating load test on loose sandin simple shear conditions (Peacock and Seed. t 968) ..
iquefaction
297
of Soils ~ 1:J \11
11'1
L-N
~Q.E 1-
60' 0
~oz4
W
~~
...
-2 Cc
~ 1-:::::> 0 0..
V
-_ ..r
- ..-
f
""
'"
.-..lr"
0 (c) Pore water pressure response
Fig. 7.15 : Record of typical pulsating load test on loose sand in simple shear conditions (Peacock and Seed, 1968)
The typical test data in Fig. 7.15 show the variation of shear stress, shear strain and pore water Iressure with time. As evident from Fig. 7.I5b, there was no significant shear strain of the sample during lle application of the first 24 cycles of stress. During the twenty-fifth stress cycle, the shear strain sud.enIy increased to a value of about 15% and become 23% in the next cycle. Pore pressure increased ;radually until the effective confining pressure is reduced to zerc. (Fig. 7.15e). At this point the resulting leformations became extremely large, and the soil had essentially liquefied. Similar trend was also oberved in triaxial test. >eackock and Seed (1968) have also studied the effects of followll1g factors on liquefaction:
Relative density. Figure 7.16 shows a plot of peak pulsatingyress (1:h) dJUsmg initial liquefaction \ith number of cycles of application for different relative densities and confll1ing pressures. From this 19ure it can be concluded that for a given value of confining pressure and number of cycles of stress lpplication, 1h increases with the increase of relative density'. A mo;"e clear presentation is shown in :ig. 7.17. 11'1
I/)
80
~
~
1-
11'1
1-
a
~N
60
90°/0
E 111"-
'\.
.c
-cnZ
'-C x
40
",
a .c.
"'" 2-""""'",-
III~
:J Q
- --
.::c
a
~ Q
0
1
800 500(kNjm
Qy
)
10
Number
-----100
1,000
10-,000
of cycles
Fig. 7.16 : Initialliqu'efaction of cyclic simple shear test on Monterery sand (Peacock and Seed, 1968)
...
'
,
298
Soil Dynamics & Machine Foundations
100 .........
N
For 100 cycles
E
~
~
.. \/I \/I
of pulsating
stress
80
60 ±ª (kN/m2)
L...
...
800
III
en
c "...
40
0 III
:J
0. oX
20
I
200
0 Cl.
0 0
40
20 Relative
100 density
(°/0)
Fig. 7.17 : Effe~t of relative density on cyclic stress causing initial liquefaction (Peacock and Seed, 1968)
Confining pressure.
From the data presented in Figs. 7.16 and 7.17, plot of'th versus aI, was pre
pared as shown in Fig. 7.18. For a given value of DR and number of cycles
of stress application, t,
increases linearly with the increase in av' \/I III ~ L............. V)N L... E 0"-..
-
75
~Z \II~
\
\
relative. density,
Initial
void ratio
eo
Cl.
0,68
DV (kN/m
I-
2
)
800 500 300
0.0
o~ ~
=
-..
~~
~
DR= 50°/0
50
enN c ......-...... :J
Initial
.!:
0
1
< .... }
'1
~
10 100 1000 Num b e r of c.y c l e s
Fig. 1.18: (a) Cyclic stresses required to cause initi~lliqueraction at different confining pre~sure
to I{t ,t 'j "If
299 uefacti~n of Soils
11\ 11\ ~
-'- ....... 75
,II\N
OR =50% eo = 0.68
'- E
0..........
~z
1I\.x 0'1c: N
10 c.ycles
50 100 cycle
.;: -0
- ~''
11\
1:)
::J
25
0.0
.x
0 .r= ~I'-J Q.
0
0
100
200 300 400
Initial Fig. 7.18:
Peak pulsating
effective
500
600
confining
700
800
pressure
(kNfm2)
(b) Effect of confining pressure on cyclic stress to cause failure in 10 cycles and 100 cycles (Peacock and Seed, 1968)
stres!> and number of cycles of stress application.
at for a given value of
o"v
From Fig. 7.16, it can be seen
and relative density DR' a decrease ~f "Chrequires an increase of number of
des ~o cause liquefaction. Further for a given value of "Ch:"number of cycles of stress application quired to cause liquefaction increases with the increase in relative density DR and confining presre 0"v .
Frequency ofload application. Tests were rformed at frequencies of 1 Hz, 2 Hz, and 4 ~, and the effect of frequency on the stress using liquefaction was found negligible. Seed and Peacock (1971) have studied the feet of coefficient of earth pressure (Ko) on e peak pulsating shear stress "Ch causing liq:faction in cyclic simple shear test. The value . Ko depends on the overconsolidation ratio )CR). Figure 7.19 shows a plot of stress ratio ,la,,) with number of cycles of stress appli.tion for different values of Ko' For a given lative density and number of cycles of stress 'plication, the value of ("Cia,.) decreases Ith the decrease of K0 .. J"
0.4
It) 0.) --s::.
~ 0
0
0.2
OCR:
~
1/1 1/1 ~ ~
~
8
ko : , OCR:" Ko : 0.75 OCR: 1 Ko : 0.4
0.1
0
1
10
Number
100
ot cycles
causing
'.000
10.000
initial liquefaction
Fig. 7.19 : Influence of overconsolidation ratio (OCR) on stress causing liquefaction in smiple shear tests
. (Seed
and Peacock, 1971)
.iJ;
.1;'\
Soil Dynamics
300
& Machine
Foundations
7.7 CO~IP ARlSON OF CYCLE STRESSES CAUSING LIQUEF~.\CTION UNDER TRlAXIAL AND SIMPLE SHEAR CONDITIONS , PeaCock and Seed (1968) performed both cyclic triaxial and cyclic simple shear tests for liquefaction
studieson Montereysandwith a relativedensityof 50%and confiningpressures (cr3or crv)of 300, 500 and 800 KN/m2. Rbults are plotted in Figs. 7.20 and 7.21. It may be seen fro~ these figures that the cyclic stress required to cause liquefaction of loose sands under simple shear condition ("Ch) is about 35 percent of the cyclic stress required to cause liquefaction in triaxial condition (crd/2).
""
150 N
-
.......
E
Triaxial
~
~
.........
z~ N
'~--~
l5' 0
.-
results OJ{kN/m2)
800
.
-~
test
,
100
500
(-.JL: lI\ lI\ ~ ~ IfI
""""--~
La ~ .J::. IfI
CJ)
c
'\
50
.
e-
" Simple
..... a
, "-
VI -:J a.
......
oX
a ~
OR = 500/0 eo = 0-68
0..
0
1
300
10
shear
test
results
DV (kN/m2) 800 500 300 lOO 1000" Number of cycles
Fig. 7.20: Cyclic stress required to cause liquefaction of Monterery sand at different confining pressures in triaxial and simpl~ shear tests (Peacock and Seed, 1968)
'i
.;quefact;on
301
of Soils
2: 150
~Le
Relati ve density} OR
.£:.
Initial
~
=
500/0
void ratio,eo =0.68
U\ U\
-~U\-100 N
E 0-~z L-
.c.~ U\-
10 cycles
CJ'I
-
.S
50
0 U\ ::) a.
100 cycles
~ 0 ~
Q. ~
100,
200
Initial
300
effective
400 confining
500
600
pressure
700
03 or
800
OV (kNfm2)
Fig. 7.21 : Comparison of pulsating shear strength of I~ose Monterey sand under cyclic loading-simple shear and -triaxial conditions (Peacock and Seed. 1968)
7.8 STANDARD CURVES AND CORRELATIONS
FOR LIQUEFACTION
For evaluation of liquefaction potential, Seed and Idriss (1971) developed standard curves between cyc lic stress ratio (crd/2/a)) versus mean grain size (Dso) for 10 and 30 number of cycles of stress application for an initial relative density of compaction of 50% (Figs. 7.22a and b). These curves were prepared by compiling the results of various tests conducted by several investigators on various types of sand.
The values of stress ratio ("th/a~.)causing liquefaction, estimated from the result of simple shear tests, have ~hown that the value of"th/av is less than the corresponding value ofad/2 a) (Fig. 7.20,7.21 and 7.22). The two stress ratios may be expressed by the relation. ad "th == ,CI ( av ) simple~hear ( 2 cr)) triax.
\,
...(7.5)
where, Cl = Correction factor to be applied to laboratory triaxial test data t? obtain stress conditions causing'liquefaction in the field "
".
~,",.' ';
"",, "'. --.:"
..,~. .~:
302
Soil Dynamics 1:5'
& Machine
Foundatiolls
-----
N '-......
0.30
1:5' ... lfo
U >u
.
0.25
.-c: c: .-0 u C
~
~
r
Triaxi cl. compression test data for Ode./203 at
9
liquefaction
.
-
0.20 0
C>i
:J
0
.--v
--
- 15
v
Field value at -Ch/ay causing lique taction esti mated tram re su Us ot sim pie sh ear tests
(J)
C 1I1
0 -10
:J d
u
G
Relative
d
density
No. of stress
0 -05
=
50%
cycles
10
=
1I1 1I1 C>i
-
L..
0
If)
1.0
0.3 Mean
grain
0.01
0-03
0.1 size
050'
mm
(a) In 10 cycles
-
N
------
tj'
0.30
...
.
1I1 C>i
-u
>u 0
'V
0-25
Triaxial compression data for 0Ct /2 CJj at liqu e ta e.tion
M
C
c: 0
0.20
test
u
d ..... C>i :J CT
15
-
(J)
.-c: 1I1 :J d
-
0
0
0
-
, Field value at7:h/r:sv causing liquefaction estimated from resul ts at simple shear tests
10
\J
Re lative
0 .....
d 0.05 ... \fI \fI C>i L.. .....
(f)
0 2.0
0.1
0-3 Me an
grain
size
=
50 °/. cycles =30
densi ty
No. of stress
0-03 050,
0.01
mm
(b) In 30 cycles Fig. 7.22 : Stress conditions causing liquefaction of sands (Seed and Idriss, 197\)
-----
Liquefactioll
303
of Soils
Seed and Peacock (1971) have proposed the followingth,reealternativecriteria of obtaininge 1 : (i) Maximum ratio of shear stress developed during cyclic loading to the r.ormal stress. The initial stress conditions of a specimen in simple shear device are shown in Fig. 7.2"?a; the corresponding Mohr's circle is shown in Fig. 7.23 b. Figure 7.23 c and d show respectively the stress conditions on the soil specimen during cyclic simple shear test and corresponding Mohr's circle. It can be seen from Fig. 7.23 d, that the maximum ratio of shear stress to normal stress in cyclic simple shear stress is 't/Ko ay. This ratio in triaxial test is ad/2a3 (Fig. 7.7). Therefore,
~
=
Koal,
ad
...(7.6)
2a3
//e 1 = 'th ai, = 'th' -- Ko I avK ad /2. a3 'th I oal,
now
...(7.7)
GV V1 tfI ~ L-
..... V1
KoOV
L-
a
(a.)
~ s:::. if)
-
KoOV .. " ,-
ay
(b)
QV NormaL s t re.ss
V1 V1 ~ L-
-rh
..... V1 L-
Ko CYy
a
~
(c)
Normal stress
s:::. VI
(d)
Th
Fig. 7.23 : Maximum shear stress for cyclic simple shear tests
(ii) Ratio of maximum shear stress to the mean principal stress. In simple shear test (Fig. 7.23 d)
Maximum shearslress. ~max= ,/~; + [~a, (1- Ko)
-...
-----_.
J
~
\
304
Soil Dynamics & Machine Foundations
Mean principal stress during consolidation (F'ig. 7.23 a)
.1 -
-
-
="3 [O"v + Ko o"v + Ko O"v]
1 =.- 0" (1 + 2 K ) 3
v
...(7.9)
0
In Triaxial test a
Maximum shear stress
=J
0"3
...(7.11)
= .:!.L
...(7.12)
Minor principal stress = "r2
h
Therefore,
2 l + - {0" (1- K )} .
2
v
0
...(7.10)
2
20'3 [~crv(1
+ 2 Ko)]
--'th
It gives
=. .:!.L
.!.(1+2K 0 )2_.!.(1-Ko)2 4
( O"vJ
( 2 0'3 )~ 9
Cl
= ,/-(I+2Ko) 9
.
1
Hence
2
1
2
--(I-Ko) 4
...(7.13)
/(ad/2a3)2
2
...(7.14)
/(ad/la3)
(Hi)Ratio of maximum change in shear stress to the mean principal stress during consolidation.
Ja;)
"rh
. It gives
,...(7.l5)
[crv(1 +32Ko)]
a3 (1+2 Ko) Cl = 3
...(7.16)
Finn et al. (1970) have shown that for initialliquefactiQn
of normally consolidated sands
(1 + Ko) Cl =
2
...(7.17)
Castro (1975 has prop
solidation. It gives the value of Cl as
.
-
.
.
,
.
.
2 (1+2 Ko)
(3.[3) ...(7.18) Values of Cl computed from the above equations are given in Table 7.1 Weighted average values of Cl
-.
C I are given in the lastcolumnof the table.In normallyconsolidatedsands,valueof Korangesfrom 0.3 to 0.5 which in turn gives the value of Cl varying from 0.45 to 0.55.
'. ..
, ..
1
-,--
II1II
Liquefaction
,.;"~'".;:,:'<,.""'".",,,;, .
'"
.,.,'i., ..'..', 0'".1)< : ",ih""".<.(,.'~:<.A.,I'A\:;t",-.,.j,~,"',','i1.
305
of Soils
Table 7.1 : Values of Cl
----------------------------Value orc!
using
Ko
Equation 7.7
Equation 7.14*
Equation 7.16
Equation 7.17
Equation 7.18
Average value
0.3
0.3
Negative value
0.53
0.65
0.61
0.45
0.4
0.4
0.6
0.7
0.69
0.53
0.5 0.6 0.7 0.8
0.5 0.6 0.7 0.8
Negative value 0.25 0.54 0.71 0.83
0.67 0.73 0.80 0.87
0.75 0.8 0.85 0.90
0.77 0.85 0.92 1.00
0.55 0.68 0.78 0.87
* For -
ad
= 0.4
2 0'3 In simple shear test equipment, there is always some nonuniformity of stress conditions. This causes specimens to develop liquefaction under lower horizontal cyclic stresses as compared to that in the field. Seed and Peacock
(1971)
demonstrated
this fact for a uniform
medium
sand (DR
= 50%)
in which the
field values were about 1.2 times the laboratory values. It can be expressed by the following relation:
--C-
-"Ch ( cr )
v field
"Ch
"
...(7.19)
~.( cr )
v simple shear
where Cz = Constant to account the nonuniformity of stress conditions in simple shear test Combining Eqs. (7.5) and (7.19), we get "C a a -1:L =C C -IL =C-L I z ( 20' ) r ( 20' ) ( cr ) v field
3 triax.
...(7.20)
3 triax.
where ...(7.21)
Cr = Cl Cz Seed and Idriss (1971) suggested the values of Cl' as given in Table 7.2. Table 7.2 : Values of Cr Relative density OR (%)
Cr
0-50
0.57
60
0.60 .
80
0.68
As evident .from Fig. 7.17, upto a relative densitY of 80%, the peak pulsating shear stress causing liquefaction increases almost linearly with the increase in relative density. Keeping this fact in view, the following general relation is suggested: "Ch
( cry) field DR .
= .5!L .C. DR triax. r 50 ( 2 0'3 ) . 50 .
'.
...(7.22)
306
Soil Dynamics & Machine Follndatiolls where ~h
( crv) field OR 5!iL ( 2 cr3J triax. 50
= Cyclic shear stress ratio in the field at relative density of DR percent = Stress ratio obtained from triaxial test at relative density of 50%. It can be determined using Fig. 7.22.
7.9 EVALUATION OF ZONE OF LIQUEFACTION IN FIELD At a depth below the ground surface, liquefaction will occur if shear stress induced by earthquake is more than the shear stress predicted by Eq. 7.22. By comparing the induced and predicted shear stresses at various depths, iiquefaction zone can be obtained. In a sand deposit consider a column of soil of height h and unit area of cross section subjected to maximum ground acceleration Qmax(Fig. 7.24). Assuming the soil column to behave as a rigid body, the maximum shear stress 'tmax at a de p th h is g iven by
rh - ( g ) . Qmax -
'tmax
where
...(7.23)
g = Accelerationdue to gravity r = Unit weight of soil
¿³¿¨ '.. ...
:
--
:
.~~::~.:':I
Unit cross sectional ¿®»¿
h
~
Tmax=("(hf9)
amax.
Fig. 7.24 : Maximum shear stress at a depth for a rigid soil column
Since the soil column behaves as a deformable body, the actual shear stress at depth h, ('tmax)actis taken as (
)
-
-
'tmaxact - rd. 'tmax-
where, rd
= Stressreductionfactor
rh rd ( g ) .omax . .'
.
...(7.24)
307
;qllefact;on of Soils
Seed and Idriss (1971.)recommended the use of charts shown in Fig. 7.25 for obtaining the values )f rd at various depths. In this figure the range of rd for different soil profiles alongwith the average value
lpto depth of 12 ni is shown. The critical depth for development of liquefaction is usually less than 12 m.
rd
0 0
0'2
0-4
0.6
Average
value
1'0
6
-
E ......
12
£.
Q. ~ 0
18
Range of different soil profiles
24
30 Fig. 7.25 : Reduction factor r d versus depths (Seed and Idriss, 1971)
The actual time history of shear stress at any point in a soil deposit during an earthquake will be as shown in Fig. 7.6. According to Seed and Idriss (1971), The average equivalent uniform shear stress 10\' is about 65 percent of the maximum shear stress 1max'Therefore 'tav
rh
= 0.65 -.amax .rd
...(7.25)
g
'.
The corresponding number of significant cycles Ns for 'tavis given in Table 7.3. Tabl~ 7.3: Significant Number of CyclesNs Corresponding to 'ta\'.Earthquake magnitude, M Ns
on Richter's scale 7 7.5 t... .~ '.~
8 . -. ..
10 20 30
-
~-
~
308
Soil Dy/tamics & Machi/te Follllda/i~ns
The procedure of locating liquefaction zone can be summarised in following steps: (i) Establish the design earthquake, and obtain peak ground acceleration amax'Also obtain number of significant cycles Ns corresponding to earthquake magnitude using Table 7.3. (ii) Using Eq. 7.25, determine 'tav at depth h below ground surface. (iii) Using Fig. 7.22, determine the value of «Jd/2
th
( crv) field
.
Multiplying 'th
for the relative density DR of the soil at site.
OR
th
( crv ) field OR
with effectives stress at depth h, one can obtain the value of shear stress
required for causing liquefaction.
(v) At depth h, liquefaction will occur if 'tav > 'th
(vi) Repeat steps (ii) to (iv) for other values of h to locate the zone of liquefaction. 'tav and 'th can be plotted as shown in Fig. 7,26,
Equivalent
peak shear
stress
¬æòææòþæóææþóþôô¢ôô -' '- - '- ." . ,- -,,,;,-,,-,,",.,.-,- - .'
.
.
-c::.I
. -
~
--
,.,.. ..'. depth for.~,~.Oepthof water
Critical
-
liquefactionJd --"',.tab~~
.
-'GWT -.::-
U
0 ~
:J \11 "0 C :J 0
Zone of initial Lique faction
~ 0'1 c::.I
L
-
£.
~
0 ....
~I £.
a. c::.I
0 ,
Equivalent
shear
cyclic
stress
developed
earthquake cycLes
\
Peak cyclic shear stress needed to cause liquefaction in Ns cycles Oaboratory)
due~
in-N°s \
'
Fig. 7.26 : Zone of initiai liquefaction in field
309
fuefaCtion of Soils
1.0VIB~TI9,N l
TABLE STUDIES
vibrationta~le studies,a large specimenof saturatedsand is preparedin a tank which is placedon a
bration' table~fig. 7.27 sho:ws atypical setup of a horizontal shake table available at University of oorkee.Ji conSlslsofa rigid platform on which the test tank (LOSm x 0.6 m x 0.6 m high) is mounted. he platform with wheels' rest~ on four knife edges being rigidly fixecl.on two pairs of rails anchored to le foundation. The platforin is connected with motor and brake assembly for imparting vibrations. Some
rnportantcharacteristicsof the table are: .. . .
"
.
.Amplitude
~
0-10 mm
.Frequency
-
0-20 Hz
-
0-20 g
Acceleration
.
Facilities are available for measuring pore/pressures at different depths in the sample placed in tank. [he procedure of carrying out test is simple. Firstly the sand is placed in the tank under saturated conjition. The table is then excited with the desired amplitude and acceleration. Variation of pore pressure
with time and number of cycles are then noted.
.
The main advantages of vibration table studies are: (i) It simulates field conditions in better way as the size of sample is'large, prepared and consolidated under anisotropic conditions. (ii) It is possible to trace the actual pore-water pressure distribution during liquefaction. (iii) Deformation occurs under plane strain conditions. (iv) Visual examination of sample during vibration is possible. 1. Vibratory system 2. Horizontal shaking table with a tank mounted on it. 3. Settlement measuring Device. . 4. Pore-Pressure measuring system
(a) General view
..
'Fig. 7.27 :.Horizontal
shaking table (..~Contd.)
,~, 3tO
Soil Dynamics & Machine Foundations ~"..,
1. Motor 2.Pulley 3. Eccentric wheels 4. Crank-shaft 5. Connecting rod 6. Fly wheel 7. Hand break 8. Revoluation counter
(b) Vibratory
system
Fig. 7.27 : Horizontal shaking table
Since 1957, many investigators have studied lIquefaction characteristics of sand using vibration table on different sizes of soil samples and dynamit: characteristics of load. The effect of the following aspects have been studied: (i) Grain size characteJ:istics of soil. (ii) Relative density. (iii) Initial stress condition i.e. overburden pressure. (iv) Intensity and character of excitation force. (v) Entrapped air. The important conclusions drawn from vibration table studies are: 1. For a given sand placed at a particular density, there is sudden increase in pore pressure at a definite acceleration. This is termed as 'critical acceleration'. Critical acceleration is not unique property of sand. It depends on the type of sand, its density, the amplitude and frequency of oscillation and the overburden pressure (Maslov, 1957; Matsuo and Ohara. 1960: Florin and Ivanov,1961). 2. If sand is subjected to shock loading, the whole stratum liquefied at the same time, while under steady-state vibrations, the liquefaction starts from the top and proceeds downward (Florin anp Ivanov,1961). .', 3. As the surcharge pressure incre~sed, the number of cycles required to cause liquefaction 'in. creased (Fig. 7.28; Finn, 1970). Tests have shown that, even small drainage surcharge wil reduce the time of the liquefied state tenfold (Fig. 7.29; Florin and Ivanov. 1961).
.'
311
le/action of Soils
;
E
70
..........
z
oX.
5.6
~
I:J VI VI ~
42
~
Frequency eo
2 Hz 0.67
thin membrane 6
I-
x
a .c
0.25 9
First liquefaction
Ia. 28 ~ 0'1
Ac c el era tron
A
First liquefaction thi c.k mem brane First liquefaction for old container
results A
\J
I:J I.f'I
1000 to first liquefaction 100
Fig. 7.28 : Effect of surface
pressure
on resistance
to Initial liquefaction
In vibration
10,000 "
table studies
(Finn, -
1972) ---
22.5
20 , .'
-
15
q
1/1
C 0
-
/-:~.~r
v
. , ..'
c 10 C>I
::I tT
q (k NI m2)
0
~
'"
C}-()
E
5
O. 2
..-. tr )f--j(
0 0
25
Number
50
20 6
50 100
75
100
of impacts
Fig. 7.29 : Influenceoftbe intensity-or dl'alnlng~w:cbarge on tbeperiod-of1lme within which the sand remain liquid (Florin"and Ivanov, i961i
-
--~
"~-,
'"
--
312
-,'-'
-'
, ,.-, , - '.
-'
- So.il
-'.-.~-
Dynamics & Ma~hine Foundaiiolls'l
4, The time during which the ~iquefiedstate lasts is much less for coarser grained soils than for fine grained' sbiiS (Fig: 7,'30.,Gupta,'l979). He carried out liquefaction studies on four sands namely (i) Ukai sanq (Dso=,l.& mDJ.}i(it) Obra sand (Dso = 1.0 mm), (iii) Tenughat sand (Dso = 0.47 mm) and Solani sand (Dso = 0.15 mm). The maximum pore water pressure developed in about 6 to 10 cycles. It started dissipating immediately after attaining maximum valu~. The total time required for dissipation was about 6s for Ukai sand, and 20s for 01;>raand Tenughat sands. The corresponding value for Sohmi sand was 12Us,it"re~ained c9nstant for ~bout 35s: thu~ ,the time required for, dissipation decreases with the increa~e in coarseness. Time,s 0'2 20
0.4
0,8
1.2 '--,.,
?
- ", '
"12
-04
200
E u
Accl. 10010 9
<:I}
L::I '::
,5Hz
'.
-/",Sotani.sand,
,
';~'~
155mm depth
10
',:-
"
,",:-::-'!,
~"
ñÌ»²«¹¸¿¬ sand Obra sand
<:I}
L-
a.
Ukai sand
<:I}
L-
e ll.
0
1 Number
40 of cvcles
200
400 600
1OC
Fig. 7.30: Pore pressure versus number of cycles for different sands (Gupta, 1979)
Since the liquid state lasts for only a short time, the liquefied masses of soil have no time for displacements, so that there is practically no indication that the phenomenon of liquefaction occurs in coarse-grained soils. 5. The excess pore-water pressuresdec!e,ases with the increase in initial relative density (Maslov, 1957; Gupta, 1979). Figure 7.31'shows a typical test data indicating the effect of relative density on an increase in pore pressure at 10 percent g for Solani sand (Gupta, 1979). In this case, no pore-water pressure increase was observed when initial density became 62 percent. Tests performed on other types of sand with different accelerations gave the values of relative densities as listed in Table 7.4 beyond which no pore-water pressure was observed. Table 7.4 : Initial Relative Density Beyond , -. which no Excess Pore Pressure Develops (Gupta, 1979) Intia/ Re/ative Density
Acceleratioll
- - - - - - - - "7",...- - - - - ~ -:-- - - - - - - - - - -
(g) Percent 10 20 40 50
So/ani,sa.~ld 65Q--(O.15m'!IL 62.5 62,S 66.0 ":','.
..0',
"'66.5'
TefJughat sand ,<:~~-{O.4!),. .,' 52.0 61.5 64.0 .) 65.0 ,;".,~...;';'; . .. " -;', " . ." '
Obra sand p.Omm) 51.5 60.0 62.5 ',..-
64'.0
Ukai sand (1,8 mm) 50.5 59.5 62.0 '63.0
t
is;;
313
efaction of Soils
1-5
360
"
32
---~- -....... --"""
280
~--
'~
"
"CA...
-
"
" 240
\-
::J Ifj Ifj
~
u
ay 0-5
'. "C\
---
200
1.0
155 250
u/CTv /> 1 complete liquefaction
E E ~
60
0 0
~, "
""'0...,
Oepth(mm)
.
0 Excess pore pressure
u/ov
160
L.-
a. ~ L.-
a Cl.
120 80 40 0 20
30
40 Initial
50 relative
-
60 density
70
80
-90
(°/0)
Fig.7.31: Pore pressure vs.initialrelativedensityin Solanisand (Gupta, 1979)
DeAlba, Seed and Chan (1976) presented the results of shake table tests in the form of stress ratio .
la) and number of cycles required for causing liquefaction as shown in Fig. 7.32. It indicates that for iven value of'th' more number of cycles are required for liquefying a sand having more relative density. is is a similar conclusion as obtaine~ by ~eed and Peacock (1971) by cyclic simple tests. In shake table IS, the value of'th is given by : . . ... W '
o'tho=-.am g where W = Tot~l p~essure exert~d OJ?the bas~ of t~nk p~ac_edon t!1eshake table 00am = P~ak ac~eleration of die uniform cyclic mQtion
...(7.26)
314
Soil Dynamics & Machine Foundations
a
+-
0.4
0 L.
OR (0/0)
11'1 111
~
~
~
¢ e--e.
0.3 '
êè 54
,
'U
.r::.
~u P 0.2 ~ L. L. a
u
0.1
1
10
100
Num be r 0 fey Fig. 7.32: Corrected
'Ch/cry versus Ns for initial
1000
cl e S J Ns from shake table studies (DeAlba, Seed and Chad, 1976)
liquefaction
7.11 FIELD BLAST STUDIES In blast tests, predetermined charge (like ammonia, gelatin etc.) with electric detonators is installed at predetermined depth in a cased bore hole. The hole is later filled with sand and the casing ~swithdrawn. Lead wires from detonators are connected with blaster so that the charges may be fired at any desired moment. Acceleration pickups are placed at regular intervals from the blast point to record horizontal and vertical acceleration at the time of blasting. Similarly porewater pickups and settlement gauges are placed at certain distances from the blast point to record the increase in pore water pressure and ground settlements, Accelerations, porewater pressure and ground settlement at the blast point are then obtained by extrapolation. The data is then interpreted to obtain the liquefaction potential. One of the main purpose of carrying out blast tests is to ascertain whether the soil at the site will liquefy under simulated earthquake loading. Field data using small explosives at some depth at the site alongwith pore pressure and settlement observations for predicting liquefaction potential are available ir literature from few investigations (Florin and Ivanov, 1961; Kummeneja and Eide, 1961; Krishna am Prakash, 1968; Prakash and Gupta, 1970; Arya et aI., 1978; Gupta and Mukerjee, 1979). For examining the chances of liquefaction at barrage site, Gupta and Mukerjee (1979) performe< blast tests in a river bed having the soil profile as given below: Depth 0-4 m 4m-7m
7 m - 20 m
.
Description of Soil
Fine sand Clay Silty sand mixed, with kankars
Average N-va/ue '5
.",
-
21-60 (increases. with depth)
Remarks ., ... ' . . - Cu ='2.47, °50 = 0.12 mm Position of water table near the surfal Cu = Uniformity coefficient
The critical hydraulic gradient of top loose sandy deposit works out,to be 0.8. Special gelatin (( percent, 2 kg) was installed at 4m depth in 150 mm diameter cased bore holes. Blasting was done wi the help of an electric exploder. Horizontal surface accelerations were measured using acceleration pie ups placed at various distances fro,mthe source point. The depth of each acceleration pickup was 200 IT below the ground surface. The porewater pressures at vari~us .dls~ancesfrom sourc~ point were measur at 2.5m depth from the ground surface.
ù
iIIIb.,
Liquefaction
of Soils
315
B7 I
10m
I
86
r
1am
I
8S
812
I
10m 10m
I
I
84
B:f , .
lOm
Srn
~0
Srn
¢º»®»²½» point
B13
.814 .
B15 816
. -
Fig. 7.33: Site layout for field blasting tests (Gupta and Mukerjee. 1979)
,
Fig. 7.34: Surface ac:c:eleration.(Guptaand Mukerjee. 1979)
Figure 7.33 shows the sketch of layout of the tests at the site. A typical acceleration record obtained lt 35 m distance from the blast point is .shown in Fig. 7.34. Variation of maximum horizontal accelerajon and porewater pressure' with dis;tan~~'from blast point are shown respectively in Figs. 7.35 and 7.36.
, 316
Soil Dynamics & Machine Foundations
On the basis of past earthquakes, the maximum possible acceleration record at the site is assumed as shown in Fig. 7.37. Using the method of Lee and Chan (1972) this earthquake record is worked out to be equivalent to 19.6 cycles of 0.075 g acceleration (Table 7.5) 24
0'1 0
;-c 0
\
20
\ \
16
0
12
(
"
;::.
0
...
-
C>I
C>I
8
.3C
4
u u 0
0
(
"
( i'-. " '. ""'(
"-- .-.....
C>I
-'--
a.
0 0
10
20 30 40 Distance tor blast
50 (m)
60
70
80
Fig. 7.35 : Acceleration versus distance (Gupta and Mukerjee, 1979)
).0
Depth of blast Pore pressure at 2.5 m
2.5 E
4m measured =
C>I
~
2.0
11\
11-74
0/1 C>I
:i
1.5
C>I
... 0
a. u E 0c
>-
1.0
0-5
0
0 0
I I 112'9 I 2'5 5,0 'Distance
1 7'5 trom
I 10.0 blast
I 12.5
15'0
17-5
(m)
. Fig. 7.36: Port: jJressure vs. distance (Gupta and Mukerjee, 1979)
",
20-0
-,"",~~",
iquefactionof
317
~oils
','""
. .. N
<-
0 .--
co
0-r-0--
.; ....
., . --If)
--
... .::0: = :::. "C c <": <":
C.
=
E -
.:
.
.2 ...:r
c
E ...
-
... '" '"
I
0"
.. r--
'-:
r-oD N
.;
0 /'
-
0
0 ,."
0
0
(3
0
0
c-
co 0 . 0
,"
0
0 .,,-"
:,;.
'C~('6 '~hruo
q DJ ~l:~"J"\1''.y"
., "'J
lP.!)J:;1
-'"
"' , "
.;:r -" . ,
>"'r ,:.; . Yh"."h..V
;~(!.,.J;"k\/t,:"J
.JVi..,.at~fl:t j. 11.'-,
t.' "
318
Soil Dynamics & Machine FoundatWd
Table 7.5 ~EquivaleDt Cy~les for Anticipated Earthquake
-
Acc!' level in percent -of-peak - acc!. - -
Average acc!. in percent
Number of cycles
- - - - - - - - - - - - - - -
(1) 100-80 80-60 60-40
(2) 90 70 50
(3) 17/2 = 8.5 8/2 = 4.0 25/2 = 12.5
40-100
20
> 1000
Conversion factor -(Fig. - -1.6)-
- -
Equivalent number of . _"J.iL -cycles - -at -0.651:
(4) 2.6 1.2 0.20
(5) 22.1 4.8 2.5
negligible Total
0.0 29.4
Total number of cycles for 0.75 1:max= 29.4/1.5 = 19.6
22 20
-~19'6
i\ 1\ ,.......
.
I
0- 16
~
z 0"1
~
r-0 0
-u
12
\
0
111
~
>u
0
8
... c
~
0 > :J er UJ
4
I
I I I
0
J
0
0
.
2.9
.
10
Distance
20 from bLast
30 (m)
Fig. 7.38: Equivalent cycles versus distance (Gupta and Mukerjee, 1979)
40
.
'319
Liquefaction of Soils
Similarly the blast records at different distances are also converted into equivalent number of cycles of 0.075 g acceleration (Fig. 7.38). From this figure it can be observed that vibrations generated due to blast at a distance of2.9 m are equivalent to 19.6 cycles of 0.075 g, the expected earthquake i.e. the blast has the same severity as the design earthquake at a distance of 2.9 m from the blast hole. From Fig. 7)6, the pore pressure developed at a distance of 2.9 m, and at a depth of 2.5 m is 1.74 m of water column. The critical hydraulic gradient for this site is 0.8, therefore at a depth of 2.5 m the critical porewater pressure or hydraulic head is 0.8 x 2.5 = 2.0 m. The pore pressure developed is 1.74 m. The actual porewater pressures developed will be larger than the measured value of 1.74 m, '~ecause there will'be a time lag in rise of water level in piezometer pipe. Hence under the above conditions, a larger pore pressure is expected to be developed and complete liquefaction of site is expected during the earthquake. 7.12 EVALUATION OF LIQUEFACTION POTENTIAL USING STANDARD PENETRATION RESIST ANCE The standard penetra~ion test is most commonly used insitu test in a bore hole to have fairly ~ood estimation of relative density of cohesionless soil. Since liquefaction primarily depends on the initial relative density of saturated sand, many researchers have made attempt to develop correlations in liquefaction potential and standard... penetration resistance. IS: 2131-1981 gives the standard procedure for carrying out standard penetration test. SPT values (N) obtained in the field for sand have to be corrected for accounting the effect of ovberburden pressure as below: NI = CN . N
0.4 0
t>I ~
:J
'.
III III t>I ~
/ 200
~
:J .D
L t>I
300
>
t>I
-U t>I
(Fig. 7.39)
tU
The correlation between ~ 1 values and relative density of granular soils suggested by Terzaghi and Peck (1967) is given in Table 7.6.
500
",
...
.
1/
/ / I
.
Fig. 7.39: Chart for correction ofN-values in sand for inOuenceof overburden pressure(pecketal., 1974)
~.
/
I I
"
t>I
~ t>I
= Correction factor
2-0
/
oX
-- 100I
cc.
1.8
/.
E
> 400
CN
--
2
0 v
NI = Corrected value of standard penetration resistance
Corr~ction factor, CN 0.8 la 1.2 1-4 1-6
'"' N
-~ -
...(7.27)
0.6
J.
320
Soil Dynamics & Machine Foundatio" Table 7.6 : NI and,
related to Relative Density
CompaCtness
NI
Relative Density
q, (Deg.)
DR (%) 0-4
Very loose Loose
0-15
< 28
4-10
15-35
28-30
10-30
Medium
35-65
30-36
30-50
Dense
65-85
36-41
Very dense
> 85
> 41
> 50
After the occurence of Niigata earthquake, Kishida (1966), kuizumi (1966), and Ohasaki (1966 studied the areas in Niigata where liquefaction had not occured and developed criteria for differentiatint between liquefaction and nonliquefaction conditions in that city, based on N-values of the sand deposit (Seed, 1979). The results of these studies for Niigata areshown in Fig. 7AO.Ohasaki (1970) gave a usefu rule of thumb that says liquefaction is not a problem if the blow count from a standard penetration tes exceeds twice the depth in meters. -
0
N
-z
I I
I
Light damage
& no liquefaction
I
E
.........
~
50
~
L~ VI \11 ~
L0.
c
100
(:,) 1J L:J .D L-
Heavy damage. and liquefaction
(:,)
> 0
(:,)
> ...
150
---
u
-~
Boundar y determined
---Boundary
-
UJ
by damage
survey (Kishida)
determined by field observation(Kuizumi)
Ohasaki
20 0
10
20
30
40
S tanda rd penet ra t ion resi stance (N blows) Fig. 7.40 : Analysis of liquefaction potential at Niigata for earthquake of June t 6, t 994 (Seed, 1979)
-'.~:
Liquefaction
321
of Soils
On the basis o(more comprehensive study on the subject and data presented by other investigators (Seed and Peacock, 1971; Christian and Swiger, 1976; Seed, Mori et aI., 1977), Seed (1979) proposed the following procedure for liquefaction analysis: (i) Establish the design earthquake, and obtain the peak ground acceleration Qmax'Also obtain number of significant cycles corresponding to the magnitude of earthquake using Table 7.3. (ii) Using Eq. 7.25, determine "Cavat depth h below ground surface. (iii) Determine the value of standard penetration resistance value (N) at depth h below ground surface. Obtain corrected NI value after applying overburden correction to N using Fig. 7.39. (iv) Using Fig. 7.41, determine ('Ch/crv)for the given magnitude of earthquake and NI value obtained in step (iii). Multiplying ('Ch/crv) with effective stress at depth h below ground surface, one can
obtain the value of shear stress N
z 0
KI~ ~
'-
C c::v
\J
0
..:.::
Q.
Cl
.-c
Cl.
0"1
where no liquefaction 0 cc urred.
-
~
.9
\J
c::v
0 5 Open points indicate sites
0
->--\J
.
L..
L..
8.-
Solid poi nts indicate sites and test conditions showing liquefaction
..x
:J 0 1.1\1.1\ 11
0 -4 86'5
-2 0.3
.6.5
C 1.1\ 'L.. 1.1\ Cl :J c::v Cl ..c. \J
for causing liquefaction.
0-6
E
.........
~
"Ch required
07.8
III
-- .-.~ .- .--
113 ~ O. 2
07.8
r== 0 ..c.
.
Cl
Extrapolated from results 11 of.large scale laboratory tests
--0
III
0
1.1\
0
c::v 0 ....
-L.. III
0
->-Cl -
.~ .-0 v U
L..
Based on field data
;}
L..
0 0
10
20
Modified1.i:)~netration
30 resistance>
1,0 Nl-blows/ft.
Fig. 7.41 : Correlation between field liquefaction behaviour or'sands for level ground conditions and penetration resistance (Seed, 1979)
iI...
,I;
50
322
Soil Dynamics & Machine Foundation..
(v) At depth h, liquefaction will occur if 'tav > 'th
(vi) Repeat steps (ii) to (v) for other values of h to locate the zone ofliquefaction. Iwasaki (1986) introduced the concept of liquefaction resistance factor FL which is defmed as R F =...(7.28) L L R is the ratio of insitu cyclic strength of soil and effective overburden pressure. It depends on relative density, effective overburden pressure and mean particle size. It is given by For 0.02 < Dso < 0.6 mm
~
0.35
R = 0.882 V~~-+70 + 0.225 10glO ( DSQ)
...(7.29 a)
For 0.6 < Dso < 2.0 mm
R
~
0.882~a, ~ 70 - 0.05
...(7.29b)
where N = Observed value of standard penetration resistance
crv = Effectiveoverburdenpressure at the depth underconsiderationfor liquefactionexamination in kN/m2 D50 = Mean grain size in mm L = is the ratio of dynamic load induced by seismic motion and effective overburden pressure. It is given by a max crv " 'd L -- -.=:-. ...(7.30. g
amax
crv
= Peak ground acceleration due to earthquake = 0.184 x 10°.320 M (Dfo.s.
g
where M = Magnitude of earthquake on Richter's scale
D = Maximumepicentraldistance in km. (Fig 7.42)
°v = Total overburden pressure rd = Reductionfactor to accountthe flexibilityof the ground g = Acceleration due to gravity, mIs2 rd = 1 - 0.015 h . h = depth of plane below ground surface in m For the soil not to liquefy FL should be greater than unity.
...(7.31
Liquefaction
323,
of Soils
9
Nu mbers represen t the earthquake numbers
8
30
.
~
.g
-
7
:J C
0'1 0
Lower bound
6
~
l0910 0 =0.77M - 3'6 Mean
5
(M>6)
line
L09100
=
0-87 M-4-S 'r
4
1
2
5
Maximum
10
20
epicentral
lOO"
50 distance
0
200
ot liquefied
500
1000
sites,D (km)
Fig. 7.42: Relationship between the maximum epicentral distanceofliquefied sites (D) and earthquake magnitude (M) (Kuribayashi, Tatsuoka and Yoshida, t 977)
7.13 FACTORS
AFFECTING
LIQUEFACTION
~
-
Although the factors affecting liquefaction have been discussed during the laboratory and field studies on liquiefaction, they are summarised below: '
7.13.1. Soil Type. Liquefaction .occursin cahesionless soils as they lose their strength completely under vibration due ta the development of pore pressures which in turn reduce the effective stress to zera. , Liquefactian daes not occur in case of cohesive soils. Only highly sensitive clays may laase their strength substantially under vibration. 7.13.2. Grain Size and Its Distribution. Fine and uniform sands are more prone to liquefaction than coarser ones. Since the permeability of coarse sand is greater than fine sand, the pore pressure develaped during vibrations can dissipate faster. 7.13.3. Initial Relative Density. It is .one .ofthe mast important factars contralling liquefactian. Bath pare pressures and settlement are canstderably reduced during vibratians with increase in initial relative density and hence chances of liquefaction and excessive settlement reduce with increased relative density. '7.13.4. Vibration Characteristics. Out .ofthe four parameters .ofdynamiuc load namely (i) frequency; (ii) amplitude; (iii) acceleratian; and (iv) velacity; frequency and acceleratian are mare impartant. Frequency .of the dynamic laad plays vital rale ,.only if it is clase ta the natural frequency .of the system. Further the liquefactian .,,', depends. .onthe . ..11 typeI" of the .dynamic""',laad i.e. whether'it isa transient laad" .orthe. ,
laad causing steady vibratians.'.'" , ' '0.' , ,. , . '" ~L'>+::' '
..
r,
.
.:
'
:
324
Soil Dy"amics
'
& Macll ille Folllldatiolls
Whole stratum gets liquefied at the same t~eunder tra~sient loading, while it inayproceed from top to lower layers u?der steaqy state vibrations (Florin and Ivanov, 1961). ,For a given acceleration, liquefaction occurs only after ~ certain number of cycles imparted to the deposit. Further, horizontal vibratiorls have more severe em~-c.tthan vertical vibrations. Multi directional shaking is more severe than one directional loading (Seed ~t al.~ 1977), as the pore water pressure build up is much faster and the stress ratio required is about 10 percent less than that required for unidirectional shaking. 7.13.5. Location of Drainage and Dimension of Deposit. Sands are more pervious than fine grained soil. However, if a pervious deposit has large dimensions, the drainage path increases and the deposit may behave as undrained, thereby, increasing the chances of liquefaction of such a d~posit.The drainage path is reduced by the introduction of drains made out of highly pervious materiaL" 7.13.6. Surcharge Load. If the surcharge load, i.e. the initial effective stress is large, then transfer of stress from soil grains to pore water will require higher intensity vibrations or vibrations for a longer duration. If the initial stress condition is not isotropic as in field, then stress condition causing liquefaction depends upon Ko (coefficient of earth pressure at rest) and for Ko > 5, the stress condition required to cause liquefaction increases by at least 50%.
7.13.7. Method of Soil Formation. Sands tmlike clays do not exhibit a characteristics structure. But recent investigations show that liquefaction characteristics of saturated sands under cyclic loading are significantly influenced by method of sample preparation and by soil structure. 7.13.8. Period Under Ssustained Load. Age of sand deposit may influen<:eits liquefaction characteristics. A 75% increase in liquefaction resistance has been reported on liquefaction of an undisturbed sand compared to its freshly prepared sample which may be due to some form of cementation or welding at contact points of sand particles and associated with secondary compression of soil. 7.13.9. Previous Strain History. Studies on liquefaction characteristics of freshly deposited sand and of similar deposit previously subjected to some strain history reveal, that although the prior strain history caused no significant change in the density of the sand, it increased the stress that causes liquefaction by a factor of 1.5. 7.13.10. Trapped Air. If air is trapped in saturated soil and pore pressure develop, a part of it is dissipated due to the compression of air. Hence, trapped air helps to reduce the possibility to liquefaction, 7.14 ANTILIQUEFACTION
MEASURES
A comprehensive study is required to find out various possible measure to prevent liquefaction. Tho.ugh it depends on -a number of factors, however, few can be controlled in field. Based on these, certain methqds have been suggested (Lew, 1984). Liquefaction resistance to some extent can be improved by :
7.14.1. Compaction of Loose Sands. As has been indicated earlier, loose saturated sands are more prone to liquefaction than dense saturated s~nds. Therefo~e!the liquefaction potential can be,redu~ed by coI?patting the loose sand deposit before any structure is constructed. The various '-methodssuggested for compaction
loose' sands . of -
in'situ
are: .
.
' '
. .' .h .
-
,
'
7.14.1.1. Rolling with rubber tyrerollers: It may be accompli~hedby excavati~g some depth, then carefully backfilling in controlled lift thickness and compacting the soil. When ruJ>bertyres are used, lifts are commonly 150 mm to 200 mm. This method, however cannot be used for compacting deep sand deposits.
325
;qllefactioll of Soils
,14.1.2. Compaction with vibratory plates and vibratory rollers: Compacti'on of cohesionless soils can c achieved using s~ooth wheel rollers commonly with a vibratory devIce inside. Lift depths upto ahout .5m to 2m can be compacted with this equipment (Bowles, 1982). Also plates mounted with vibratory ~semblycan be used; however, small thifkness of soils ca~ be compacted by these methods and they can' ot be used for large deposits.'
.
,.
.14.1.3. Driving of piles: Piles when driven in loose deposits, compacts the sand within an area overed by eight times around it. This concept may be utilized in compacting the site having loose sand \eposits. As pile remains in the sand, the overall stiffness of the soil stratum increases substantially. '.14.1.4. Vibrofloatation : The method is most commonly used to densify- cohesionless deposits of sands L11d gravel with having not more than 20% silt or 10% clay. Vibrofloatation utilizes a cylindrical penetrator. t is an equipment of about 4m long and 400mm in diameter. The lower half is vibrator and upper half s stationary part. The device has water jets at top and bottom. Vibrofloat is lowered under its own weight vith bottom jet on which induces the quick sand condition, when it reaches the desired depth, the flow )f water is diverted to upperjet and vibrofloat is pulled out slowly. Top jet aids the compaction process. \s the vibrofloat is pulled out a crater is formed. Sand or gravel is added to the crater formed. 7.14.1.5. Blasting: The explosion of buried charges induces liquefaction of the soil mass followed by ~scape of excess pore water pressure which acts as a lubricant to facilitate re-arrangement and thus leading the sand to a more compacted state. The earliest use of detonating buried charges of explosive for compacting loose cohesionless soils in [heir natural state has been reported by Lymari. (1942). He concluded that (i) Lateral distribution of charges should be based on results obtained from a series of single shots. ,~ (ii) Where loose sands greater than 1Om thick are to be compacted, two or more tiers of small charges are preferred. (iii) For deposits less than 10m thick, charges placed at 2/3rd depth from surface will generally suffice. (iv) There is no apparent limit of depth that can be compacted by means of explosive.
.
Later Hall (1962) reported that (i) Repeated blasts are more effective than a single blast of several small charges detonated simultaneously. (ii) Very little compaction can be achieved in top 1m.
(iii) Small charges are more effective than large charges for compacting upper 1.5m of sand.
.'
(iy) The compaction gained by repeating the blasts more than 3 times is small. (v) The relative densities can be increased to 80%. .
.
. -
..
7.14.2 Grouting and Chemical Stabilization. Grouting is a technique of inserting some kind of stabi- 'lizing agent iI:1tothe soil mass under pressure. The pressure forces the agent into the soil voids in a limit space around the injection tube. The agent reacts with the soil and/or itself to form a stable mass. The most common grout is a mixture of cement and water, with or without sand. Generally grout can be used if the permeability of the deposit is greater than 10-5 m/s. Chemical stabilization is in the form of lime, cement, flyash or combination of these.
......
"","""""""
"'"
Soil Dynamics & MJlcmne Ftnmdations
326
7.14.3. Application of Surcharge. Application of surcharge over the deposit liable to liquefy can also be used as an effective measure against liquefaction. Figure 7.43 shows a plot between rise in pore pressure and effective over burden pressure at an acceleration of ten percent of g. It indicates that pore pressure increases with increase in overburden pressure till a maximum value of pore pressure is reached, after which it starts decreasing with further increase in surcharge. Thus an overbuden prssure above this value, depending upon the situation, makes the deposit safe against liquefaction.
24 Dead weight surcharge 250 mm depth _Acct. 10% gNo. of cyc les 10
Solani sard
"E z-"
-
201-
DR=20 15Hz
%
16
Cl! L-
:J
~ I Zone ot liquefaction ~ 12 .
L-
a. Cl! L-
0
a. III III -.CI!
v
8
-
-
4
)(
UJ
0 0
4
8
12 Effective
16
20-
overburden
24
28
pressure
32 (kN/m2
36
40
44
48
)
Fig. 7.43: Excess pore water pressure versus initial pressure on Solani sand (Gupta, 1979)
7.14.4. Drainage Using Coarse Material Blanket and Drains. Blankets and drains of material with higher permeability reduces the length of drainage path and also due to higher coefficient of permeability, speed up the drainage process (Katsumi et a1. 1988 and Susumu et a1. 1988). 7.15 STUDIES ON USE OF GRAVEL DRAINS Yoshimi and Kuwabara (1973) were first to introduce gravel drains to stabilize a potentially liquefiable sand deposit. Seed and Brooker (1976) have proposed an analytical procedure for designing such drains (Fig. 7.44). These drains are considered fully effective if the permeability of material of drains, ktJ' is about 200 times the permeability, ks' of the soil in which they are installed i.e. (kd/ks) > 200. The effectiY,e drainage path is reduced by ,the introduction of number of artificial drams. Seed and Brooker (i97~) developed nondimensional charts as shown in Fig. 7.45 .'for determining the spacing of drains. Vario~5 . '1 terms shown on this figure." are '.as below: ., . .' .< . ill( ..
, ,
.. ~ .~
'--~q~~e!aftioll;
°t'~~Lf
C,',
\, "
'.,
.
327 f:~;,
.
. '-",'"
',~
,".',
c,:
, "
e'
,,-'. .e'
..
.
'"
'-"",'
"
(0 ) Plan , :.
','...";- :;""",-,-,
- ':
-
: . .' - t ,~
~ "'.:
',.',-,-.-..
,-,..'
/'~:'"
1
i
,
':::'{. ".
"',
"
":-',~-,','~":",.',,-.
",'
~ "::~~LG_WT ~
,..-.
'r"
'
. ':,,',
"':, '..',' ~~~I
',.-',-
, ,', - ,. ,,','
,
c
'
, ..
.
"
'..
J, -' '" , '.' .' , "r
Re' ,
.
,
, :,', t" " "" 'N
. ..
.:~-;.. " '" .,-
; " ~
,..
".':
.:2"::::.,
(b)
.'~';;':!"":~',~~
,:;.,,-y,~~:,-J'.:-)
:: 'G
Rd~
;'"",","'{',,"-,"
$" ""',
" .;
~
,".
.,.,
,~'
;')
Sec tior.' ot, S-S ..
, ,;"::,.
J-
:,/
.'
¥' \ ,,-,
: i."".', .
Re
\~\
t' '"It> -".' ,fi.g,.7.4,4 :,.:Gravel drains ,-.".",..:',.,'"""c.<,"..,-,,,
-";;'"""",,"'-!"~'~;,,,,~;'\"""
I:',
,~"'-::-
328
SoU Dynamics & Madine
Founda~s
<-
0-6 rg
0.4 0 .21-1f:J°
Tad-:' '. '
0 0
0.1
0'2
0'3 Rd
0-4
O'S
0.4
0-5
0-6
/ Re
(a) iNl = 1 1.0
rg
0-4
0-2
0 0
0.1
0.2
0-3 R.d / Re (b) NiNl
-
0-6
-
=2
Fig. 7.45: Relation between gr~3test perewater pressure ratio and drain system parameters (...Contd.)
-:.
I!
.
I
o~
lefaaion- of Soils
~19
,.0 0-8 0 -6 t-
- 200
110
Tad-
rg
0.4. 0.2
0 0
1.0
3-0
2.0
~.O
5.0
6.0
/ Re
Rd
(c) N/Nt = 3
1.0
0.8 0,6 rg
0.4 0.2 "
0
0
.:.0-1
0'2
0.30,4
Rd
.' . 0,5
0.6
/ Re
'J:
(d) .NslNl = 4 Fig. 7.45: Relation between greatest porewater pressure ratio and drain system parameters
,
. .'
'" I " .' c' .:.
330
Soil Dynamics
for design r = Limiting value of,- Ugchosen , .. g
cry
.'
-~
-
& Machine
FiJII;idat;o;u
-<'~
-'~'-
Uu = Excess porewater pressure build up in ~.cyclic 'simple sh~ar test (Fig. 7.46) , 0
= Initial consolidation pressure Ns = Number of cyclic stress applications cry
.
NI = Number of stres~cycles needed for liquefaqiion Rd
= Radius of rock or gravel drains,
:
'
iUSof the rock or gravel drains Re
T
-
h
,d
ad - '¥ro 'my ( Kh
1d
'
'
R~
J-
= Coefficient of permeability of sand in horizontal direction'
'¥ro= Unit ~eight mv
-
= ~fec tiV; rad
of water'
,'-'
= Coefficient,of volume compre~~ibili~_.~fsand,
= Duration
of earthquake"
,
1.0 0.8 0.6
-Ug
ay 0.4 0.2 0 0
0.2
,
0.4
0~6
N/N,
0.8
. 1.0
,
Fi,g. 7.46: Rate of pore water pressure buildup in cyclic simple s,hear test
Yosufumi et al. (1~84) had,developed a method to evaluate liquefacti~~resistance under partia drained condition, They assumed that the dissipation of excess pore water pressure induced by an ea! quake will occur according to Darcy's-'law. Dynamic triaxial apparatus was used to conduct tests un perfectly undrained and perfectly drained conditions. In case of perfectly uildrained condition, liquet tion resistance was neither influenced by permeability of sample nor by the frequency of loading. Thi
reasonable, because, no drainage of water is involved and hence permeability of soil does not have a I to play, It was also concluded that in case of p'artial~ydrained condition, the effect of drainage frequency is remarkable for soils with relatively largef1relative density, OR' and not so significant soils with smaller relative density: But for loosesoils'rat~ of generation' of pore pressure depend' number of cycles of stresses which in turn depends on frequency. Rate of pore pressure built up depl on the rate of dissipation of pore pressure, which is based on draInage.
'luefactidn of Soils
",'
.
331
Yasushi and Taniguchi (1982) carried out large scale model tests to confirm the effectiveness of 'avel drains for preventing liquefaction of sand deposits. The purpose of the tests, as stated by them, as : .
(i) to know the generation and dissipation of ' pore water pressure, ,. " characteristics , "',-'
'".-
',-,
"
,
,0"
",
"
(ii) to clarify the effective are~foithe g~aveldra~fromthe view-p~int of preventing liquefaction and (iii) to know whether the grayel drain.is effective in preventing the liquefaction of subsoils under a road that is partially buried. ' :' - ,,"", .
,
They performed tests on shaking table of 12 m x 12 m x':3ffi(deep) size, filled with cohesionless soiL 'he acceleration of loading was 200 gals, the duration of shaking w,as,one minute and the frequency was cps. They concluded that pore water pressure within 500 mm from the edge of a gravel drain is much maller than that, for away from the gravel drain. '
Wang (1984) made experimental study on liquefaction inhibiting effect of gravel drains. A shaking )oXof size 1.5 m x 0.28 m x 0.5 m was used. He used gravel drains walls under the foundation and it was lssumed that under plain strain condition the walls ar~ referred as drains. It was noted by him that the ,ection of non liquefied zone of deposit was basically a trapezoid in which pore pressure ratio (i.e. the "atio of excess pore w
'
,
.
As the number of d!'ainsin~talled is increased, the non liquefied zone increases. As the acceleration increases, the zone reduces gradually but the increase in time does not reduce the non liquefied zone. The angle of trapezoid was found to be 15° to 17° in the 'direction of depth. The zone is about 40 mm outsid,e the drains. It was also observed by him that the surface drains may effectively prevent foundation settlement. In order to obtain good effect in reducing foundation settlement it must be ensured that adequate dept.I1,~ndwidthQf drains qe inst~lled when ~n~tallingshallow dr~ins and outside drains. a-hara and Tamamoto (1987) pi"ese'nteda fundamental 'study on gravel pile for preventing liquefaction. They used a shaking box of size 1.0 m x 0.35 x 0.65 m (deep). Radii of gravel piles were 0.75 m, 0.15 ~ etc. The flow of pore water was assumed to be horizontaL They measured the pore water pressure at points near the dr~iris a~d away from the 'drai~s.They concluded that liquefaction occured at points too far from the drain and that at points close to the gravel drain, liquefaction did not occur. Results obtained by them presented -in form of -optimum radius of pile and optimum spacing between gravel piles. Figure 7.47 shows the effective circle whiChis defined as the circle with area equal to area of square with sides equal to the line joining mid p~ints of the ~pacing between adjacent gravel piles. The sides of the squares are tak~n as optimum spacing between gravel piles. They found that the effective area of gravel pile increases in proportion to the diameter of the gravel pile and the permeability. For a fixed diameter of pile and permeability of'soil, as the optimum distance decreases, the pore pressure ratio decreases. As the permeability increases, pore pressure ratio decrea~es very sharply. Highly permeable gravel are much more effective even at higher optimum distance and smaller diameter of drains. ,
,
AJlexible ve~tical drain formed by us'ing organi,c fibres like jute or coir has been used in several projects. The 'most important properties of.such drains are permeability and tensile strength, The jute filter cover h.~spermeability better than 10-5m/sec. This facilitates the flow of water from pervious lenses present in the seams and layer of sand and speeds up the pore water pressure dissipation. They have the advantage of decaying and getting mixed with the soil without harming the environment. When the filter permeability 'is large, the clogging of the drain has to' be considered. - - -
,
,
332
Soil Dynamics & Machine Foundations
C ire le of et f e cti ve a re a
..
-'-EL:~
i"
l'77'r~'.
~I
Fig. 7.47: Radius and distance (or spacing) of gravel pi/es (O-hara, 1987)
Geotextiles are used fairly widely in surface and subsurface installations, (Krishnaswamy and Issac, 1995), Crushed stone wrapped in geotextiles have often been used as surface and subsurface drains. Perforated plastic pipes too may be used for this purpose. They may be filled with crushed stones, if necessary.
I
ILLUSTRATIVE EXAMPLES
I
Example 7.1 At a given site, a boring supplemented with standard penetration tests was done upto 15.0 m depth. Tht results of the boring are as given below: Depth
Classification
(m)
of soils
1.5 3.0 4,5 6.0
SP SP SM SM
0.18 0.20 0.12 0.14
7.5 9.0 10.5 12.0 - 13.0 15.0
SM SP SW
0.13 0.16 0.20
' SW
SW SW
N-Value
D50 (mm)
3 5 6 9
19 30 35 40
12 17 20
45 52 52
0.22
18
46
0.22 0.24
24 30
60 65
'.
Remarks
DR (%)
(i) Position of watt table lies 1.5 m below the grour surface
(ii)
'YmOiSI=
19 kN/r
'Ysub = 10 kN/m
.'
333..
iquefaction' of Soils
The site is located in seismic ally ,active region, and is likely to be subjected by 'an earthquake ,of
1agnitude 7.5. Determine the zone of liquefaction using
,
(a) Seed and ldriss (1971) method (b) Seed (1979) method (c) lwasaki (1986) method >olution :
(a) Seed and Idriss (1971) method (i) From Fig. 7.42, For M = 7.5, D = 106 Km amax = 0.984 x 10(0.302x 1.5) x 10~.8 = 0.083 g 9.81 Number of significant cycles (Table 7.3), Ns = 20 (For M = 7.5)
From Eq. 7.31,
amax
(ii) From Eq. (7.25)~
'tav
.
= 0.65yh. - g . rd
= 0.65 x Yh x (0.083) rd = 0.054 Yh rd
It may be noted that y h represents the total stress at depth h below ground surface. Value of rd are read from Fig. 7.25. Values of total stress, ad and 'tavat different depths are given in Cols, 3 and 5 of Table 7.7. (iii) For 50% relative density, the stress ratio ad/2a3 is read from Figs. 7.22a and'b for given values of,Dso' Average of the two values is the stress ratio for number of significant cycles equal to 20.
The stress causing liquefactionat any depth is then, c9mputedusing Eq. 7.22. . DR
ad 'th=
'.
-'
( 2a 3 ) 50%"50.Cr.av
Values of Cr are .obtained using Table 7.2. The details of computations of 'rh are summarised in
Table 7.8.
.
Table 7.7 : Computation of 'ray S.No. (m)
Depth (kN/m2)
Total stress
(I)
(2)
(3)
1. 2. 3. 4. 5. 6. 7. 8. 9. 1p.
1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 .15.0
28.50 58.50 88.50 118.50 148.50 178.50 208.50 238.50 268.50 298.50
rd
---------------------------------
r
tal' (kN/m2)
(4)
(5)
0.99 0.98 0.97 0.97 0.96 0.95 0.94 0.92' 0.91 0.90
1.52 3.10 4.64 6.21 7.69 9,14 10.54 11.90 13.23 14.51
3:J~;, Id
Soil Dynamics
.),'
& Machine
Foundatiolls
:"
';1'able S.
No
7.8 Computations'
Depth (m)
of 'thfrom
Effective stress. cr
Seed and Idriss
DR
,
(1971) Metbod"
Cr
.rh
, .rh/a!,
(kN/m2)
;,
(kN/nO
I.
1.5
28.50
19
0.55
0.2198
1.31
2.
3.0
43.50
30
0.55
0.2260
3.24.
3,
4.5
58.50
35
,0.55,
0.2012 . '
4-.53,
4.
6.0
73.50
40
0.55
0.2074
6..71
5.
7.5
88.50
45
0.555
0.2043
9.03
6.
9.0
103.50
52
0.573
0.2136
13.17
7.
10.5
118.50
52
0.573
0.2260
15.96
8.
12.0
133.50
46
0.556
0.2298
15.69
9,
13.5.
148.50
60
0.60
0.2298
24.57
1O,
15.0
163.50
65
0.61
0.3336
30.29
.'
(b) S,eed (1979) Method. " . '(i) In. this method, the value of shear' stress at any depth induced 'by the earthq",lake is obtained , ,.;'. exactly in th,e same manner as illustrated in Seed and ldriss (1971) method (Table 7.7) (ii) To de,termine 1:hfirstly N-values are corrected for effective overburden pressure using Fig. 7.39. The stress rati
Depth
N- Va/ue
of'th by Seed (1979) Method
Corrected
,
Effective
.rh/av
.rh
(6)
(7)
óóóóóóóóóóóóóóóóùóóóóóóóóóóóóóóóóóóóó (m)
(1) 1.
>
",
N
stress (5)
(2)
(3)
(4)
1.5
3
4
28.50:,
0.045
'
'1.28
2.
3.0
5
6
43.50
0.067
- 2.91
3.
4.5
6
7
58.50
0.079
4.62
4.
6.0
9
10
73.50
0.113
8.30
5.
7.5
12
13
88.50
0.146
12.92
6:-
9.0
17
17
103.50
0.191
19.77
7..
10.5
20
' 18
118.50
0.194
22.99
8.
12,0
18
17
133.50
0.183
24.43
9.
13.5
24
" 20
148.50
0.225
31.93
163.50
0.269
43.98
. ."
I
10.
15.0
30
25 .
"
l'UeTattwn
ofSolls:
I Iw-asaki's
,'"
,<""
Method
.--'( i) Fir'stIy the'value of factor R is obtained using the following Eel.7.29a ; .
,
.
, ,,'
..
'
0.35
.,
R .= 0.882~
V~:+ 70,
..
+0.225 loglO
(
J
°50
The details of computations of factor R are given in Table 7.10. "
Table 7.10 :' DetaH~ of Computations
S,No,
Depth
ay
(I)
'
(m) ,(2)
(3)
(4)
28.50 43.50 58.50 73.50 88.50 103.50 118.50 133.50 148.50 163.50,'
0.18 0.20 ' 0.12 ,0.14 0.13 .0.16 0,20 0.22 0.22 0.24
4.5 6.0 7.5 9.0 10.5 12.0 13.5 15:0
5. 6. 7.',' 8. 9. "1'0. -,':,. ".
R
N
'D5o
2 (kN/m )
1.5 3.0
4,
Method
'- - - - - - - - - - -(mm) - - - - - - - - - - - - - -
- - - - - -- 1. 2. 3.
for obtaining Factor R in Iwasaki's
(ii) The factor L is then obtained ,
'
.'
using Eq. 7.30 :
L = .
a
(5)
"
(6)
3 5 6 ,9 12 17 20 18 24
0.2189 0.2398 0.2950 0.3104 0.3394 0.3525 0.3419 0.3077 0.3377 0:3464
" 30
/.
a
max '~'r-
d
gall
The details of computations are given in Table 7.11. The ratio of factor of safety RlL is listed in the astcolumnofTable7.11.
",'
,
'
Table 7.11 : 'Details :of Computations S.No.,
-
., ,Depth
(m). ,
--------------------------------.
'.,
.,"
Liquefaction
Potential
rei
L
FL
'. (3),
(4)
(5)
(6)
,
L. 2. 3. 4.
.
5,
. 6: 7,
, 8. ., 9. 10,,- , ::" i'C,:
,).5., 3.0 4.5 6.0 7.5 . ; 9.0
10.5 12.0., 13.5 15.0. ,'L
,
by Iwasaki's J\1ethod
. crJcrv'"
(2)
(I)..
of Obtaining
0.98 0.97 0.97 0.96 0.95 0.94 0.92
i,758 :,,-,'~."'L787 1.808 ; ...;,;',1.8,26 . ;:..\:; ."
,
0.0820 0.1092 0.1215
0,9,9
1.000 1.345 1.513 '!' .' .1.612 . ~, '1.678 ' '1.725
-
...
,
2.670 2.196 2.427 2.397 2.544 2.597 2.498
',0.1295 0.1334 0.1357 "
. . , 0.1369
0.91
0.1362 0.1363
0.90
, 0.136,1
,
2.260 :
2.4 78 2.545
336
Soil Dynamics & Ml1ChineFoullda
.
In Table 7.12, summary of different methods are given. It is evident from this table that liquefaction ,tl occurs only uptoL5 m depth according to Seed and Idriss (1971) method, 3.0 m depth according to Seed $' (1979) method; and no liquefaction according to Iwasaki's method. Tabl-e 7.12 : Summary of Different Methods S.No,
Depth
(m)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
t'h
th
Seed and Idriss
Seed (1979)
method (1971) (kNlm2)
(kN/m2)
t'av
(kN/m2)
1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0
FL (R/L)
1.28 2.91 4.62 8.30 12.92 19.77 22.99 24.43 31.93 43.98
1.31 3.24 4.53 6.71 9.03 13.17 15.96 15.69 24.57 30.29
1.52 3.10 4.64 6.21 7.69 9.14 10.54 11.90 13.23 14.51
/wasaki method
2.670 2.196 2.427 2.397 2.544 2.597 2.498 2.260 2.4 78 2.545 'I
ÎÛÚÛÎÛÒÝÛÍ Arya, A. S., ~andakumaran,
p" Puri, V. K. and S. Mukerjee, (1978), "Verification of liquefaction potential by field
blast tests", Proc. 2nd. International Conference on Microzonation, Seattle-, USA, Vol. II, p. 865. Bowles, 1. E. (1982), "Foundation analysis and design", McGraw Hill Book Co. SIngapore. Casagrande, A. (1965), "The role of the calculated risk in earthwork and foundation engineering", 1. Geotech, Engg. Div., ASCE, Vo\. 91, No. SM4, pp: 1-40, Proc. Paper 4390. Castro, G. (1969), "Liquefaction of sands", Harvard Soil Mechanics Series No. 81, Harvard University, Cambridge,
Massachusetts:
.
.
.
Castro, G. (1975), "Liquefaction and cyclic 'mobility of saturated sands", Journal of the Geotechnical Engineering Division, ASCE, 10L(GTS), pp. 551-569. Castro, G. and S. 1. Poulos (1976), "Factors affecting liquefaction and cyclic mobility ", Symposium on Soil liquefaction, ASCE, National Connvention, Philadelphia, pp. 105-138. Christian, J. T. and W. 'F. Swiger (1976), "Statistics of liquefaction and S.P.T. results", 1. Geotech. Engg. Div., ASCE, Vol. 101,No. G T 11, pp. 1135-1150. . Corps of Engineers, U. S. Dept. of the Army (1939), "Report of the slide of a portion of the upstream face at fort peck dam", U. S. Government Printing Office, Washington, D. C. DeAlba, P., H. B., Seed, and C. K. Chan (1976), "Sand liquefaction in large scale simple shear tests", J. Geotech. Engin. Div., ASCE, Vol. 102, No. GT9, pp. 909-927. Dunn, J. A., 1. B. Auden and A. M. N. Ghosh (1939), "The Bihar Nepal earthquake of 1934", Mem. Geol. Surv., India, Vo\. 73, p. 32. . Finn, W. D. L., P. L. Bransby and D. 1. Pickering (1970), "EffeCtof strain history on liquefaction of sands", 1. Soil Mech. Found. Div., ASCE, Vat. 96, No: SM 6, pp. 1971':J934.
.
.
-'
337
Liquefaction of Soils
Finn, W. D. L, J. J. Emery and Y. P. Gupta (1970), "A shaking table study of the liquefaction of saturated sands during earthquakes", Proc. Third Europ. Symp., Earthquake Engin., pp. 253-262. Florin, V.A., and P. L Ivanov .(1961), '~Liquefactionof saturated sandy soils", Proc. Fifth Into Conf. Soil mech. Found. Engin., Paris, V01. 1, pp. 107-111. Gupta, M. K. and S. Mukherjee (1979), "Blast tests for liquefaction studies", Proc. International Symposium on Insitu Testing of Soil and Rock and Performance of Structures, Roorkee, India; vel. I, p. 253. Geuze, E. (1948), "Critical density of some dutch sands", Proc. 2nd ICSMFE, vol. 1Il, pp. 125-13{). ... Gupta, M. K. (1979), "Liquefaction of sands during earthquakes", Ph. D. Thesis, University of Roorkee, India. Hall, E. C. (1962), "Compacting of a dam foundation by blasting", ASCE Journal, Vol. 80. Hazen, A. (1920), "Hydraulic fill dams", ASCE Transactions, Vol. 83, pp. 1713-1745. Housner, G. W. (1958), "Mechanics of sand blows", Bull. Seismol. Soc. Am., Vo1.48, No. 2, pp. 155-168. lshibashi, 1.and sherif, M.A. (1974), "Soil liquefaction by torsional simple shear device", Journal of the Geotechnical Engineering Division, ASCE, 100, G T 8, pp. 871-888. Katsumi, M. M. Maraya and T. Miteuru (1988), "Analysis of gravel drain.against liquefaction and its application to design", IXth WCEE, Tokyo, vol. Ill, pp. 249-254. Kishida, H. (1966), "Damage' of reinforced concrete bu,ildings in Niigata city with special reference to foundation engineering", Soil Found. Engin. (Tokyo), Vo\. 9, No. \, pp. 75-92. .~, . . Koppejan, A. W. ,Wamelen, B. M. and L J. Weinberg (1948), "Coastal flow slides in the dutch province of seeland", Proc. 2nd lCSMFE, Vo\. 5, pp. 89-96, Rotterdam. Krishnamaswamy, N. R. and N. T. lssac (1995), "Liquefaction analysis of saturated reinforced granular soil", ASCE, Vol. 121, No. 9, pp. 645-652. Krishna, J. and S. Prakash (1968), "Blast tests at obra dam site", J. Inst. Engin. (India), Vol. 47, No. 9, pt. CI5, pp. 1273-1284.
..,'
Kuizumi, Y.{1966), "Change in density of Sand subsoil caused by the Niigata earthquake", Soil Found. Engin. (Tokyo), Vol. 8, No. 2, pp. 38-44. Kuribayshi, E., Tatsuoka, F.and Yoshida, S. (1977), "History of earthquake induced soilliquefactiori in Japan", Bulletion of PWRI, 31. Kuwabara, F. and Yoshumi, Y. (1973), "Effect of sub surface liquefaction on strength of surface soil", ASCE, JGE,.
VoI.19,No.2.
.
Lee, K. L , and C. K. Chan (1972), "Number of equivalent significant cycles in strong motion earthquakes", Proc. , First lnt. Conf., Microzonation, Seattle, Vol. 2, pp. 609-627. Lee, K. L. and H. B. Seed (1967), "Cyclic stress conditions causing liquefaction of sands", J. Soil Mech. Found, Div., ASCE, Yo\. 93, No. SMI, pp. 47-70. Lew, M. (1984), "Risk and mitigation of liquefaction hazard", Proc. YIIlth WCEE, Yol. I, pp. 183-190. Lyman, A: R. N. (1942), "Compaction of cohesionless foundation soil by explosive", ASCE Trans., Yo\. 107. Maslov, N. N. (1957), "Questions of seismic stability of submerged sandy foundations and structures", Proc. Forth Int. Conf. Soil Mech. Found. Engin., London, Vol. 1, pp. 368-372. Matsuo, H. and S. Ohara (1960), "Lateral earth pressure and stability of quay walls", Proc. Second World Conference on Earthquake Engineering, Tokyo, Vol. 1, pp. 165-182. Middlebrooks, T. A. (1942), "Fort peck slide", ASCE Transactions, Vol. 107, pp. 723-764. Ohasaki, Y. (1966), "Niigata earthquake 19'64,.buildingdamag~ and soil conditions", Soil ~ound.(Tokyo), Vol. 6, No. 2, pp. 14-37. ,', : Ohasaki, Y. (1970), ;"Effectsof sand.c~mpacti~n on liquefaction during the Tokachioki earthquak~",,S()il.fQ.und. ,
'
. (Tokyo),Yol.;.lO,No.-2,'pp.";t112-J28...
';,
'.I'
;
! i. ',!
,
"',
338
Soil Dynamics
& Machine .Foundation
O-hara, S. and T. Tamamoto (1987)" "Fundamental study on gravel pile.method for preventing liquefaction", ECEE 87, pp. 5.3/41-48. Peacock, W. H. and H. B. Seed (1968), "Sand liquefaction under cyclic loading simple shear conditions", 1. Soil Mech. Found., Div., ASCE, Vol. 94, No. SM 3, pp. 689-708. Prakash, S. (1981), "Soil dynamics", McGraw HillBook Co. Prakash, S. and M. K.Gupta (1970a), "Final report on liquefaction and settlement characteristics of loose sand '" under vibrations", Proc. International Conference on Dynamic Waves in Civil Engineering, Swansea, .
pp. 323-328~'
Prakash, S, and M. K. Gupta(1970b),
.
.
"Blast tests at Tenughat dam site", 1. Southeast Asian Soc. Soil mech. Found.
Engin (Bangkok), Vol. I, No. 1, pp. 41-50. Prakash, S. and M. K. Gupta (1970c), "Liquefaction and settlement characteristics of Ukai dam sand", Bull Indian Soc. Earthquake Tech. (Roorkee), Vol. 7, No. 3, pp. 123-132, Prakasn, S. and mathur, 1. N. (1965), "Liquefaction of fine sand under dynamic loads", Proc. 5th Symposium ofthl Civil and Hydraulic Engineering Department, Indian Institute of Science, Bangalore., India Seed, H. B. and K. L. Lee (1966), "Liquefaction of saturated sands during cyclic loading", ASCE, JGE, VoL 92, No SM 6, pp, 105-34, .
.
.
Seed, H. B. and Idriss, L M, (1971), "Simplified procedure for evaluating soil liquefaction potential", Journal of So; mechanics and Foundations Division, ASCE, 97, SM9, pp, 1249-1273. . Seed, H. B" Lee, K, L. , Idriss, L M. and F. L Makdisi (1971), "The slides in the San Fernando dams during tlearthquake of Feb, 7, 1971", Journal of the Geotechnical Engineering Division, Proceedings, ASCI VoL 101,No.GT7. .
Seed; H. B. and Booker, 1.R. (1976), "Stabilisation of potentially liquefiable sand deposits using Gravel dra system", Report No. EERC 76-10, Earthquake Engineering Research Centre, University of Califc nia, Berkeley. Seed, H, B. (1976a), "Some aspects of sand liquefaction under cyclic loading", Conference on Behaviour of 0' shore Structures, The Norwegian Institute of Technology, Norway. Seed, H. B. (1976b), "Evaluation of soil liquefaction effects on level ground during earthquakes", State of the f Paper, Symposiumon Soil Liquefaction,ASCENationalConvention,Philadelphia,pp. 1-104. Seed, H. B. (1979), "Soil liquefaction and cyclic mobility evaluation for level ground during earthquakes", 1.Geote Engin. Div., ASCE, Vol. 105, No. GT2, pp. 201-255. Seed, H. B., I.Arango and C. K. Chan (1975), "Evaluation of soil liquefaction potential during earthquakes", ReJ: on EERC, 75-28, Earthquake Engineering Research Center, University of California, Berkeley. Seed, H. B. and L M, Idriss (1967), "Analysis of soil liquefaction Niigata earthquake", J. Soil mech. Found. D ASCE, yoL 93, No. SM 3, pp. 83-108. ' Seed, H. B. , K. Mori and C. K. Chan (1977), "Influence of seismic history on liquefaction of sands", J.Geolt Engin. Div., ASCE, VoL 103, No. G T 4, pp. 246-270. Seed, H. B., and W. H. Peacock (1971), "Test procedures for measuring soil liquefaction characteristics", 1. Mech. Found. Div., ASCE, Vol. 97, No. SM 8, pp. 1099-1199. . Susumu, I. A. I. , Koizimi, K., Node, S. and H. Ysuchia (1988), "Large scale model tests and analysis of Gr drains", IXth WCEE, Tokyo, Vol. III,pp. 261-266. Terzaghi, K. and R. B. Peck (1967), "Soil mechanics in engineering practice", John Wiley and Sons, Inc., New 'r Wang, S. (1984), "Experimental study on liquefaction inhibiting effect of gravel drains", Proc. VIII WCEE, Cal . nia, Vol. 1, pp. 207-214. . .
WaterwaysExperimentStationU. S. Corpsof Engineers(1967),"Potamologyinvestigations,report 12-18,\ cation of empirical methods for determining river bank stability", 1965Data, Vicksburg, Missis'
׬å墳
339
Liquefaction of Soils
Yasushi, S. and Taniquchi, E. (1982), "Large scale shaking table tests on the effectiveness of Gravel drains", Earth-
quake Engg,Conference,Southamopton,pp. 843-847.
"
.Yosufurni, T., Kokusho, G. and Matsui (1984), "On preventing liquefaction of level ground using Gravel pnes", <'
',.
' Proc.
JASCE,
No.
352,
pp. 89-98.
,Yoshimi, Y. 'and H. Oh-aka (1973), "A ring torsion apparatus for simple shear tests", Proc. 8th International Con'- . -' ference on Soil mechanics and Foundation Engineering, Vol. 12, Moscow, USSR.. . Yoshimi, Y. (1967), "Experimental study of liquefaction of saturated sands""Soil Found. (Tokyo), Vol. 7, No. 2 pp. '.",
,20-32.
,,'
"
, Y~~h{~i,:' Y. ;a~d H. Ohaka (1975),- "I~fl~ericeoi ", ..
,
"'..
'
,
"
:
dJgT~e'of sh~~t str~ss revers~l on the liquefaction potential of
saturated sands""" Soil FolJnd.(Tokyo), Vol. 15, No. 3, pp. 27-40. '
'-
,."'"
.
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'
'
,',
'
..'
','
:-,
,..',,',;
,---:
.."
,
,--,-
L,
'
PRACTICE PROBLEMS
!'..'"
I
,
:-
,'..
7.1 Explain the terms 'initial liquefaction', 'liquefaction' and 'cyclic mobility'. Illustrate your an.
,s~e-r 'with
7.2 listand,
"
neat
~ketches.
'
'
,
"...
discuss the factors on which liquefaction of,saturated sand depends. ~, ,
7.3 Give the salient features of the liquefaction studies made by (a) triaxial tests, (b) shake table tests, and (c) blast tests. '
7.4 Describe briefly the following methods of predicting liquefaction potential (a)
Seed
and ldriss
(1971)
-method
",'
,'"
(b) Seed (1979) method (c) lwasaki (1986) method
'
'/>'
7.5 At a given site boring supplement with SPT was done' upto 20.0 m depth. The results of the boring are as given below': " . '. ~ . : ,
,
Depth
Classification
(m)
of soils
(mm)
;'.2,0
D50
'
N-Value
'DR"
SM ,
0.20
SM SP SP SP
0.18 0.16 0.18 0.19
4 5 7 9
30 33, 40 43
12.0
SP
0.19
I0
~' 45
SP'
0.19
16,0
,
SW ,
0.30
,
Remarks"
, ,(%)
4.0 ~.O 8.0 10.0
14.0,
'
. ,,4
','
, ,"I,
25
12 "
52
60
14
(i)
SW
0.32
16
20.0
SM '.
0.20
18'
of water
(ii) 'YmOiSI = 20 kN/m)
56 .'
.f
tabie lies 2.0 m below the ground surface"
::'
18.0
Position
;,
'Ysub= 10 k~/rn
,~3
,
.
3
,
.~
This ~jte is 10catedTD.s~{smically,actIve region, ,and the .likely to be subject~4 by an ~aitliquak~,of magnitude liquefaction ,by the three methods mentioned in Prob. 7.4. ;, ' 7.5. Determine',:ih~"zoneof , ,"',. " . ' " ,.. <':1 ':'.. t'
, ;
,, -.; , '.
:.
.,..'" , , . '; "
DLJ u
GENERAL PRINCIPLES OF MACHINE FOUNDATION DESIGN 8.1 GENERAL For machine foundations which are subjected to dynamic loads in addition to static loads, the conven. tional considerations of bearing capacity and allowable settlement are insufficient to ensure a good de sign. In general, a foundation weighs several times as much as machine (Cozens, 1938; Rausch, 1959) Also the dynamic loads produced by the moving parts of the machine are small in comparison to th static weight of the machine and foundation. But the dynamic load acts repetitively on the foundatior soil system over long periods of time. Therefore, it is necessary that the soil behaviour be"elastic undt
the vibration levels producedby the machine, otherwisedeformationwill increasewith each cycle ( loading and excessive settlement may occur. The most important parameters for the design of a machir foundation are: (i) natural frequency of the machine-foundation-soil system; and (ii) amplitude of ID! tion of the machine at its operating frequency. 8.2 TYPES OF MACHINES AND FOUNDATIONS There are various types of machines that generate different periodic forces. The main categories are: 8.2.1. Reciprocating Machines. These include steam, diesel and gas engines, compressors and pum; The basic mechanism of a reciprocating machine consists of a piston that moves within a cylinder connecting rod, a piston rod and a crank. The crank rotates with a constant angular velocity. Figure shows the outline of a typical Gangsaw in which the out of balances forces may lead to vibration pr' lems. The operating speeds of reciprocating machines are usually smaller than 1000 rpm. Large feci! cating engines, compressors and blowers generally operate at frequencies ranging with in 50-250 r Reciprocating engines such as diesel and gas engines usually operate within 300-1000 rpm. ~
.
The magnitude of the unbalanced forces and moments depend upon the number of cylinders iD machine, their size, piston displacement and the direction of mounting. The mechanism developiDf of balance inertia forces for a single crank is ~hown in Fig. 8.2. It consists of a piston of mass mp~ 'within a cylinder,"a connecting rod AB of ma~smr and crank AO of mass mewhich rotates abou~JI at ~freq~ency 00.The centre of g~avity of the connecting"rod is lo~ated at a distance L) from PO1~' "the rotatmg masses are to be partially or fully balanced, counterweights of mass mwmay be locat~ their centre of gravity at point c"
c~
341
Genel-lll P,inciples. 0/ Machine Foundation Design
,,'
,
"
.
, '..
Uppu slide
-
b ao-ck
log teed
,E In C 0 lit C
l
Saw 'Modes
., E
lower
said~ block
" -
... '
,'r',
0
c.onn~ctin9
u
rod
a. >- c.ounter ~ weight
. .,. . .. .'"
'
Fly wheel
Foundation block
~
. "'.
.
- " . . . '. .. .
Fig. 8.1 : Outline of a typical Gang-saw machiDe
Piston
mp," "'.,
'"
,-, ,
0,
..-
'..
"""
"
'-v COUnt,,-...~ttt
->
z Fig. 8.2 : Crank mechanism
L
1RUP...&4i.
GeneJ!.al,P,inciples- of Machine
Foundation
341
Design
,,' "
UppuSl~,de., blO'ck
-
Logteed
,E In
c: 0 ,
'
l.
Saw 'blad~s
lit
c: ., E
Lower
slide block
'0
- -'-0'1' u
c.onnecting
Q. >- c.ounter ~ w~ight
-.
. .'" .
':: :
rod
Fly whul
foundation block
..-
~
. "'.
-
- '" ,..',".
Fig, 8.1 : Outline of a typical Gang-saw machiDe
Piston
mp," "'.,
".
,', "
0,
..
"
'-'J' ..
"-v Count~..
,~'
z
Fig, 8,2 : Crank mechanism .".'>i ~,"C L
.,
'7~t
342
,-"
.~oil Dy"ami~. -& ~a.c~i"e F,°'!"d.f!..tiO/,'s "~
In order to simply the analysis of the motion of the connecting rod, the mass mr is replaced by two equivalent masses; one rotating with the crank pin A, the other translating with the wrist pin B. The inertia forces can then b~ e~pre~sed in.t(:OI1sof thtHotal rotating 'mass (mrot)and the total reciprocating mass (mrec)'The total rotating'mass is a'ssumedto be con~entrated at the crank pin A. r2 L2' r;'
m
=,--'-me+ - mr~- mw
. rot. - "i . .' '
' -L
==
mrec
+~
LI
mp
...(8.2)
L mr
I
' C .'
F: (m rot:+: mrec ),,~1~2c:,os "'t,
Jt
...(8.1)
,ri'
'
r. 2CJ:) 2
F : mre c: 1
L
cos 2 u.J t
F y : mro t r1",2sin u:>t
",
'. ~
.-
';~
0./
0
1r
2'1r
'
..
00'
31T -
L.1T .'
"
Fig. 8.3: Variation ofinertiaJorces with time
,:,,;.'
. ' ...\
,.'
,. ;; ~:!'~
'" -
r!ral Principles of Machine Foundation
343
Design
The inertia force (Fz) in the z direction may be shown to be F~
2
2 '~ mrec) r} 0> cos (0 t + mrec
L
= (mrot +
,
'
2 (0 cos 2 0> t,
'
" ..,.(S.3)
lchhas a primary component (F1 acting at the frequency of rotation, and a secondary component (F") [ng at twice the rotation frequency. Fz = Ft + F" ...(S.4) And in the y direction
.
2
. ,
,,':-
-,,'
,
""'-' I
Fy = mrot . rl 0> sm 0>t .,.(S.5) The time variations of these inertia forces are illustrated in Fig. S.3. If. the rotating mass is balanced, the inertia force in the y direction disappears and that. in the z '
,,',',
'
ectlon becomes
"
'
-, '1
-,
.'
,
,-
F; = mree r 1 ,.,2 VJ
(
COS Cl) t
+ rl cos 20) t L'
..'
-
"
',:
-." '
"
0'
"
'-,"
"'
)
..,
-
"
( S, 6)
The amplitude of the primary (F:nax)and secondary (F;ax) inertia forces are then relat~da~J.ollQWs' F"
max
r, = ~ Ft L max
.
'; ..:(8,7f
The preceding development relates to a single cylinder machine, which possesses unba~ancedpriary and secondary forces. As more cylinders are added the unbalanced forces and couples are modified shown in Table 8.1 (Newcomb, 1951). With a six cylinders machine complete balance is achieved. Different crank arrangements pertaining tot,able S.l are shown in Fig. S.4.
~
x
In-tincz
~
cytindczr
Qpposczd
lM
cytindczr
(b)
(a)
y
(c)
~ (d)
1~
h1u(I)
(e)
+~
0
Cranks at 180
C ran ks at 90
A
0
(g)
J:m,1;t (h)
Fig. 8.4 Different crank arrangements: (a) Single cr;ank(b) Two cranks at 180°(c) Two cranks at 90° (d) Two cylinders at 90° on one crank (e)T,io'op~s;d ~yllnders ~~one crank (g) Four cylinders (h) Six cylinders -', .".. ...:\ , .! 1 ~..:. :_--~,-~
.' ,-"
.", ,,-.. -",!}",'1-..t"'.'"."'l't';.
-
"""""~>""">"';";""';""'.
"
"i"
:C
:If' j
,
344
Soil Dynamics
& Mac/,ine
Foundations ,~, " ,.
Reciprocating machines are very frequently encountered in practice. Usually the following two types of foundations are used for such machines: ," ; (a) Block type foundation consisting of a pedestal of concrete on which the machine rests (Fig. 8.5). ; "Cb) Box or Caisson type foundation consisting a hollow concrete block supporting the machinery on its top (Fig. 8.6). " ;
':
,
',..., .
Fig. 8.S : Block type foundation
Fig. 8.6 : Box type foundation
8.2.2. Impact Machines. These include machines like forging hammers, punch presses, and stamping machines which produce impact loads. Forge hamIners are divided into two groups: drop hammers for die stamping and forge hammers proper. These machines consist of falling ram, an anvil, and a frame (Fig. 8.7). The speeds of operation usually range from 50 to 150 blows per minute. The dynamic load~ attain a peak in a very short interval and then practically die out. '
Anvil
Fig. 8.7 :Dro'p biu~~er wit" frame mounted-on anvil
---
:etll!rilf'Principles 'ofMddtitte'Ft1tIiidation
'345
Design
:able' 8.1 : Unbalanced Forc:esalid Couples for Different CFank Arrangements (Newcomb, 1951) Forces --------------------------,
Crack arrangements
Couples
'
(Fig. 8.4)
Primary
Secondary
l. --Single 'crank
..F' without counter wts. (0.5) F' with counter wts. 7. Two cranks at 1800 0
2F"
Opposed cylinders
0
0
... Two cranks at 900 (1.41) F' without counter wts.
0
(0.707) F' with counter wts.
, :-Secondary
0
:0
F' 0 without cou~terwts.' "
0
{ (0.5) F' 0 with counterwts.
:0 0
0
(1.41) F' 0 without counter wts. E" 0 (0.707) F' 0 with counter wts.
F' without counter wts.
1. Two cylinders on
,Primary
Fit
In-line cylinders
.
1.41 F"
0
0
0
0
0 with counter wts.
, one crank, cylinders at 90°
e. Two cylinders on
2 F' without counter wts.
-, . 0
F' with counter wts.
one crank, ,'opposed cylinders
r
0
Three cranks at 1200
0
(3.46) F' 0 without counterwts. (3.46) F 0 11
(1.73) F' 0 without counterwts.
g. Four Cylinders Cranks at 1800
,-
Cranks at 90°
0
0
0
0
0
0
(1.41) F'O without counterwts.
'(4.0) F" 0
(0.707) F' 0 with counterwts.
0
h. Six cylinders
0
0
0
F' = primary force; F" = secondary force; 0 = cylinder-centre distance Impact machines may also be mounted on block foundations, but their details would be quite different from those of reciprocating machines.
8.2.3. Rotary -Machines. These include high speed machines such as turbogenerators, turbines, and rotary compressors which operate at frequencies of the order of 3000 rpm to 10000 rpm. Associated with these machines there maybe a consider~ble amount of auxiliary equipment such as condensers, coolers and pumps witl1connectingpipework and ducting. To accomodate theseauxiljary 'equipments a common foundation arrangement is a two storey frame structure with the turbine located on the upper slab and 'the auxiliary equipment placed beneath, the upper slab being flush with the floot .level of machine ?all (Fig. 8.8). ~
,0,
.'
~--""
,.__'0'
i +.tt~JJ\;
'0'-.
eo
"-'
,
+
._-
-,", -"~, . ,> r+,.I,~I':"',"
0.)
,
'...
0"
0',.
'0'" ..
0./ '
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-
-
Co<
.-.
.~.. ~
346
SoU Dynamics & Machine Foundatiolls
Turbin~
G ~ n ~ ra tor
. "....
Floor
L1m~
==:J
c::.
Upper stab
u ':
-: ::.
--, ,.
':-:';.'~':: ..
- '0"' ",
. .:'
. :.':"'"
'-,.
Base stab Fig. 8.8 : Concrete frame turbogenerator foundation
Rotating machinery is balanced before erection. However, in actual operation some inbalance always exists. It means that the axis of rotation lies at certain eccentricity with respect to principal axis of inertia of the whole unit. Although the amount of eccentricity is small in rotary machines the unbalanced force may be large due to their high speed. Figure 8.9 a shows a typical rotating mass type oscillator in which a single mass me is placed on a rotating shaft at an eccentricity e from axis of rotation. The unbalanced forces produced by such a system in vertical and horizontal directions are given by Fy
= me e ol sin rot
FH
= me e cos rot
...(8.8 a) ...(8.8 b)
...
3 c
In UN /
F::me~w2
'
F( , f .)F /
(b) Double shaft
(a) Single shaft
Fig. 8.9 : Rotating mass type oscillator
F ::
me
~(A)l
""
347
meral Principles of Machine Follndation Design
Figure 8.9b shows two equal masses mounted on two parallel shafts at the same eccentricity, the 1aft rotating in opposite directions with the same angular velocity. Such an arrangement produces an ;;cillating force with a controlled direction. For the arrangement shown in Fig. 8.9b, horizontal force )mponents cancel and the vertical components are added to give ...(8.9)
F = 2 me e 002 sin 00t .3 GENERAL REQUIREMENTS OF MACHINE FOUNDATIONS or the satisfactory design of a machine foundation, the following requirements are met:
1. The combined centre of gravity of the machine and foundation should as far as possible be in the same vertical line as the centre of gravity of the base plane. 2. The foundation should be safe against shear failure. .
3. The settlement and tilt of the foundation should be within permissible limits. 4. No resonance should occur; that is the natural frequency of the machine-foundation-soil system should not coincide with the operating frequency of the machine. Generally, a zone of resonance is defined and the natural frequency of the system should lie outside this zone. If 00represents the operating frequency of the machine and oonas the natural frequency of the system, then
(a) In reciprocating machines (IS: 2974 pt 1-1982) For important machines: 0.5 >
~
> 2.0
(On
For ordinary machines:
0.6 > -
"
(0
> 1.5
(On
(b) In impact machines (IS: 2974 Pt 11-1980) (0
0.4 > -
> 1.5
(On (c) In rotary machines (IS: 2974 Pt 111-1992) (0
0.8 > ;- n > 1.25 It may be noted that where natural frequency of system oonis below the operating frequency of machine 00,the amplitudes during the transient resonance should be considered. For low speed machines, the natural frequency should be high, and vice versa. When natural frequency is lower than the operating speed, the foundation is said to be low tuned or under tuned, when the natural frequency is higher than the operating speed, it is high tuned or over tuned. 5. The amplitude of motion at operating frequencies should not exceed the permissible amplitude. In no case the permissible amplitude should exceed the limiting ampli~~e of the machine which is, prescribed by the manufacturer. . . . . 6. 'The vibra'tiOli~must not be amloying to the persons working in the fact~iy or be damaging to other precis;.on machines. Th~ '~ahiie 'of vibrations that are perceptible, -annoying, or harmful depends on the frequency of the vibrations and the amplitudes of motion. Ri~rt (1962) devel-
ùæ{æ
,.c.
~-~~":"-.:,.':;;;:';';;"';":,:";::,L_:~,_-':"'".:1:L'-'~U:t
"'"
" '";,,,
."",
;
,"""
"
348
Soil Dy",a,mics &. Machine. FollndationS"
oped a plot for vibrations (Fig, 8,10) that gives various limits of frequency and amplitude for different purposes, In this figure, the envelop described by the shaded line indicates only a limit for safety and not a limit for satisfactory operation of machines,
0
0
Disp la c.emen t ampl i tude AI mm 0 0 ,0 0 . 0 0
.....g 0
0
0
0.
0 0
0
N
0
0
U1
.....
0
0, N
0
.
0 0
0.....
U1
0
.
'-/
0 N
0
A
'"
0
-
o.
"o 0
r
') ,o
'1»-oCO
,
0.<> 'b" 0
tJ1
-0'1> 0; "D ""-
q.;>0 )( ..... 0 0 0
0
.0
,,0 ')',
)"D
"'rig ""I ~
.D C ~ ::J n '< '" n
0"
\--':'
0"
'" )"D
,'" \--,' 0
", 1")' "D &>
"D"""
"
,>0
0
0"
)"D
N
" 0 3 00
0'" ",Y>
V'1 0 0 0 ..... ..0 0 0 0
Fig, 8,10 : Limiting amplitudes ofvibratlon
for a particular
frequency (Richan, 1962)
8.4 PERMISSIBLE AMPLITUDE For the design of machine foundation, the values of permi~sible amplitudes suggested by Bureau Indian St~ndards for the foundations 'of different typ~s of~chines.are given:"", in Table 8,_2,' , ,t, -'i' "," -,. "',;', ",. ',"",','" ,x/A
(~,N~
;:
:~
"
,.,; ',"
'0 . - .
,.
.'
-, ., ..", .
349
eral Principles of Machine Foundation Design Table 8.2: Values of Permissible Amplitudes for Foundations of Different Machines ,
No.
I. 2,
3.
.
Type of machine
Reciprocating machines Hammer. (a) For foundation block (b) For anvil Rotary machines (a) Speed < 1500 rpm (b) Speed 1500 to 3000 rpm
(c) speed> 3000 rpm
.
Permissible amplitude mm
Reference
0.2
IS: 2974 (Pt - I)
1.0 to 2.0 1.0 to 3,0
IS: 2974 (Pt - 11)
0.2 0.4 to 0.6 Vertical vibration 0.7 to 0.9 Horizontal vibration 0.2 to 0.3 Vertical vibration 0.4 to 0.5 Horizontal vibration
IS: 2974 (Pt
- IV)
IS: 2974 (Pt
-III): 1992
Permissible amplitude dependents on the weight of tup, lower value for 10 kN tup and higher value for the tup weight equal to' 30 kN or higher. ..
,5 ALLOW ABLE SOIL PRESSURE he allowable soil pressure should be evaluated by adequate sub-soil exploration and testing hi accormce with IS: 1904-1978, The soil stress below the foundation,s,hallnot exceed 80 percent of the allow)le soil pressure. When seismic forces are considered, the a1f6wablesoil pressure may be increased as ~ecified in IS: 1893-1978. .6 PERMISSIBLE STRESSES OF CONCRETE AND STEEL or the construction ofthe foundation of a machine MI5 or M20 or M25 concrete in accordance with IS: .56-1978 shall be used. The allowable stresses of concrete and steel shall be reduced to 40 percent for oncrete and 55% for steel, if the detailed design of foundation and components is limited to static load )f foundati'Jn and machine. Considering temperature and all other loadings together, these assumed ,tresses may be exceeded by 33.5 percent. Alternatively, full value of stresses for concrete and steel as ;pecified in IS: 456-1978 may be used if dynamic loads are separately considered in detailed design by lpplying suitable creep and fatigue factors. The following dynamic moduli of concrete may be used in the design: Grade of Concrete
Dynamic elastic modulus (kN/m2)
MI5
2.5 x 107
M20
3.0 x 107
M25
3.4 x 107
M30
3.7 x 107
"',
.....
'--~"'--"'-
"'-"""-'-""--'---'-'
'--'
"""'-"
'-"'-
'-,..""
(~~.
-~ SoU Dynamics & Machine Foundatioll$
350
'"-.
8.7 PERMISSffiLE
STRESSES OF TIMBER
The timber is generally used under the anvil of hammer foundations. Grade of timber is specified accord--\'> ing to the size of defects like knots, checks etc. in the timbe~ Timber is thus classified into three grades." Select, Grade I and Grade H. The best quality timber having minimum or no defects at all is of the select ~rade. ?rade I timber is one hav~g. defects not larger than the ~peci~ed ones. Grade H t~mberi.spoore~ In quahty than grade I. The permissible values of stresses are given IDTable 8.3 for species of timber of grade I. In machIne'foundations timber of select grade is used. The permissible stresses of timber given in Table 8.3 may be multiplied by 1.16 to get the permissible stress of timber of select grade. t Table 8.3 - Minimum PermissibleStress Limits (N/mm2)in Three Groups of Struc,turalTimbers (For Grade I Material) S.No.
(i)
(ii)
Strength Character
Bending and tension along grain
Location of Use
Inside(2)
Shear(I)
Group A
Group B
Group C
18.0
12.0
8.5
1.05
0.64
0.49
Horizontal
All locations
Along grain
All locations
(iii)
Compression parallel to grain
Inside(2)
11.1
1.8
4.9
(iv)
Compression perpendicular to grain
Inside(2)
4.0
2.5
1.1
(v)
Modulus of elasticity (x 103 N/mm2)
All locations and grade
12.6
9.8
5.6
(I) The values of horizontal shear to be used only for beams. In all other cases shear along grain to be used": (2) For working stresses for other locations of use, that is out side and wet, generally
factors of 5/6 and
2/3 are applied
~
The permissible bearing pressures on other elastic materials such as felt, cork and rubber are gener-.J ally given by the manufactureres of these materials. No specific values are recommended here since theY4 vary in wide limits. . , 1 ...
.. . ", "'~ } - . .I; ..
~~J ""'1 <-J
i
-
'
- ,
ó
ùþôòþôæôùôùþù¬ùô¢åþùæùâþôæùåôôôôùôòþùô
òþôþùþ
ò
îÿåô
Õù
FOUNDATIONS OF RECIPROCATING MACHINES
..
9.1 GENERAL Reciprocating machines are common in use. Steam engines, internal combustion engines (e.g. diesel, and gas engines), pumps and compressors fall in this category of machines. Block type or box type foundations are used for reciprocating machines. For the satisfactory performance of the machine-foundation system, the requirements given in sec. 8.3 should be fulfilled. For this, one has to obtain (i) the natural frequency of the system, and (ii) the amplitude of foundation during machine operation. In this chapter, methods have been presented to obtain these two parameters in different modes of vibration. The basic assumptions made in the analyses are: (i) the foundation block is considered to have only interial properties and to lack elastic properties, and (ii) the soil is considered to have only elastic properties and to lack properties of interia. Design steps and illustrative examples are given at the end of the chapter. 9.2 MODES OF VIBRATION OF A RIGID FOUNDATION BLOCK A rigid block has, in general, six degrees of freedom. Three of them are translations along the three principal axes and the other three are rotations about the three axes. Thus, under the action of unbalanced forces, the rigid block may undergo .vibrations as follows (fig. 9.1) : 1. Translation along Z axis
- Vertical
2. Translation along Y axis
-
vibration
Longitudinal or sliding vibration
3. Translation along X axis - Lateral or sliding vibration 4. Rotation about Z axis - Yawing motion 5. Rotation about Y axis - Rocking vibration 6. Rotation about X axis
-
Pitching or rocking vibration
The vibratory modes may be 'decoupled' or 'coupled'. Ofthe six modes, translation along Z axis and rotation around the Z axis can occur independently ot any other motion and are called decoupled modes. However, translation along the X or Y axis and the corresponding rotation about the Y or X axis, respectively, always occur together and are called coJlpled modes. Therefore the dynamic analysis of a block foundation should be carried out for the following cases: (i) Uncoupled translatory motion along Z axis i.e. vertical vibration. (ii) Coupled sliding and rocking motion of the foundation in X- Z and Y - Z planes passing through
the common centre of gravity of machine and foundation. (iii) Uncoupled twisting motion about Z axis.
353
oundatiQns of .Recip~oCQtingM achines
A rigid block being ~ problem of six degrees of freedom has six natural frequencies. The natural requency is determined in a particular mode (decoupled or coupled) and compared with the operating requency. Similarly, amplitude is worked out in a particular mode and compared with the permissible
~.
'
vczrtical
Rocking
~--',
~
+X
Latczral /
Fig. 9.1 : Modes ofvibration
"
oh rigid block foundation
9.3 METHODS OF ANALYSIS The following two methods are commonly used for analysing a machine foundation: (i) Linear .elastic weightless spring method (Barkan, 1962) (ii) Elastic half
- space method (Richart, 1962)
In the first method which is proposed by Barkan (1962), soil is replaced by elastic springs. In developing the analysis the ,effects of damping andpartidpating soil 'mass are neglected. Damping does not affect the natural.frequency of the-system appreciably, but. it affects resonant amplitudes considerably:
Since the zone of" resonance is avoided in designing !I1achinefoundations,the effect of dampingon amplitudes computed at operating frequency is also $mall. Hence neglecting, damping may not affect the design apPI:eciably,and if any that on the conservative side. Empirical methods have been suggested to obtain the soil ma~s ,participating'in vibration. '
.
,
In the elastic half-space method,;:,!he"machine:foundation is idealised as a vibrating mechanical oscillator with.a circular' base resting ~Il,the~surface;ofgroumLThe .ground is assumed to bean elastic, homogeneous, isotropicf;semi~infiriit~ody~~whichis referredto.asanelastic half-space. This approach is apparently more rational, but relatively more complicated.
'-"',' .,"'.
'
.
>i
354
Soil Dynamics & Machine Foundations
In the above two methods, the effect of side soil resistance is not considered that is the foundation is assumed to rest on the ground surface. 9.4 LINEAR ELASTIC WEIGHTLESS SPRING METHOD Barkan (1962) has given the analysis of block foundation in following modes of vibration: (i) Vertical vibration (ii) Pure sliding vibration (iii) Pure rocking vibration (iv) Coupled sliding and rocking vibration (v) Yawing motion Machinq;
Fou ndation Ground
L
Of
-L Soil Fig. 9.2: Block foundation
Let us consider a block foundation of base contact area A placed at a depth Dfbelow the ground level (Fig. 9.2). Neglectingthe effect of side soil resistance and considering soil as weightless elastic material, the machine
- foundation
soil system can be idealised to mass-spring system shown in Fig. 9.3 a to 9.3 e
for different modes of vibration. Barkan (1962) had introduced the following soil parameters which yield the spring stiffnesses of soil in various modes: (a) Coefficient of elastic uniform compression (C,) : It is defined as the ratio of compressive stress applied to a rigid foundation bl~ck to the 'elastic' part of the settlement induced consequently. Thus Cu =!!.L (Fig. 9.3 a) Sez It is used in vertical vibration mode. From defmition, spring constant Kz K= z
Load
.
elastic deformation
.-
qz A =-=C.A Sez
u
...(9.2)
...(9.3 )
355
""datioilS of ReCiprocating Machines
t Fz(t)
.1 Fz
-- --------Fz
m
K.Z = Se2
L--
--
--
J
~~. f
S
ez
(a) Vertical vibration
.., I I I I I
Fx (t)
. .
I
I I
I I I I I I
.
I I
, ,.
I -"'7jN...-
.... I
I I I I
I I I I I
'--.... 'U:{f
,
-+I~'Sex I
I I
..
.
I
I
--.
m
---- --My(t)
. ..
Fx
Fx (t),
KX = Sex
.'
Kx (b) Pure sliding vibration
---""""
My(t)
-'"7 I I I
f:\
My
F:)
I I
K,=
I I
-""-.... rp'
K~ (c) Pure rocking vibration Fig. 9.3 : Types of motion of a rigid foundation (...Contd.)
T
356
, ';Soil"'vDymnnics-'>& ,
'i = Angle
nMZ{t)
,
.
~Mfld,ine
'-Fl1wuJtltions
ot
torston ot
~MZ{t)
toun'da tier. (not shown)
/
/
(d) Yawing vibration
Fd (t)
I I
,,
I
Fd (t)
.
... ......
....
, , ,
/
,
i., .... '-.~"'-.
,
r, a....
.... ...
....
, "
,,I ,, I '...
tSex
-i 'r' ~~
"...,
-
I
'.
,"
..
'Kx
Kx
>."'"
K~ (e) Coupled slidding and rocking vibration
'. Fig~'9.3': Types of motion of a rigid foundation
(b) Coefficient of elastic uniform shear (Cr): It is defined as the ratio of the average shear stress a the foundation contact area to the elastic part of the sliding movement of the foundation. q Ct = Sex x (Fig. 9.3 b)
...{9.4
It is used in analysing sliding vibration mode. The spring constant Kx is given by . K = Shear load = qx .A = C . A x t Sex
Sex
" ... (9 ".
(c) Coefficient of elastic non-uniform Compression (C~ : It is used in rocking vibratic (Fig. 9.3 c). In this case the elastic settlement of the block is not uniform over the base. It defined as the ratio of intensity of pressure at certain location from the centre of the base of bloc to the corresponding elastic settlement. If cpisthe angle of rotation of block, then at a distaDI from the centre of the base of block, the elastic-deformation will be I cp.Taking the intensity , pressure at this location as q, C. is given by q C~ =[cp
...(9.
J57
F oLlm/lIlilJl;,. (~r J:ecfproclItillg Machines
The sllj},:f.::';sKIj>is'defil1E".1as the motnent per unit rotation,a~d M K =-=C .1
where,
is given by
...(9.7)
cp
J = Moment of irf'rtia of the base of block about the axis of rotation M = Moment caused due to soil reaction
(d) Coefficient of elastic non-uniform shear (CljIJ: It is used in yawing motion. If a foundation is acted upon by a moment with respect to vertical axis, it will rotate about this axis (Fig. 9.3 d). Tests have shown that the angle of rotation \j/ of the block is proportional to the external moment. Therefore, M z =K 'I' \j/ where K =C .J 'I' 'I' z J- = Polar moment of the intertia of contact base area of foundation.
...(9.8) ...(9.9)
In the rotation of a' foundation around a vertical axis, the base of the foundation undergoes nonllniform sliding, hence the term "Coefficient of elastic nonuniform shear". is applied to the Coefficient C IjI.
Barkan (1962) derived the £q. (9.10) for determining -. the value of CU . It is based on theory of elasticity. . 1.13£
1 ...(9.10)
Cu=I-~2'JA where,
£ = Young's modulus of soil ~ = Poisson's ratio A = Area of base of the foundation
,
.'
He also developed the relationships between Cu' Ccp'Ct and C'I" For analysis and design of machine foundation, he recommended that
Cu=2Ct
...(9.11)
Ccp=.2Cu. ...(9.12) Ct = 1.5 C'I' ...(9.13) For preliminary design, Barkan (1962) recommended the values of ClIas listed in Table 9.1. The procedure of determining th.evalues of cu' Ct £ and G have been given in detail in sec 4.3 of chapter 4. As discussed in that section; dynamic elastic' constants depend on (i) base area of foundation, (ii) confining pressure and (iii) strain level. The method of converting the value of adynamic elastic constant obtained from a field test for using in the de~ign of actual foundation has been illustrated in examples 4. 2 and 4.3.
j
. .,'... ,..:-: "
, U, !' " .
.~,~. i;.'. -I
',. ~
c :".; r . ,t. "': . n.!,
~.
358
Soil Dynamics & Machine Foundations
Table 9.1 : Recommended Design Values for the Coeficient of Elastic Uniform Compression Cu 2* for A = 10 m Soil
Permissible static load,
Soil group
Cu' kN/m
3
kN/m2 4
3
2
up to 150
4 up to 3 x 10
150 to 350
(3 - 5) x 104
350 to 500
(5 - 10) x 104
Weak soils (clays and silty clays with sand in plastic state; clayey and silty sands; also soils of categories II and III with laminae of oraganic silt and of peat) 11
Soil of medium strength (clays and silty clays with sand close to the plastic limit; sand)
III
Strong soils (clays and silty clays with sand of hard con-
IV
sistency; gravels and gravelly sands; loess and loessial soils) Rocks
> 10 x 104
> 500
* After Barkan (1962) 9.4.1. Vertical Vibrations. For the purpose of analysis, the machine -foundaton-soil system shown in Fig. 9.3a is represented by the idealised mass-spring system shown,in fig 9.4. Let the unabalanced force is representd by (91.4) Fz.(t) = Fz sin rot
³
³
ã
Ó¿-- of machine plus
Kz
=
foundation
Cu.A
Fig: 9.4 : Equivalent model for vertical vibration
If the centre of gravity of the foundation and machine and the centroid of the base area of the foun dation in contact with the soil lie on a vertical line that coincides with the line of action of the excitint force Fz, then foundation will vibrate vertically only. The equation of motion of the system is
where,
mz + Kz' Z = Fzsin IDt' M = Mass of machine and foundation
(9.15
Kz = Equivalent spring constant of the soil in vertical direction for base area A of the foUl dation = CuA
.,
359
,tidaiiiJns of Reciprocating' MaCih;n~ ;.
= Coefficient of elas~c uniform compression.
Cu
.
,
Therefore, the natural frequency IDnz of the system is (9.16)
ID.z = ~~z =r~
A.
The amplitude .of motion Az is given by .
or
. Fz sin IDt Fz sin 0)t A = = 2 Z . K -m0)2 C A-mO) z Il A
=
Z
(9.17a)
Fz sin 0)t 2
m ( 0) nz -
0)
2
'
(9.17b)
)
Maximum amplitude of motion Az is given by Az .
Fz
2
m (Ct>nz.-ro
.'
(9.18)
2
)
.
.
3.2. Sliding Vibrations of a Block. In practice, rocking and sliding occur simultneously. But if the bration in rocking can be neglected ,then only horizontal displcement of the foundation would occur tderan excitingforce Fx(t) on the block of area A (Fig.9.3b). This system can be indealised as shown Fig.9.5.
-,. I I I I , I I I I
m
I
~x ~/ Fig. 9.5:
,\..
(a)
Block foundation in pure sliding vibration
(b)
Equivalent model
The equation of motion of the system is'
mx + K.~ = Fx sin rot vhere,
(9.19)
x = Sliding displacement of,the foundation Kx
= Equivalent spring c~nstap.t of the soil in sliding for base area A of the foundation = Ct. A,
Ct
= Coefficient of elastic uniform shear.
'",
:
is
Therefore, the natural frequency Ct>nxof the system :
'
~;',~ nx"
fKX; =' rc;A f;; v-;;-
: (9.20)
q
Maximum amplitude of motion . A x is given by .
'~"-'..'>;:).,:' ",
,
,.1,
""":~:':,'-:I:A ..
"
l_~I"'-<'Fx":. x'~'~~;x'_~2)"""'J"_:
.,.;
,:~f':,
,. '."
",
....(9)J) . . .. ",
,-,
360
SoU Dynamii;s ~,Mac;4,ine, Found~tiof}s.
9.4.3. Pure Rockmg Vibrations of a Block. Consider only the rock~ngy~bri,J,tions in~uced in a foundation block by an externally exciting moment Mv(t) (FIg. 9.3c). this "isalso a hypothetical case as rocking vibrations are coupled with sliding vibrations. Let the unbalanced moment be given by My (t)
(9.22)
= Mysin IDt --
where, My = Moment acting in the X - Z place' At any time t, considering that the applied moment is actIng in clockwise -direction the"displaced position of the block will be as shown in Fig. 9.6. In ~achine foundations, as the rotation
radians. The equation of motion can be obtained by applying Newton's second law of motion. My (t) = My Sin GJt
~'
( '-,-
"
...
-----x
~ ( a)
qst I dA
Elem
Fig. 9.6 : Block foundation under pure rocking vibrations
The various moments acting on the foundation about the centre of rotation are obtained as descri below: (i) Moment MRdue to soil reaction: Consider an element dA of the foundation area in contact' the soil and located ar ..: :..tance / from the axis of rotati~n (Fig. 9.6 b). At any time, the soil wil
ndations . . ' '" of Reciprocating 1>."
npressed nonuniformly.
Machines -",
"
361 .
From t~e de~n~t~on of ~oef5cient of dastic un}form compression, ,C~ ~s d R I dA C~ =
.ere,
14>
...(9.23 a)
d R = Soil reaction force acting on dement dA cl>
= Angle of rotation
I~~hefoundation does not lose contact with soil, then the soil reaction will be as shown in Fig. 9.6 b. le total reactive moment MR against the foundation area in contact with soil is given by ,
A
...(9.24 )
= J C~.l~dA.l ==C~.cI>JPdA = C~ I
MR
here,
'
<\>
I = Moment of inertia of tpe foundation area in contact with the soil with respect to the axis of rotation. .
This moment acts is the anticlockwise direction.
'
(ii) Moment Mw due to .the displaced position of centre of gravity of the block: As shown in ig. 9.6 a, the centre of gravity of the block is shifted from point 0 to 0' . As angle of rotation <\> is small, :le moment Mw of Weight W will be Mw =WLti\
'
.
...(9.25 )
't'
vhere, L = Distance between the centre of gravity of block and axis of rotation. This moment acts in the clockwise direction. (iii) Moment Mj caused by intertia of foundation: It is given by .vhere,
Mj
= Mmo ~
,
"
.
...(9.26)
.
Mmo =; Moment of inertia of the'mass of the foundation and machine \"":"ith respect of axis of rotation. '
This moment acts in the anticlockwise direction.
.
The equation of motion can be written by equating clockwise moments to anticlockwise moments Therefore, or
,
My sin' (tJt + WL cl>=
c cl>14> +
Mmo
~
...(9.27)
Mmo 4> + (C~ 1- WL) <\>= My sin Cl)t
The natural frequency wn«/> of this system is given by ...(9.28)
wn4>_,t.lM mo WL and maximum displacement A . . . ' ~ i~ given, by . '. .
A
=
. ,
My 2
2 . '.
l
...(9.29)
".
~. Mmo(ron,~ro) "",
.'
.' .
,--"
.'
-- .-'-
362
Soil Dynamics. &Mac1iin~
Foundatiolt/; '.0;, 'l
In practice, C. I is many times WL ; hence Eq. (9.28) may be written:'
~
.I
con.=I Mmo
...(9.30) ,;~
If the dimensions of the footing at the base are a and b in the X andY direction~, respectively, ba3 1=, ...(9.31) 12
,
-.
Cell ba3
conI/!
= "MU /nO
...(9.32)
Y
It is seen from Eq. (9.32) that the linear dimension of the contact area perpendicular to the axis of rotation exercises a considerably greater effect on the natural frequency of rocking vibrations than the other dimension. This principle is sometimes used in proportioning the sides of the machine foundation undergoing predominantly rocking yibrations. ,
'
The amplitude of the vertical motipn of the edge of the footing is Azr = ~xA, Mya/2 = Mmo(O)~'- 0)2) Similarly, the contribution of rocking, towards the horizontal amplitude is Axr = h. A, where, h = Height of the point above the base where amplitude is to be determined.
...(9.33'
...(9.34
Azr and Axr are added to Az (Eq. 9.18) and Ax (Eq.9.21) respectively to obtain total vertical an sliding amplitudes when rocking is combined with vertical and sliding vibrations. 9.4.4. Yawing Vibrations of a Block. A foundation is subjected to yawing motion if it is subjected to torsional moment Mz (t) about Z-axis (Fig. 9.7a). The positionof the foundationat any time t mayt defined in terms of angle of rotation "'. Let the unbalanced moment is given by ...(9.3
Mz(t) = Mzsin (J)t As explained in Sec. 9.4, the resistive moment due to soil is C'II'Jz "'. The equation of motion is written by taking moment about Z- axis. It gives Mmz iV
...(9.:
+ C'IIJ z '" = Mz sin rot ,
where,
Mmz
~~
= Mass moment of inertia of the machine and foundation about the axis of rotat (Z-axis)
Jz = Polar moment of inertia of foundation contact area C1jf = Coefficientof elasticnonu~iformshear
t
,. '4 ,~1 '.."i .-
óóóòóóó
Foundations
of Reciprocating
363
Machines
'.
z
x
v
(a)
Isom~teric
vi
,
( b)
,
"
.'
pia n
Fig. 9.7: (a) Yawing motion ora rigid block
.
(b) Development of non-uniform shear below the base
The' expressions' for natural, frequency an{ma~imux:n angular displacements are as follows: ...{9,37)
(J)nl/f-
Mmz - tw J,
,A
Mz 2 = M 'I' mz( IDnljl.
.
2 CD
)
The horizontal displacement AhlJlc~used, by .t,orsion is
where,
Ah'l' = r A'I' ...(9.39) r = Horizontal distance of the point on the foundation from the axis of motion (Z-axis)
'9.4.5. Simultaneous'Vertical, Sliding and Rocking Vibrations. In general, a machine -foundation is subjected to time dependent vertical force, horizontal force and moment, and therefore it simultaneously slide, rock and vibrate vertically. In Fig. ~.8, a foundationblock subjected to a vertical force (Fzsin 00I),
....
,~
i: ~\ 364
SoU-Dynamics
& Machine
Foundations
a horizontal force (Fx sin (J) t) and an oscillatory moment (My sin (J)t) is shown. These forces and moment are considered to act at the combined centre of gravity 0 of the machine and the foundation, which is alsotaken as the origin of coordinates. At any time t, considering the vertical force acting in downward direction, horizontal force in right-hand side direction, and moment in the clockwise direction, the foundation block will be displaced as shown in Fig.'9.8. It is therefore subjected to (i) displacement z, in the vertical direction (ii) displacement Xo in the horizontal direction at the base and (Ui) rotation <1> of the base.
z I
'
Fz Sin C.Jt Initial
position
x
~ z
Fig. 9.8: Block foundation subjected to simultaneous vertical, sliding and rocking vibrations
The equations of motion can - be written by evaluating the resisting and actuating - forces , and.moment acting on the foundation in the displaced position. These forces and moments are obtained as give: below: . (i) Upward soil reaction Rv due to vertical displacement z : Rv
= Cu A z
...(9.4(
(ii) Horizontal soil reaction Rx due to horizontal displacement xo: Rx
= C't
A Xo
As the origin is at 0, Xocan be expressed in terms of x and 4>as below: xo=x-Lcp where L = Height of Centre of gravity 0 from base of the block (tU) Moment MRdue to resistance of soil induced by rotation:of the foundation by $ : The MR about point 0 is given by MR-=C .'""'I
...(9.4 ..,(9.4 jJ
+ .-
.:.(90'
--" ,
'
,
'<,
','
,
","
uni/ations of Reciprocating (iv)
.
,
365
Machines
Moment Mw due to displaced position 'of the centre' of gravity of block :
The moment Mw about point 0 is given by ..(9.44)
Mw = W L
(v) Moment MxRdue to horizontal resisting force Rx : Moment MxRabout point 0 is'give~ ~y . .L MxR = Rx . L = C~ A (X - L"' ~) (vi)
...(9.45)
Interial forces and moment:
(a) In the Z - direction
Fiz
(b) In the X - direction
Fix
(c) In the rotational mode Mi$ "here'
= mi = mx
...(9.46)
= Mm ~
...(9.48)
...(9.47)
M m = Mass moment Of inertia of the machine and foundation about an axis passing through '
combined centre of gravity 0 and in the direction of Y-axis
.,
Cheequations of the motion 'can now be written as below: In the Z - direction:. mi + Cu Az = Fz sin cot
...(9.49)
In the X - Direction: mx + CtA Xo= Fx sin cot
...(9.50)
Substituting the yalue of Xo from Eq. (9.42) in Eq. (9.50), we get ...(9.51)
mx + Ct A (x --L
,
Mm + C$ I - WL - Ct A (x - L
..
or
"
= My sin cot
2
Mm - Ct A Lx + (C$ I - WL + C~ AL ) = My sin COt
Equation (9.49) contains only the terms of z, therefore the motion in Z
...(9.52)
- direction
is independent of
any other motion. The solution of this equation is already given in Eqs. 9.16 to 9.18. Equations (9.51) and (9.52) contain both x and and are interdependent. Therefore, sliding and rocking are coupled modes. A solution for simultaneous rocking and sliding vibrations is presented below.
9.4.5.1.Naturalfrequencies of coupledrocking and sliding.The systemrepresentedby Eqs. (9.51) and (9.52)
is a two-degree-of-freedom system. The solutions for natural frequencies are obtained by consid-
ering the free vibrations of the system.
Hence,
mx + C~ Ax - c.~ AL =0 ,
...(9.53)
'
and Mm ~ - Ct A Lx + (C$ I - W L + C~A L2 ) ==0 Particular solutions of these equations may be assumed as x = xI sin (con t + ex)
...(9.54) ...(9.55)
and
= <1>1sin
(cO n t
+ ex).
...(9.56)
in which x I' <1>1 and exare arbitrary constants whose values depend upon the initial conditions of motion.
,
i::~
*
I.',
366
Soil Dynamics &I,M~chine Ft!""~
By substituting Eqs. (9.55) and (9:56) into Eqs. (9.53) and (9.54) and dividing by sin (con t + a), we get, 2
"., :"
-mOOn Xl + Ct A xl - Ct A LI = 0
2 -moon)-CtAL1
or Xl (CtA
=0
...(9.57)
,
~
and -Mmoo~1+I(CtAL2+CcpI
-WL)
-C'tALxI
...(9.58)
=0 CtAL1
From Eq. (9.57),
xl =
...(9.59)
2 CtA -mOOn
By substituting the value of xI from Eq. (9.59) into Eq. (9.58), we get
...(9.60)
<1>1 [-C~ A 2 L2+ (Cc!>I - WL + Ct AL2 - Mm oo~)( CtA -m oo~)] = 0
For a nontrivial solution, <1>1 can not be zero. Hence the expression within the parentheses mu~tJ)( zero, This gives' 2
,
I ~
2 2
2
L + (C. I óÉÔõݬßÔ
-CtA
2
2
...(9.6~
óÓ³×ܲ÷øݬß-mOOn) =0
The term (On'which represents the natural frequency in combined sliding and rocking, is the onI ,unknown in Eq. (9.61), which can now be solved. Equation (9.61) may be rewritten as follows: -C~A2L2+C~A2L2+CtA(C.I
=
-WL) -CtAMmoo~-CtAL2moo~-(C.I-WL)moo~+Mm1nro:
...(9.6 By dividing by mMm and rearranging, one obtains
CA.I-WL
4
2
n
n [(
00 -00
+
'f
Mm
C A(M t
]
m
C A CA.I-WL
+mL2)
mMm
+-L-
m
]
=0
'f
(
Mmb
...(9,(
)
By definition, the quantity (Mm+ mL2) is the mass moment of inertia of the foundation and mach about an axis that passes through the centroidof the base contact area and is perpendicular to the'~1: of vibrations. This is denoted by Mmo. Thus, , ij~ 2 ,M I Mmo = Mm + mL .:;t9. '
M Further, by denoting
~
Mmo m = r where 1 > r > 0
,
,
.(1~ -.J .~. '~
,
Equation (9.63) may be rewritten as 4
r
Now,
C.I-WL
oo~
OOn--
(
Mmo
CtA +m
)
h
CtA C.I+m
WL
'
=0
"
.t't.;.(S ,
tMmo
'i 'i 'f~i ,.~ '4 ,,
CtA
-
.',.: ,c
m
=
2 00nx
"' :0' ~,".. , ,\,
~,~f:' "
",
.
C.I-:-WL: 2 M' mo '; ~ OOn,
~, ""
~
Foundations
---
..
--. 367
of !leciproC/lting Machi,!es
'2y writting the Eq. 9.66 in terms of OOn,and CJ)n~we ?et 2
,
0>
n
2 2 0>nx + 0>ncp
-
r
(
2
2 2 0>nx Cl)n+
n
r
0> +
)
=0
...(9.69)
.
The Eq. (9.69) has two positive roots, CJ)nland CJ)n2' which correspond to two natural frequencies of the system: The ro~ts of Eq. (9.69) are: .
0>2
nl,2
2
=1.2
2
'2
O>nx
+ O>nq,:t
r
2
(Onx+ O>nq,
J
[(
2
(
r
J
2
- 4(J)n."(J)ncil r
...(9.70)
]
, Equation (9.70) may be rewritten as
.
2
2
I
(J)nl.2=
2
2r
2
)+
2 2 ,2 7"~ (O>nx+(J)ncpo
[( (J)nx+(J)ncp
4
2
2
r(J)nx (J)nq,)1
...(9.71)
From the property of a quardratic equation: , .
2
2
O>nl+O>n2 2
2
-
2
...(9.72)
r 0>2
(J)nl x (J)n2
2
0>nx + 0>ncp
=~
0>2
...(9.73)
r ncp.
22-1. 22_"221/2 (J)nl-(J)n2 - r [«(J)nx+ (J)ncp) 4r(J)ncp(J)ncp]
and
It can be proved that
OOnx
...(9.74)
and 00 ncpwill always lie bet~~en limiting natural frequencies 0001and 0002'
9.4.5.2. Amplitudes of coupled rocking a,nd sliding. The amplitudes of vibration are determined in the following three cases: Case I.
If only the horizontal force Fx sin ootis acting: Eqs. (9.51) and (9.52) may be rewritten as follows: ' .
-
mx+ Ct
Ax - Ct AL~ = Fx sin
...(9.75)
(J)t
2"
Mm~+ ~(Ct AI.: + Ccp1- WL)-Ct ALx = 0 Assume that the particular solution to these equations are
x
,
~
Ax'sin
~ ::;: Acp sin
(J)
...(9.76)
t
(J) t
in which Ax and Acpare the maximum sliding and rocking amplitudes respectively. By substituting these solutions into the above equations, we get ,
.
Ax (Ct A
2
- moo) -
Ct ALA.
= Fx
-Ct A LAx + A. (Ct A L2+ C. I - WL - ~m 002),= 0 or
~
...(9.77)
...(9.78)
- (C~L2 +C.I-WL-Mmoo2) Ax .A. ...(9.79) CtA~
~~' (~
:' ,.t~
368
SDi/Dynamics '& Machine FtiuiidQiio~ «i
."
.~ ~
~~~ "
By substituting for Ax in eq. (9.77), we get 2
.,
2
2
(Ct A L + C, 1- WL-Mm 00 )(CtA-moo CtAL .
.
A~
=..
.
.
mM m
OOIlX 00 Il~
r
[ -:-':--~-By
using the relations A
~ -
2
t
[
(C",I-wt) 't'
'.
.
+00
Mm
]
4 x.~t'f";.;". t.
]
- ~ ( 2 + 2 )+ 4 00Il~ 00n.l. 00 r
.
.
't'
]
C AL t 2
mMm [ oon1 oon2 -00
,.. 2
2
4
(oon1+oon2)+00
CtAL
(9.80), we get xF x
] "
F
...(9.81)
2 2 2 2 x mMm(OOn1-OO )(OOn2 -00 ) Let,
.('. ...(9.80)
FX
Ct~L '
2
)
m +
mMm
gives in (9.72) and (9.73) into'Eq. 2
.
2 C tA(mL2+M
-00
= 2
):'j
C AL'
't' mMm
m[
. "
- . A~ -Ct ALA, - 0
"
CtA+(C",I-WL)
mM
)
2 2 2 2 mMm(OOn1 -00 )(OOn2 -00 )
2
= ~(oo )
A = ~
C AL ~
t (00)2
...(9.82)
F
...(9.83)
x
By substituting for A~ in Eq. (9.79), we get A = Ct AL2 +Cq. I -WL-Mm x ~ (002)
Case 11.
...(9.84) ,.v
If only moment My sin rot is acting: Eqs. (9.51) and,(9.52) may be rewritten as . ,
mx + Ct and
002F x
...(9.85'
Ax- Ct ALcp = 0
.. .., Mmlj>-Ij>(CtAe+C4»I-WL)-CtALx
=Mysinrot
...(9.86
By assuming solutions as for Eqs. (9.75) as (9.76), it can be shown that the following expression hold: '.
,'.
.
CAL
My t Ax = ~ (002) -
.
'.
11.
..:(9.B'; -
2
and , ,
,
A.
- Ct A-moo 2 - ~ (00.)
My
...(9.B' 'i.
.. .. .. Poundations
Case Ill.
of Reciprocating
:
- --oii
369
Machines
If both the unbalanced force Fx aJ:?dmo~ent MJLare acting, the ~plitudes
of m~tion are
determined as follows: 0
-
02
A=
0
(C'tAL
x.
0 0
0
.
A 4>-
0 -
+ CtPI-:WL-Mm(O 2 )Fx + (c.~) 0 A L.l( (0 )
My
...(9.89)
0
0
0
0
0
2
.
"
2
0
0
and
.
(CorAL)Fx+ (CorA -mm)
My
...(9.90)
oL\(ro2)
The total amplitude of the vertical and horizontal vibration are given by a .
and
where,
Ay
= Az+2A~
...(9.91)
Ah
= Ax + h ArpH__,__-
...(9.92)
h = Height of the top of the foundation above the combined center of gravity.
In foundations with two degrees of freedom, specific forms of vibrations correspond to the frequencies (0/11and (0112'These vibrat~ons are characterised by a certain interrelationship between the amplitudes Ax and A.pwhich depends on the foundation size and the soil propet:t~es,but does not depend on the initial conditions of foundation motion. ..- .. Let us examine the case when the foundation is subjected to exciting moment My only. The ratio of amplitudes Ax and A4>obtained using Eqs. (9.87) and (9.88) is given by . 2 Ax CtA L ronx L ...(9.93) p = -= 2 = 2 2 AA'I' Cor A - m ro «)nx - «) 0
,
o'
z
0
,,, ,
z
I I
---
..."\ ",/
,.
\ \
:\ \
v
..."\ \
-- ' )
..J
....
\
L
--
--- --
\ /
~
\
(
/
\
",/
V
(a)
(b) Fig. 9.9:
(a) Rocking and sliding in phase with each other (b) Rocking and sliding In opposite phl1se
The following cases are important for consideration of form of vibration: (i) If (0 «
(Onx'then p ==L . It means that the.-.foundationrotates about an axis that passes through
the centroid of the base contact area and slidi~g is absent!
~
370
v
Soil Dynamics & Machine Fou"datiol
'(U) If 00'= (On2' (O~2 being the lower limiting natural frequency, then oo~ - 00;2> 0 . It m~ans that during vibration at frequency (0 n2' when the centre of gravity deviates from the equilibrium
position, for example, the positive dir~ction of the X - axis, the rotation of the foundation will also be positive, and changes of amplitudes Ax and A, will be in phase. The form of vibration will be as shown in Fig, 9.9 a, i.e. the foundation will undergo rocking vibrations with respect to a point situated at a distance PI from,the centre of gravity of foundation, The value of PI is determined by the absolute value of expression (Eq. 9.93) if 0>n2 is substituted for O>n' (Ui) If 0> = O>nl then oo~ - 00;1< 0, P will be negative, and Ax and A~ will be out of phase, Figun
9.9 b illustrates the form of vibrations around,a point which lies higher than the centre of gravit; and at a distance P2 determined from expression (Eq. 9.93) if 0>nl is substituted for O>n' 9.5 ELASTIC HALF-SPACE METHOD
F (t)
. . ,r
;
"
.
. ' " .-4, ':::.:'~":','.
~':~~':.:::::,',::~,..': G
pe . :':'::':~"""':;"~::',<;,
G
p
P
)J
JJ
(a)
(b)
Fig. 9.10: Oscillating force on the surface of elastic half-space
9.5.1. Vertical Vibrations. Lamb (1904) studied the problem of vibration of a single oscillating fon (Vertical or horizontal, Fig, 9,10) acting at a point on the surface of an elastic half space. Reissner (193. developed the analysis for the problem of vibration of a uniformly loaded flexible circular area (Fig. 9,1 by integration of Lamb's solut~on for a point load, Based on his work; the vertical displ
F0 = Magnitude of oscillatory force 0>
= Forcing frequency, rad/s
G = Dynamic shear modulus of the medium
r0 = Radius of the footing /1' /2 = Displacement functions i=H
,.., "
,..(9.9
.
III
Fountiatitins '0/
Rlc;procatin'g"Mai:h
'",.., ,
, .j:" ,
371'
in iii
. ,-:,-;,-;
~ -: .', ::-,: -'
,-
F.- i - 0 (Z
-
__
(..;t
-
,:
'r
,
.-""
'
'-
o':! i: ;
.
.>-.
f
óТýññ¢
'-
"
",-,-.':, - .: .-, :.i??//(/4 ~,.
: . .'
Go
'0
'00'0"
p
'. -"," ,
~
~
,
l°":::"-:':
load
p(Zr
unit or(Zo
=
)J
F. (ZiG.)t
~
11"r20
z Fig.9.11: Vibrationora uniformlyloadedcircular flexiblearea
Reissner introduced following two non-dimensional tenus : (a) Mass ratio, b : It is given by ,. m W ..,(9,95) b=~=~ pro yro It describes the relation between the mass of vibrating footing and a certain mass of the elastic half-space.
(b) Dimensionless frequency, ao : It is given by a =00 r °
,
Jp = Cllr
,oVa..
Iv
...(9,96)
O,S_,
,
m = Mass of the machine and foundation
where,
= Shear wave velocity y = Unit weight of soil p = mass density of soil
Vs
Using the displacement Eq. (9.94), and solving the equation ofeguilibrium forces, Reissner obtained the following expression for the amplitude of foundation having circular base: fl2 + f22 2 2 2 2 (~-:-~ao 11),.+, (bao f2)
Az = Azn ~. ,(FofG ro) where;
,Az ,'A
"'"
zn ,
= Amplitude
of foundatiQn,.>
= Dimensionless amplitude; . """:"',""""")"':""'j"'.'",
' " i ; ,::;
' ,',
. , ,,'
:, ;,
...(9.97) : .., : ,':' i
, ,",
'.1 : i .. {J':.."
"'"",,,"
. V~lue,s,of disp~a~einentlu~ctio~fi'ai!2f2 ~~ ~o.~d dep~~dien(op. P~is~o~~sr,a~o ~..an~ ~~~~.~n~i?~~ less frequency factor a '.Their~values fo~"f1exible'"'circular' foUndation are' giveIi'iii Table 9.2:;:'.' :-'-'.,,J! 0
. '.-it';~. ,:.[
372
SoU, DJ!n.am.ics.&, M.~~hine Fou~d~tion~;~
Table 9.2 :Values of Displacement Function 'of Flexible Foundations (Bowles, 1977} Poisson's Ratio J.1
Valus of(fl)
a/ 1-0.07405 a04
0
0.318310 - 0.092841
0.25
0.238733: - 0.{)59683'a02 + 0.004163 a04
0.5
0.159155", 0.039789
a/
+ 0.002432 a04
Values of (- 12) 0
0.214474'ao'~'.
0.25
0.148594 Go-
0.5
0.104547 Go-
0.029561 'a03+ 3 0.017757 Go + 3 0.011038 Go +
0.001528 a05 0.000808 0.000444
5 Go 5 a0
The classical work of Reissner (1936) for circular loaded area was extended by Quinlan (1953) and Sung (1953) for the following three contact pressure distributions: F e;w/ (i) Rigid base (Fig. 9.12 a),
fz = ..
(ii) Uniform (Fig. 9.12 b),
I~ = ~
~-
O
2rrr0 r0
r
2
: for r::;; ro
...(9.98)
F e;w/ ...(9.99)
for r ::;;ro
2 rrro'
0
It is the same as considered by Reissner i.e. for flexible foundations with circular l>ase 2 (r2 - r2) F iwt (iii) Parabolic (9.12 c),
f = Z
4
.0
:
1t ro
0
...(9.100)
for r ::;;ro
~
~
Cl
I
,
/'
. ~-d'
(b)
(0)
Fig. 9.12: Contact pressure distribution
(c)
~ .j
under a circular foundation
In the above equations, fz is the contact pressure at a distance r measured from the centre of foun,tf dation. Equation (9.97) holds good for all the three types oJ conta9t pressure distributions with change4: val~es of/l a~d/2. The v~lues off} an4/2f~~ rig~dbas~ foundatio;nswere comp~ted by Sung (19~~)?Jt1l/ '~A the assUinpt~o!l~at the.press.ur~.di~~b~~~~ri:r~ma,in unchang'e~ with,frequency. Their values ~re ~~v , ' .
in Table 9.3.
.
.
.,'
'"
.
.
.
~~~ ...'r'~
.,;.I. ..
datio/fs o/,Reciprocating {-;
373,
M/lcMnes ,
,": ',: ',' '.. .
':
, ,'-:
"',.
i ' "
.: -. "."
.
'-',
' ", " ,
.,.
,
~';
-
0.7 poisson's
ratio)
,',
j.J = 0.25
,
0.6
O.S c:
N
,0.4
c:,I 'U ;:) +'
,,-.
a. E 0
0.3
\11 VI
-c
c:,I
0
1/1
C tI
E 0.2 b
0.1
0
~
0,
. .'
,
"
,
.
0; S
'.
a
.
0
,-'
1.S
1.0,' ..
"f..
',',
: ,-
,
"
>\'
Fig. 9.13: Plot of AZAversus ao for a rigid circular foundation subj~te~ to:constantexcitati.on,r~rce (Ric~art,I,962) , , i. .r' ',', ;, .. ,
--
,--
,~
374
Soil Dy,u,inics-&' Machilre -Fourrdations
Table 9.3 : Values of Displacement Functions for Rigid Foundations ~'.'<'
Poisson's
..'
'-~'-"'-(Bowres~"f977r-'--'-
ratio J.1.
0 0.25 0.5
0 0.25 0.5
0'0
.",-
.
V alues' of 11
2 4 0.250000 - 0.109375 ao + 0.010905 ao 0.187500 - 0.070313 a02 + 0~006131a04 0.125000 - 0.046875 a/ + 0.003581 a04 Values of -/2 3 5 0.21447 ao - 0.039416 ao + 0.002444 ao 0.14894 ao - 0.023677 a03 + 0.001294 a05 3 5 0.104547 ao- 0.014717 ao + 0.00717 ao
Figure 9.13 shows a typical plot of Azn versus a0 for various values of mass ratio b for a rigid base . circular footing subjected to a constant force excitation F0 e/(JH. A high mass ratio (greater height of footing and smaller contact radius) implies a large amplitude of vibration for a given set of conditions. Manytimes foundations are subjected to a frequency dependent excitation (Fig.2.1~). The amplitude of the external oscillating force is given by 2 F=2m em ...(9.101) 0 e where, 2 me= Total rotating mass For this condition, the amplitude of vibration Aze is given by 2 2 me e CO Aze=
Ora
2 2 I1 + 12
2
Az
or Azen
where,
Azen
...(9.102)
(l-ba~/I)2+(ba;fz)2 2
I1 + 12
= (2meero2)/Oro ~ (l-ba; 11)2+ (ba; 12)2
:..(9.103)
= Non-dimensional amplitude in frequency dependent excitation .
.
Figure 9.14 shows a typical plot of Azenversus ao for various values of mass ratio b for rigid base circular footing subjected to frequency dependent excitation. It may be noted that the curves shown in Figs.9.13 and 9 .14 are similar to the frequency-amplitude curves shown in Figs.2.13 and 2.16 resp~<::tively. Richart and Whitman ( 1967) have studied the effect of the shape of contact pressure distribution and Poisson's ratio on amplit~d'~'~freq~en~y;esponseof rigid circular footing ~~bjectedto frequency dependent excitation.Figure9.15 demonstratesthenatureof variationof Azenwith ao for three types of contact pressure distribution; i.e. uniform, rigid and parabolic. Parabolic and uniform pressure distributions produced higher displacement than a rigid base. The effect of Poisson's ratio on the variation of Azencap be seen in Fig. 9.16. The peak value of Azendecreases with the increases in the value of J.1.;but the corresponding value of ao increases with increase in J.1.. . i'
l ;.4 ",
\,
375
"daiions ut Reciprocating' MilChin~s "--"--""'-"""""""'-'-""'-""""-'-
"-'.--
'~.
0.28 poisson'5
ratio,
).J = 0.25
.'....
0.24
0.20 c
~ N
«
... ~
0.16
"0 ::::J
....
0-
E 0 III III
0.12
"
C 0
--
III C ~
E 0.08 (:)
0.04
0 -
0.5
0
°0
1.0
1.5
Fig. 9.14: Plot of AzcDversus.o for. ~igid ~in:ular foundation subJ,ectedto frequencydependentexcitation . " ... (Rlchart,1962) "" . ", "
-..
-- -.. .- -,
-_~_::c,i:- .---'-<-':":' ~'--
-'- ;-
-
--------
376
Soil Dynamiq_&Machine,Founda.(~(Jf3
0,6 b :'5
c:
~ O.S
parabolic
J.l : 0.25
pnz ssu rq distribution
~
-
-g
,
0.41,-
a. E
0.3
0 U1 U1 b.I
C 0.2 0 U1 C \:11
E
'0
0.' 0-
0
0.5
°0
1.0
1.5
Fig. 9.t 5 : Effect of c~ntact pressure distribution on the variation of Aze,nwith ao (Richart and Whitman. 1967)
c
~
0.4-
... ~ "U
~ 0.3
Rigid basq b : 5
a. E 0U1
0.2 0.5
III ~
c:
0 'jA 0.1 c: ~
E 0
0... 0
0.5
°0
Fig. 9.16 : Effect of Poisson's ratio on the variation
1.0
1.5
of Ann with ao (Richart and Whitman, t 967)
F~omthe amplitude~frequencycurves.in,Figs.9.J3 ,and 9,.14,on~can obtain:-(i)~axffI1um amI tude, and (ii) value of ao for maximum amplitude (i:,e;.resonance condition) for different values 01 Richart (1962) have plotted this data in graphical form as shown in Figs. 9.17 and 9.18 which convenient for design use. -
',dations of Reciprocating
Machines
100 Rotating
maS5
Constant
forc(Z
.£)
. .-0
Rigid
bas(Z
... 0 ....
10
III III
0 :--.......
= 0.5
'....... """'....... """'0
<,
10 .2
0.4-
0.6
0.8
ya lu (Z of
1.0 ao
at
1.2
1.4
1.6
r (Z50no nc (Z
Fig. 9.17 : Plot of mass ratio b versus ao for reso!,ance condition for vertical vibrations (Richart, 1962)
100
r
\ '\
\'\ \ \,
, \\
\" .£)
, \\ \ \\ \ \\ \ \\ \ \\
I
0
.;:
,
,"
10 0
'\ fJ
Rigi d
\\
ba C;(Z
Ji< \ =0.5 "
0.25
Rota'ting
ma5S
. .
. ",
0 ..~
Con s'tant"
torc(Z
. . ,,,;
1-
-.0
1.0 . ~. Dim(lnsio'rde-ss',,:omplitude
ot
1.2
re5.0nanC(l
'1.4 ..~
Fig. 9.18: Plot of--;)""\"";*~;,'J..~.~, mass ratio bversus, dim~nsio.nless~mplitude at resonance (Ricbart, 1962) ~'", ,
"'.."
'.~\!;,A
,~,{~,:'H"...;,,~) ':!l~t:...n~iv~ir~l1'i'
~(1'i
'r,;',,~~ ~,
1r~ Q...(m'\\i
,,~4 {
'.f,,<'"
378
Soil Dynamics & Machine Found4ti.o., óó¢þò¢ù
ùòù
æþþùóù¢ùå¢öôº
Lysmer and Richart (1966) propose4 a .siplplifi~dmass-~pring-dashpot analog foL-calculating,the response of a rigid circular footing subjected to vertical oscillations. The values of spril1gconstant It and damping constant Czwere taken as giv~n ~elow :" : .
K=z
, 4Gro ...(9.104)
I-J.t 2
3.4 ro er;G z I-J.I. The equation of motion may thus pe written as
and
C = -vpv
2
...(9.105)
. ...(9.106)
-
iwt .. 3.4 ro er;. 4 G ro mz+-vpG.z+_.z =F e I-J.i. I-J.i. Z Lysmer and Richart (1986) also suggested the modified mass ratio as 1-J.I. I-J.I. m ~.b 3 B -z
-
4
-
4
"
...(9.101'
pro
.The damping ratio; z is obtained as C
C
3.4 r;
,
JPG
...(9.108
to
~, C: ~ 2J~, m ~ îøïóñ´÷ù I-J.I. To m Putting the value of m in terms ofBz from Eq. (9.107), Eq. (9.108) becomes ~
;
- 0.425 z
-
...(9.t'0'
ùÖÐææ
The response of the system can be studied using Eq. (2.58). The dashed curves in Fig. 9.19 illustrate how well the response curves for the analog agree with t! response curves for half space method. The derivation of magnification factor, Mz is given below. '3 }J = 1/3 Halt-space H
:E ~
L-
theory
--- - Si mptitied
-
ono log .1
Bz :: 5
2
Constant
e
....
torce excitation
u 0
-
'" ',1 ~ ~J
-) Z
c
0
.... 0
-uc
(JI 0 ~ -..0 ... :t!?
."I!. ..",
0
0.5
. L0
1.S
~,
, Dimen'sioritesS' freq"uency, 'cro" "l , " , ~ Fig. 9.19: Response of a rigid circular footID ' for vertical vibrations (Lysmer and Rlchart, 1966) ""-"i ",~' ~ ~: 1 .I
379
lions of ReciproCllting Machines
e natural frequency of the system described by Eq. (9.106), i.e. .
',".K'
. .
<.0
m
""
= ri-L.
'."
~
m
2"
~l-~
...(9.110)
.
tting the valu'e~'of Kz, ;n a~d~~zfrom-Eqs:' (9.104), (9.107) and (9.109) into Eq.(9.110), we get <.0
nz'
~
f(Bz
...(9.111)
- 0.36) 0 .~B r 0'. z 0
Id at resonance the amplitude is given by Eq. (2.61). i.e. 2 Fo / Kz
...(9.112)
(Az)max = 2 ~~ 1-~z 2 utting the values of ~z from Eq. (9.109) into Eq. (9.112), we get
Fo (Az)max
Fo.(I-J.1)
Bz
40'0
. 0.85~Bz - 0.18
(Az)max= .
vlagnificationfactor,
...(9.113a)
= Kz 0.85 ~Bz - 0.18
"
r
Bz
M- =
(A)
.4G,
z max
"-
.
0
=
Fo(1-J.1)
...(9.113b)
B
z 0.85.JBz-0.18
~or a frequency - dependent excitation ,the resonant frequency and the maximum amplitude are 1 by
:
H
0.90 <.onz
(Az)max
...(9.115)
= -;;;- .0.85~Bz- 0.18
Mze = 3mee
m
:re
Bz B' z
(Az)max -
Magnification factor,
...(9.114)
= ,/(Bz - 0.45)P . '0 2 me e
and
-,
1
...(9.115 a)
- 0.85~Bz-0.18)
2 me = Unbalanced rotating mass e = Eccentricity of mass from the axis of rotation
.2. Pure Sliding Vibrations. Amold et at. (1955) have obtained theoretical solutions for sliding
rationsof rigiq circular foundations(Fig. 9.20) subjectedto an oscillatoryhorizontalforce F0
ei(J)(.
~y have presented the solutions for two cases namely (i) constant force excitation, and (ii) frequency )t:nderit excitation.
In constant force excitation (F0 = constant), the amplitude Ax is gi~en by . .
-
.
'F.
.. ...
-"'
.
...(9.116)
Ax = O~0 Axn
lere,
" ..
Axn = No!1-dimensional amplitude factor
.~-
"~
--. ., .""
...
"""'-'
-=.""---==~
;;:;;~
Soil Dynamics & Machine ,Eo. .
380
.. F""(Z 0 iQt
...
otT f
Foundation f
x
s.t :~'"
'
,
- -.' --:-.,-, ~. ;.:,\':'" .;"-,~': -",-,...,-'". -. . "-' ,..' " .' f ., , G
P
r0
1
I
I
",
1}\
, I
1.S f
}.J
-h'J' J.'I..~;
z Fig. 9.20: Rigid circular foundation subjected to sliding oscillations
";J
The variation of amplitude versus frequency is shown in Fig. 9.21 by dotted lines. The envelop
f-
10
t, ~
ti
}J = 0
"0 ::J ....
a. E 0 0'1
---
III
..~,
ߨ»² c
ti x
.= <{ 1:) .-
- Axn
....
~"~ '":I-
, "I it
1
0
~I '"
C )(
T'
××Ää¥ III ti C 0 III C ti
,-E c 0.10.2
1.0
°0 Fig. 9.21 : Plots of AsDand AscIIversus ao for sliding vibrations (RIchart, 1962)
tionsof
íèï
ReCiprocating Machines
ïðð
£òÖãÑ
³¿--
Eccentric oscillator
ïð × ô
Ö ó
ݱ²
¢¬¿²
¬
ï ðòî
ðòìò
ðòê
ðòè
ïòð
ïòî
ïòìò
ïòê
pð Ú·¹ò çòîîæ
Plot of mass ratio. b versus a 0 for resonance condition for sliding vibrations (Richart. .
For the case Fo = 2 mex.e
ci ,the
value of A is given by:
/,
1962)
.
- 2 me e Ax - -rAxen pro
...(9.117)
The firm line in Fig. 9.21. shows the envelop of Axenversus ao for resonant condition. The variation he mass ration b versus ao for resonant condition is given in Fig. 9.22. Hall (1967) propos~d a simplified mass-spring dashpot analog for calculating the response of a rigid :ular footing subjected to sliding vibrations. The values of the spring constant Kx and damping con1t Cx were taken as given below.
;
K .
and
x
=
32(1-J.1) G ro
...(9.118)
. 7-8J.1
C = x
18.4 (1 7-8J.1 ~
J.1)
c:;:;
ro vPuo 2
...(9.119)
The equation of motion thus can be written as
.. 18.4 (1-J.1f
mx+
7 - 8J.1-
r
2
0
' P .x+ JPG. . .
32(l-J.1) G r .x -
7 - 8J.1
F e;CJ)l
0
..,(9.120)
0
He also suggested the modified mass ratio B -for sliding vibrations as .
.
\. "
B x
".',.,..".."",J,,'I,
=
X
7 -8J.1
'..
-m
32(1- Ji) pr3 "O,,}.
...(9.120 a) '.'
"".
.
382
Soil Dynamics &: ,Mllchine Foundlltitt~
1
The damping ratio ~x is given by .
~ - Cx Cx x - Cc - 2 ~kx m Putting the values of Cx and kx from Eqs. (9.118) and (9.119) in Eq. (9.121), we get
~
I
...(9.121)
I
- 0.2875
x-
...(9.122)
Fx
I
Figure 9.23 illustrates how well the response C\lrvesfor the analog agree with the response curves for the half space model.
4 Bx = 5 Exact
solution
- - -Analog solution 3 x ::E
I
...
L-
I
\
2
-
0
.....
u 0
..... c
0 ......
2
0
u ......
It
.-
C 0'1
i1~.j
0 ::E
.' T.:!i; "
..
tii.
.,'
",'
0
0.5
1.0 ,°0
Fig. 9.23 : Response of rigid circular footing for pure sliding (Hall, 1967)
.~.~~ ,:;,,
1.5
(1 /1.1' ~.
~
,J' '1i~;r jI.,.~
ùïïä¥ù
383
,dotions -of RecipI'ocatingMachines
The natural frequency <.olltand maximum value of amplitude (Ax}m~ can be' computed using Eqs. 23) and (9.124) respectively. . . . <.0
m
nx
.
and
(AJ
g
= f-: -2.. 1-~
X
.
...(9,123)
x
Ft / kx
=
" ma
2
2
...(9,124)
~x .R1- ~x
,2. Pure RockingVibrations. Amold et al. (1955) and Bycroft (1956) have obtained theoretical soons for rigid circular foundations subjected to pure rocking vibrations (Fig. 9.24). The contact pres~ below the foundation is varied according to .
ere
My
3 My r cosa. ejro( 3~ 27tro 'Vo -r
. q --
...(9.125)
(for r ~ ro)
= Exciting moment about Y -Y axis
a = Angle of rotation \
/
I
\rjJ
fJ /
I
\~I -
Footing,
I
i\T7\ \ , , " '. ., ,..' ., " .". ."" , . G'. .' . '"', . . -
"
-:
~
/
i
My Cli~,t
L
' ',,'
I-'
.,... r
-
-
~ ~:,
0 ,
.~
/"",
1I
':"',:"
,'"
p }J
x
.
~
,;"
"
. "'"~~~-~ptali~;bt 1..1 ; Fig. 9.24 '.;."
y ';} ,
,t'
tootin9'~
.~..,
:
....~
: Rigid clrcuJar foundation subjected to rocking oscillations " ;,I':(V;.t.~'H;tL""'i
4;':'
,.)"~:;>
Ö¼
i
.
384
Soil,Dynamics
&:.Machine
Fou"dtltioM
Borowicka (1943) gave the following equation for computing static rotation of the foundation under the static application of moment My' 3(1-J.l) Mmo A~ =
8
...(9.126)
~pro
Under dynamic conditions the amplitude of rocking is a function of the inertia ratio B. which is given by 3(1-J.l) Mmo ...(9.127) B. =8 pro5 where, Mmo = Mass moment of interia of machine and foundation about the axis of rotation For the dynamic moment My, the amplitude of angular rotation A. can be expressed as A. = where
...(9.128)
MY3 'A,n
( Gro )
= Non-dimensionalrotational amplitude
A.n
Fig. 9.25a shows the variation of A.n with dimensionless frequency ao for various values of inertia ratio B.. The envelop curve shown by the flfm line can be used to define the relation between ao at maximum amplitude (resonant condition). The plot of inertia ratio B. versus ao for resonant amplitude is shown in Fig. 9.25 b. .
Hall (1967) proposed an equivalent mass-spring-dashpot analog for calculating the response of a rigid circular footing subjected to rocking vibrations. The plot of spring constant k. and damping constant C. were taken as given below: 3
8G 1~
...(9.129)
K. = 3(1-J.l)
0.8r: ,fGP
...(9.130)
Ccp= (l-f.l)(I+B,)
'"
40
't) :I
..
a. E
-
'f :'
0 0
c
" -
, ,, 'I'I ,\, 1I ,I
0
,
\
~
0
c 0
"
"
, \
I
'\
,
'" C tII
0
JJ = 0
11 ,I
10
.. c 0"6+-4 0
E
'
I
I
1 0.2
,I
I
"
20 \ I I
\1 1\
0.4
,
'\
/
\ \
I
1\
10 \ I \I (
0.6
\
\
\
S\ \
1.4
0.8
°0 Fig. 9.25 : (a) Plot of A." versus 10 (Richlrt.
1962) (...Contd.)
'j .j...
.
H'I\!~
Foundations
of Reciprocatipg
385
Machines
60
-e.
m
)J = 0
... 0
10 ..... 0
....
...... 0
L..
C
1 0.2
0.4
0.6
1.2
1.4
°0 (b) Fig. 9.25: (b) Plot of B. versus. ao for resonance condition for rocking oscillations of rigid circular foundation (Richart. 1962),
/-
The equation of motion can be written as 4 ~ 3 i(f)/ ;j, 0.8ra "Gp .h 8Gra '" _ M ",+ '",+ '", - M e m$ (l-fl)(l+B~) 3(1-fl) Y For critical damping,
...(9,131)
...(9.13'2)
c~c = 2~K~Mma Therefore damping ratio ~~ will be C~ = 0.15 ~~ = -c ) In
..,(9.133)
(1+ B$ "B~
~c
Figure 9.26 illustrates how well the response curves for the analog agree with the response curves for the half space model. The undamped natural frequency ron~and amplitude A$ in rocking vibrations are
givenby
.
ro = n~
~ ~
...(9.134)
Mina My
A.
1
~
.
..
k'[{l-:~r+~'ro:ni
...(9,135)
386
".
Soil Dynamics & Machine Foundations
50
B~=S
Exact solu tio n
- --
20
Analog
solution
9-.. 10
-)
J -:)
-:>
5'-
0
u -
+c: 0\ 0
2
z
0.5 0
t.' "
1.0
0.5
°0
l<
Fig. 9.26: Response of rigid circular foundations subjected to pure rocking vibrations (Hall, 1967)
Vibrations. Reissner and Sagoci (1944) have obtained theoretical solutions for rigid circular foundations subjected to torsional vibrations (Fig. 9.27). The variation of tangential shear stress is given by 9.5.3. Torsional
3 't 9 Z
Mr
41t
3
. = -:-.~ro
where
'tz9
2
ro
2
- r.
0
for 0 < r < r
...(9.136) "'. ~ I>
= Tangential shear stress
Wo
= Mz ei(J)( = Moment at any time t Mz = Maximummomentabout Z-axis M
'if. "!
For a static moment Mz.the angle of rotation A'IIsis given by A
-
3
",s ~[ 16G r;]
M z
~
~ .u(9.I31~i .,.
~. "
~,. c"' ":
---------..-.
'j'
387
Foulidiltions '6/ Ri!~iprdcating 'Mai:htnes
The amplitude of the angle of rotation Av can be expressed as M Av = ~Awn Gro .-"-""
'-',"'.
"-"'--'."-"-~"'""
'.
where
...(9.138)
Awn = Non-dimensional amplitude factor
Under dynamic condition the am~litude'of torsion is a function of inertia ratio Bw which is given by Mmz Bw = -S ,..(9.139) pro where Mm= = Polar mass moment of inertia of the machine and foundation about the' axis of rotation .
.(
Footing TZ9
-.
,"'00"'-
G " :', :,,~(,::::~(~<'<~
z'
: ...""."r,
',_..
ro ~
,
.
,
., "
Flaxibl
ro
,,
(b)
Ize
Rigid to undat ion
--'..
- -.. (a ).-
,
ro (c)
,.,..,-,,-'"
~'
Fig. 9.27: Rigid circular foundation subjected to torsional vibrations
Figure 9:~8 _ashows the ~ariation of Awn with~ime,n~ionless frequency aofor ~~rious values of inertia ~atioB",:'~e' ~nveIop'cur~f's,ho~'~y !~~,:~~..lirie'c~n'~.~.~s~ato,defme 't~e relation between ao at maXimum amplItude (resonant condItion) ana the'values of Intena ratio B", (FIg. 9.28 b)
-,
388
Soil Dynamics & Machine
F:oulldatioRS
-~.
~,,--
10
tII '0
, "-
::J ....
""
Co
E
II
1\
0
,
I1
rI I
0
c 0 ....
,
I I
\
1
\ \
\
I
'" '" to'
/ \,
C 0 '" C tII
.
/
/ /
E Q
0
/ /
//
/./ ññ¢óóþùþ
0.2
/;\
0..4
"'"
\ \ ~;.\./ I
"
Rigid"
5, \,'
B4' = 10!
Flczxiblcz
"V'.......
, I'
, I
.~ c '" r ...
.
/
\
\ \
\
\
I
0.8
.......
.-
\)/ /'
/'
"
\
\
1.2
1.6
2.0
00
(a)
60
7 m
.~ .... 10 0 L. '" '" 0
~
1
0.4
0 (b)
Fig. 9.28 (a) Plot of
~
0.8
1.2
1.6
2.0
00
versus ao; (b) Plot of B. at resonance versus ao for torsional oscillations of a rigid circular foundation (Rlchart,1962)
'. .,'f;' 'w
." iO' ~
",. .......
389
FiJundlltions of ReCiprocating Miichines ":;;-~~_'d
Richart and Whitman (1967) propose~ an equivalent mass-spring-dashpot analog for calculating the response of a rigid circular footing subjecred to torsio~al vibrations. The values of spring constant kw and damping constant CIjIwere taken as below;
= 16 G r3
~
,
J
~
...(9.140)
,
0
~::
',.'"
and
...(9.141)
The equation of motion can be written as M
.. III
mz 'Y
+
;m( 1.6ro4.Jp G." 16 3 . \V + - G r. \V - M e,
I+B
IjI
.
"",
3-:
0
...(9.142)
Z
1
The damping r~tio ~1j1~ili be
,"
,
C,o.s ,..(9.143) ~1j1
= 2 ~K:
.m
=
(1 + 2 BIj1)
The undamped natural frequency wnlj1and amphtude AIj1of the torsional vibration are given by
'K
"
...(9.144)
vM:;
wnlj1=
A
,
=
Mz
W " KW[{I-
...(9.145)
~:r +~Wro:Jr
9.5.4. Coupled'Rocking and Sliding'Vibrations. Figure 9.29 a shows a rigid circular foundation resting on the surface of elastic half-space and subjected to an oscillatory moment Mv /(JJlan a oscillatory horizontalforce Fx e,m{.The sign convention chosen is illustrated in Fig. 9.29 b, which indicates that + x and + F act to the right and that + 4>and + Mare clockwis'e.Figure9.29c showsthe motionof the footing when both translation of the e.g. and rotation about c.g. are positive (i.e. in phase). In this case. the centre of rotation lies below the base of the footing, and the motion ,is termed as first mode of vibration. If the translation is positive while the rotation is negative, then the centre of rotation lies above the e.g. and the motion is designated as the second mode' of vibration. ': "',1/ ,~}') ,/1' Nl
." ..
J
" >
1t
390
Soil, Dynamks
" ,Milehine FOUIIdtltions
I J
My (l iwt
I I
I
!.
I
... ~ Fx(li6.>t
I
I
IL
m
Er MR ~+Fx
.. Fx
,
fMR
( b)
(a) r
I
I
1
I
/:[1 1-.-
~
I I
,
=
/
II
+
I
I
--r I-- x Xo1r-
+x
'
~~ leA]
I :
(c)
Fig. 9.29 : Coupled rocking
Lc,&
and sliding vibrations of a rigid circular block on an elastic half space (After Richart
and Whitman,
1967)
From Fig. 9.29a x 0 = x - L<\>
where
..,(9,146)
XO = Displacement of base x = Displacement of c.g, <\>
Therefore
= Angle of rotation in radians
Xo = x- L
...(9.14'n
The equations of motion are written in terms of x and <\>.The equation of motion for sliding is ~ , .
or or
..
m x + Cx
.
Xo
+ Kx Xo = Fx eirot
.1 ~I. ~4
m x+ Cx (x- L~) + Kx(x- Lcp)= F.tirot
mx+C x x +K x -x-LC x'fd.-LK x'f'" =F .t irot The equation of motion for rocking'is .. . ,
irot
= M.ve
Sustituting the value of Xo from Eq, (9.147), the above equation simplifies to Mm ~+(C~ +L2tx) ci>+(K~+ L2K:c}cp- L Cx x- L Kx'x
...(9:148 , .[)"\Or .,.1~aiJ
'
Mill cp+C~ <\>+K~ <\>+C:c,xo'L+Kx,xo.L
r
= My
irol
., ~:y;t
,
.,;, ,<:; ...(9,Jf~ ':: ''~:r . :"., ..t~
'~:',;~!
391
'Idations of Reciprocating~Machines
The natural frequencies of coupled rocking and sliding are obtained by putting forcing functions in
(9.148) and (9. 149) equal to zero. Thus
,
³¢õÝ x ¨õÕ x ¨óÔÝ x n.-LK 'I' "
.
2
(
2
) (
x .n. 'I'
)
Mmcp+ C.+L Cx cp+ ÕòõÔÕ¨ cp-LCxx-LKx'x
=0
...(9.150)
=0
...(9.151)
The solutions of above equations can be obtained by substituting ...(9.152)
x = A eiCiJndt cp
= B eiCiJndt
...(9.153)
which A and B are arbitrary constants. By doing this the Eqs (9.150) and (9.151) become A LKx +i LCx,oond B-2 -m OOnd+ K x + I. c x OOnd A
d
...(9.154)
-Mm OO;d+(KcjI+L2 Kx)+i(CcjI+L2
Cx)OOnd
...(9,155)
B L Kt, -i Ct- OOnd#.' Equating Eqs. (9.154) and (9.155), and substituting
if
oonx
= -,m oonq» x
ff.
x =~ ~= ~ Cx M'mo r M mo ' x 2 vL'\o.x rH
C
,d ~4I 2 2 roncP+ronx
4
rond-
(
[
4 ~xronxrond r [
-
r
.
= 2 ~ K. xMmo , on simplification we get
4~x~~ronxron~
2,
~.
(ro~2 - ro~2.)+
)
rond+
~~ron~rond
r
2
22 roncpron..t
2(ronx
r
1
+
J
2 2ro~ -0 )]
...(9.156)
It may be noted that Eq. (9.156) reduces to Eq. (9.69) when ~x = ~cjI= O.As the effect of damping on laturalfrequency is small, the undamped natural frequencies for coupled sliding and rocking vibrations an be computed using Eq. (9.70). . The damped amplitudes of rocking and sliding of a foundation subjected to a horizontal force For/°)1 aregiven by lfJ
A xl
~
ÅøóÓ³®±îõ¢õÕ®Ôî÷î +4ro2(~,,~KrMmo+L2~x~Kxm)2 m Mm, ~ (002)
- Fr
(
2'
2
F L ronx ronx+4~x ro
and A =--L.1 Mm
...
~ (ro2)
]
-
...(9.157)
112
)
...(9.158)
392
Soi/Dynamics.& 2
,
2
2
Machine Foun ;~
2
2
'." '
where lI. (CO2)~ [ ",4_",2
{ '" ,+ :"'M
'" ,+} + "',. r"'M
4 ~x~+;M
,
,
]
~
2 ln
+ [ +x m"; m (m~ -m2)+ ~.m~.. m(m;. -m2)} ] ,
~I
., "
't
,.
...(9-.IS~~ "':'
"
,
The dampedamplitudesof rocking and slidingof the foundationsubjectedto an excitingmoment iw . M, e are gIven by '$I ,c
:4
2 Mv
A-'
x2 -.
MIII
[ (OOnx)
2 1/2
2 +(2;x
OOn.t) ]
2 A
-
."(9.Jf~1 ",
-Mm
iI !I1'r
t
2 112
2 2
- My [ (OO"x-OO)
and
,
" (00 2 ) D. +(2;xoo/lxoo) f..
(00
"' "
]
...(9.1~!
2
,
,
~,,
)'.
When a footing is subjected to an oscillatory moment Myirot and a horizontal force Fx im/ simulb-
neously, then the resultingamplitudesof sliding and rocking are
. "j
Ax = Ax1 + Ax2 A<1>
= Acjll +
...(9.162
A<1>2
...(9.163
9.6 EFFECT OF FOOTING SHAPE ON VIBRATORY RESPONSE
fy Elastic half-space theory was developed for a footing with circular contact area. Response of a footing~1 influenced by the shape of contact area. The usual practice is to transform area of any shape to an equjy: lent circle of same area (for translational modes) or equivalent moment of interia (for rocking and)b sional modes) (Richart and whitman, 1967 ; Whitman and Richart, 1967). Thus ~'
For translatory vibrations:
r0 =
b
i!-
..(9.U
7t
,;;:.1 1/4
3
For rocking vibrations:
ro
~~~1. .:~(~~t
= ( b3: J
~aj,yt ~ .. ",./I
114
ab (a2 +b2) For torsional Vibraf .ns : where
ro
ro
=[
61t
, "
~.j91,
]
'. ~'.~if
tt'
= Equivalentradius
t" ,1 '
a = Width of foundation (parallel to the axis._ofrotation for rocking)
'
b = Length of foundation (perpendicular to the axis of rotation for 'rocking)
-
"
d1,, , ,
.'' " ,,
i<)
I ["'~
"'
,~,
"4!t"l:'1 ,,
' ~~.h.. '' '
,5' " .'' ~ 't\', """ ,
, 'ft, , , '"r",~,
394
Soil Dynamics & Machine Foun~
,.
9.7 DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS - ~i For an embedded foundation, the soil resistances are mobilised both below the base and on the sides. Th~ additional soil reaction that comes into play on the sides may have significant influence on the dynamit response of embedded foundations. Typical response curves showing the effect of embedment are pre~ sented in Fig. 9.30. It gives that as a result of embedment, the natural frequency of the foundation- soil, system increasesand the amplitude of vibration decreases (Novak, 1970, 1985;Beredugo(1971) ; Beredugo and Novak,
1972;
Fry, 1963 ; Stokoe,
1972 ; Stokoe and Richart,
1974 ; Chae, 1971 ; Gupta,
i
1972
Vijayvergiya,1981). . The problem of embedded foundations has been analysed by both linear elastic weightless spring approach (Prakash and Puri, 1971, 1972 ; Vijayvergiya, 1981) and elastic half-space theory (Anandkrishan and Krishnaswamy, 1973; Baranov, 1967; Berdugo and Novak, 1972; Novak and Beredugo, 1971).Th~ analysis developed by Vigayvergiya Cl981) is simple and logical, and therefore selected for presentati9P here. On the basis of theoretical analysis, he had recommended the.equivalent spring stiffnesses in different types of motion as given below: .
9.7.1. Vertical Vibrations
(Fig. 9.31) ¢
! Fz5in
6>1
Machin
~,~
Ú¦ Í·²
êâù
³
-
Foundation
h
ø±¨ bxh)
0
1
~
a
f..
Õ¦ä¦
1
Fig. 9.31 : Embedded block foundation subjected to pure vertical vibration
...(9.! ã Ý«Ü ßõîݬ¿ªø¾Üõ¿Ü÷ Õ¦» ã Û¯«·§¿´»²¬ spring stiffness of the embedded foundation Ý«Ü ã Coefficient of elastic uniform compression obtained at the base of foundation Õ¦»
where
ݬ õÝ¬Ü Ý¬¿ª= Averagevalue of coefficientof elasticuniformshear =-
2
Ct ݬ Ü
= Coefficient of elastic uniform shear at the ground surface ã Ceofficient of elastic uniform shear at the base of foundatiqn
D = Embedment depth b = Width of base of foundation a = Length of base of foundation ~
d
.
..
~"" ... ..."".. . ~.."L... . .
395
Foundations of Reciprocating Machines
The equation of motion will be ...(9.168)
= Fzsin rot
mz+Kze'z
The natural frequency oonzeand maximum amplitude Aze of motion are given by
fK: oonze
Aze
...(9.169a)
= v--;;; Fz
=
. m 9.7.2. Pure Sliding vibrations (Fig. 9.32)
2 ronze
(
...(9.169b)
2
- ro )
Machin
.
-
Fx Sin wt
...-.
h
~
m
..
0
ø¿¨¾¨ h)
~
Õ¨ä¦
FxSin wt ~
Foundation
1
, ,
a
-i
1
'.
Fig. 9.32: Embedded block foundation subjected to pure sliding vibration
Kxe = Ct D A + 2 Cllavb D + 2 Ctav aD
...(9.170)
where Kxe = Equivalent spring stiffness of the emebedded foundation
C +C Cl/a v
= Average value of coefficient of elastic uniform compression =
!!.-
2
uD
CII = Coefficient of elastic uniform compression at the ground surface.
Similarly, the equation of motion will be
m x + Kxe
...(9.171)
..i = ~t sin rot
The natural frequency and maximum amplitude of vibration are given by
fK:: oonxe
=
A xe
...(9.172)
= v--;Fx 2 m (ronxe
2
~.
...(9.173)
- ro )
.. ~
.
396
Soil Dynamics & Machine -
Follndatioi;.\ -,
9.7.3~ Pure Rocking Vibrations (Fig. 9.33)
,. Machinq
My Sin
GJt
m
-::.'\\,: ×
¸
-
×
0
Foundation (axbxh)
ï
1 ./
a
I..
~
r
Fig. 9.33 : Embedded block foundation subjected to rocking vibration
C~av b Kljle =C~D.I-W.L+ where
24
3
(16D
f Db a2
2
-12hD
)+2C~avlo+Ctav
2
-t
...(9.£7< :1 ,"1l
Kljle = Equivalent spring stiffness of the embedded foudation
CIjID=
Coefficientof elasticnon-uniformcompressionat the base levelof foundation
Cljlav= Averagevalue of coefficientof elasticnon-uniform compression= CIjI
C~+2C~D
= Coefficient of elastic non-uniform compression at the ground surface
L = Height of the combined e.g. of machine and foundation from the centre of the base J W = Weight of foundation b 3,
,
I=~ 12
'~J
,
3
I = aD 0 3 The equation of motion will be
.,.
MI1IO«\>+Kljle.$
...(9.I
= My sin rot
The natural frequency and maximum amplitude of vibration are given by Ct)lIljIe =
~
~e M mo
...(9.,. } -- i
My A
= :ce
J';J
2
2
M",O ( ro119- ro )
..~t} ~, ,I .i;J .-;i' ,. f j.
~ IJI ~
Foundations
,.Fo/llldatiolls
.
"
of Reciprocating
397
Machines
9.7.4. Coupled sliding and rocking Vibrations. The equations of motion in coupled vibrations are given as ...(9.178)
mx+K.u,x+Kxtp' = Fxsinrot and
...(9.179)
Mm~+KcI>'+Ktpx'x = My sin rot
}\There
]
= Ctpavb
Kq,tp
= CtpD1+
...(9.181)
(D2 - 2 DL) - C'tD . AL 2
.
a2
C'tD AL - WL + 2 CljlavIy + C'tav bD
2
2
3
3
+ 3' Ctpav[L + (D - L )]
> Ktpx
~K~t2
~ ~
where
= - [ C'tD
CljIlIV =
AL + 2 Cl/a v bD
( L- ~ ) +
2 C'ta v
...(9.182) ...(9.183)
( L - ~ ) aD]
Average value of coefficient of ~Iastic nonunifonn shear
ly = Moment of inertia of area a x Dlying in the plane of vibration about axis of rotation Da3 =-+12
,..(9,174)
(a) Only the horizontal force Fx sin ootis acting: m x + Kxx . x +
the base
Kfttp
.
= Fx sin
...(9.187)
=
...(9.188)
Atpfsin 00t
By substituting x and from Eqs, (9,187) and (9. 188) in Eqs, (9.185) and (9.186), we get (Kxx - m 002) Axl + Kxtp Atpl 2
KtpxA.d + (Ktptp- Mm 00 ) Atpl
d.
sin 00 t
I tlI F t j
...(9.189)
=0
...(9.190)
By solving Eqs, (9, 189) and (9,190), we get
,..(9)75)
.
Axl ...(9.176\
= Fx
I '11 . . .
,..(9.186)
x=Adsinoot
I
~ I 1'1 . I ~\j
...(9.185)
00t
The solution of the above equations can be represented by
t
I I I , ,
The equations of motion will be :
Mm ~i + Ktptp. + ~tpx X = 0
l
I ,'1 I
JI ...(9.184)
m Mm co~e- (r:zKtptp+ Mm Kx.~)co~e+ (K~~K~ - K~x x Kxtp)= 0 The amplitudes of vibration ofthesystem can be obtained as below:
.
D
aDb2 4
1'he natural frequencies of the system can be obtained by solving Eq. (9,184)
on
-::. 11. , 411 I
...(9,180)
Kxx = C'tD A + 2 Cl/aVbD + 2 C'tav aD Kxtp
f' '" ~I'
2
-
-
2
(Kxx-moo
'" Atpl -
2
)(Ktp~-Mmoo 2
(K.n-moo
...(9.
Q,
.(K~~- Mm 00 ) Fx
...(9.191)
r.
...(9.192)
)-KxtpK
-K~
'., 2 ) (Ktptp-Mm 00 )-KxtpK
I I I~ .I
...
~ O//II~
""i"'""
~
----
u
cl! ~
~-
ó¢ùó±
íçè
Soil DYllamics & Machifle Foufldatiofls
(b) Only the moment My sin cot is acting: The equations of motion will be : 11
I.i
m x + Kxx . x + Kcpx=
...(9.193)
0
1I
...(9.195)
= A.p2 sin co t ...(9.196) the values of x and from Eqs. (9. 195) and (9.196) in Eqs. (9. 193) and (9.194), we get 2 (Kxx - m CO) Ax2 + Kx.pA.p2 = 0 ...(9.197) 2
;.
Substituting
4:
K.px' Ax2 + (K.p.p - Mm CO) A.p2 = My By solving Eqs. (9.197) and (9.198), we get
,f'
...(9.198)
-Ad and
A.p2=
I~
(K,X,X-
-Kx.p
2
2 My )(K.p.p-Mmco )-KcpxKx.p 2 (Kxx -m CO)
2
2
(Ku-mco ~" I
as:
X =Ax2 sin COt
f.i
I
...(9.194)
~
Mm + K.p.p . + K.pxx = My sin COt The solutions of the above equations can be represented
m CO ) (K.p.p- Mm ro ) - KcpxKx.p
My
If both Fx sin cot and My sin cot are acting simultaneously, then Ax = Axt + Ax2 H:
ïï åô ®
ô
...(9.199)
...(9.200)
...(9.201)
A.p = A.pl + A.p2 ...(9.202) Sometimes to screen the vibrations, some .air gap is left between the pit and the foundation block (Fig. 9.34). Figure 9.35 shows the comparison between the response of embedded foundation with air gap and without air gap. From this it can be concluded that if air gap is provided around the foundation the amplitude of vibration increases whereas the natural frequency decreases when compared with corresponding foundation with no air gap around it. The response of embedded foundation with air gap can be obtained by analysis given in sec. 9.4 by using CuD' CtD,C.pDand C",Din place of CII'Ct' CeI> and CIjI respectively.
ô
Machinq × ô
òº
'"
×æ
..,\ ..//..
Foundation (axbxh)
h
×ù
li
Ai r gap
¬
×
t-
º
a
~
ïïæ
Fig. 9.34 : Embedded block foundation with air gap
¬
¶ò
III
u w
~
1111
!Ill,
III Ill!
~
,..
I!!
Ii
'1111
da
tio
Foundations
11 s
399
of ReciprocatillgMacltines 0 L.()
I .(9.193)
I i:
~
.(9.194)
t,
L.()
I
--.j'
I
..(9.195)
-
f
..(9.196) ), we get ..(9.197)
QC)
'"
0 --.j'
..,.,
..... ... ...
r" 1:1
:; Cc. '" .. ... '"
~ ~!
>
..(9.198) I L.() M
...(9.199)
0
::
,~
...(9.200)
1
...(9.201) ...(9.202) lion block ith air gap .dation the vith correir gap can
I Ill' 11
'i '0 c
0 M
I
i
tf) cV ...
>-
I
L.() ('4 L.() N
. 0
j ,I
II
m
1
CQ>and
~
...... 0«
C'v
I
L.() N. 0
cCl C1 L..
v c ::J
c-
er 01 lLL
Cl C7) l-
" .CD Cl ...... 0 0' Z
0 N
'" c. '" OJ)
... '" ::
1,\
'i"' -
f
c
"
,~ ';... .Q '> '" ,;: t:... >
~ ~ : u=
\I
"u
L.()
c '-0 0
~ >
-
Cl v .....
Cl ...Cl
L.. ...Ln "'Cl
0
C
Cl tf)
L.. >01 ...-
> ::: (/)
11
C ... :::! 0' ... .:: ,/, 'C B
c. a «!'. or, ~ 0-: .. ~
L.()
I 0
0 0
0 r.f)
0
SUOJ:>!W c apn~!ldwV'
(
..
-'i ,..'
--
--
~
~,..
~"'--"
~~
-
..- ~'" ,-
,
~
I'
Soil DYllamics & Mac/lille Foulldatiolls
400
9.8 SOIL MASS PARTICIPATING IN VIBRATIONS In both the methods of analysis and design of foundations of reciprocating machines described in sec. 9.4 and 9.5, the effect of soil mass participating in vibrations has not been considered.
.,
J! ~I
I:
;1 'f
.
,
Fig. 9.36 : Stress distribution in soil mass
Pauw (1953) developed equations for the apparent soil mass by equating the kinetic energy of the affected zone to theCkineticenergy of a mass assumed to be concentrated at the base of the foundation. He gave the following expression for apparent soil mass ms for translatory modes of vibration:
11
i
11
.
t
b3
If
m =Lc s ga.
P
,
where
...(9.203) m
a. = Factor which defines the slope oftnincated pyramid (Fig. 9.36). It is generally taken'unity. Cm = Functionwhich dependson sand r ~=- a.he . b
j
a r="b he = Equivalent surcharge defined by the ratio of foundation pressure to unit weight of soil. For non-cohesive soils, Cmis obtained from Fig. 9.37 (a). No graphical data is suggested by Pauw for cohesive soils. The expression for mass moment of inertia of soil in rotational vibrations is given by
ij
r
,J~.
'Yb5C. M =---'ms 12g a.
iI
whereC.I canbe obtainedfromFig . 9.37
.
..(9.204)
b to e. In these fi gures, C~, C~ I r and c~I denote the factors of mass
moments of inertia about X, Y and Z axes respectively. These factors can be obtained from Figs. 9.37b to d for cohesionless soils and from Fig. 9.37e for cohesive soils.
I
'I. .,
"'I
~---
-.'-
~'.
-
._-,-,
-
-
--
-, --~
"-'---~
'Ill IJI
~~---'
-
ìðï
Foundations of Reciprocating Machines
ݱ¸¬-·±²´¬-0.2
0.2
"' -óóÝ - 'tb3 gJ-
0.5
I.a
'""6. 11
Cl)
soil Mmz = )f'b5 CX 5 12g-" i
0.5
m
1 0
1.0 - - - - - --
- - -- -- ---
1
-- - -.\"'&.a
2.0
-11
Cl)
2.0
S~O
5.0
r=12JLr (Ca)
0:6
0
J Cm r
0
. 0~8
Cb)
0.2
0.2
;rh 5 CY .
0.5
Mm ys
--I - 12'3 aC..
5 z rJL ,Mmzs- 129cl Cj
0.5 ~'1.a1.0
I.a 1.0
V) 2.0
11 2.0 Cl) 5.0
5.0 0
0.5 Cc)
r::1 --'-0.5 cf
CjY --;:-3
1.5
rJ+r ¥b5
x
Yb5
Y
M.mx$-= 12 gel Cj 2 Mmys= CJI.a ," ...
1290£ ݶ
5
Cj
(t)
Fig. 9.37: Apparent mass factors for horizontal contact surfac£..> -' :.,' ' ; I';. '
L
402
Soil Dynamics
& Machine
Foundations
Balkrishna (1961) has developed the following expression for the apparent soil mass in vertical vibrations: 3/2
m = ~ 0.4775 Q s 3( 4 P) where Q = Sum of the static and dynamic load .
1t P
...(9.205)
..
Barkan (1962) has suggested that the apparent soil mass may be taken as 23% of the mass of machine plus foundation. Hsieh (1962) gave the expressions for getting apparent soil mass as given in Table 9.9. Table 9.9 : Effective Mass and Mass Moment of Inertia for Soil below a Vibrating Footing (Hsieh, 1962) Effective mass or mass moment of inertia of soil Mode .of vibration
ïïãð Vertical translation Horizontal translation
11 = 0.25
11= 0.5
3
, 3 2.0 -p ro
3 0.5 pro
\.0 pro 3
3
0.2 pro
0.2
3
1.0 pro
pro
5
Rocking
0.4 pro
Torsion (about vertical axis)
0.3 P r5 0
0.3 P r5 0
0.3 pr50
The apparent soil mass/mass moment of inertia is added to the mass m/mass moment of inertia Mm or J0 to get the natural frequency and amplitude of vibration. .
9.9 DESIGN PROCEDURE FOR A BLOCK.FOUNDATION The design of a block foundation provided for a reciprocating machine may be carried out in following steps: 9.9.1. Machine Data. The following information shall be obtained from the-manufactures of tl:1emachine for guidance in designing: (a) A detailed loading diagram comprising the plan, elevation and section showing details of con-
nectionsand point of applicationof all -loads on foundation;
-
(b) Distance between axis of the main shaft of the machine and the top face of foundation; (c) Capacity or rated.output of machine; (d) Operating speed of machine; and (e) Exciting forces of the machine and short circuit moment of motor, if any. 9.9.2. Soil Data. The following information about the subsurface soil should be ~own : (a) Soil profile and data (including soil properties generally for depth equal to twice the width of ;; the proposed foundation or up to hard stratum). ~ (b) Soil investigation to ascertain allowable soil pressures and to determine the dynamicproperties 4 . 'it of the sOIL..
,>1
(c) The relative position of the w~ter tablebelo~'-g~o~nd ~t differe~t times of the year.
'J
The minimum distance to any important foundation in the vicinity of the machine foundation should,
alsobe accertained.
..
-
}I
-, ~,~-
Foundations
403'
of Reciprocating .Machines
9.9.3. Trial size of the Foundation.
'.
Area of Block-The size of the foundation block (in plan) should be larger than the bed plate of the machine it supports, with a minimum all-round clearance of 150 mm. Depth- In all cases, the depth of foundation should be such as to rest the foundation on good bearing strata and to ensure stability against rotations in vertical plane. Centre of Gravity":: The combined center of gravity of the machine and the block shall be as much below the top of foundation as.possible, but in no case it shall be above the top of foundation. Eccentricity-The eccentricity shall not exceed 5 percent of the least width in any horizontal section. Sharp corners shall be avoided, whenever possible, praticularly in the openings. 9.9.4. Selecting Soil Constants. The values of dynamic elastic constants (Cu' Ccfj1 Ct' C'V'G, E and J.l) are obtained from relevant tests and corresponding strain levels are noted. ,These values are reduced to 10m2 contact area and 10 kN/m2 confining pressure. A plot is then prepared between dynamic elastic constants and strain level. The value of dynamic elastic constants are picked up corresponding to the strain level expected in the actual foundation. These values of dynamic elastic constants are then corrected for the actual area of the foundation (if < 10m2), and confining pressure. The details of this has
alreadybeen discussedin illustrativeexample4.2
,.-'
.
9.9.5. Centering the Foundation area in Contact with Soil. Determine the combined center of gravity (Table 9.4) for the machine and the foundationin X, Y and Z planes and check to see that the eccentricity along X or Y axis is not over 5 percent. This. is the upper limit for this type of analysis. If eccentricity exceeds 5 percent, the additional rocking due to vertical eccentric loading must be considered in the
analysis (Barkan, 1962)
"
The static pressure should be checked; it should be less than 80 percent of the allowable soil pressure under static conditions. This condition is met in most practical foundations. . Table 9.4 : Determination of CG of the System Element of system
Coordinate e.g. Static moments of of the element mass of elements
Dimensions a.t
ay
az
Weight of element
Mass of element
Xj
. Yj
Zj
mjXj
mjYj
mj
2 3
9.9.6. Design Values of Exciting Loads and Moments. The fmal values of force and resulting moments are now obtained with respect to the combined center of gravity of the system. The relative magnitudes of the unbalanced forces and moments will decide the nature of vibrations in the block foundation. 9.9.7. Determination of Moments of Inertia and mass Moment~' of Jnertia. The moments of inertia and mass moments of in~rtia may be obtained using the formulae given in Tables 9.5 and 9.6.
.
404
Soil Dynamics & Machine Foundllliolls
Table 9.5 : Moments of Inertia Formula for Figure
Shape of the area
Ix
~
Iz
ab3 12
ba3 12
ab(a2 +b2) 12
1td4
1td4
64
32
,
ffi "
Rectangle
T
Y
I
--f::.G. X
I
1---0
y
b X
Jl
~
y
T.... Circle
X
X
N 11 '0
.!!:...ct 64
J..
)
Table 9.6 : Mass Moments of Inertia
'1;1
Formula for Shape of Elements
,
Figure
MmT
Mm:
Mmy
""-.
z
1'J'
m
Rectangular block
--
2
12 (b + h2)
m
m
12 (a2 + h2)
12 (i
+ b2) ~ "t !j " .""1
z
,.~ f'
'"
X
Circularblock
1/
1-
y,u t-d-j
T h
1
!!!-
3d2 + h2 12 ( 4 J
!!!- 3d2 + h2
12 ( 4
)
Foundations
of Reciprocating
405
Machines-:M mo-
where
-
Mm + m L2
= Mass moment of inertiaof machineand foundationabout the axis of rotation passing
Mmo
through base. L
= Distance of combined
r=-
centre of gravity above base.
Mm Mmo
.
9.9.8. Determination of Natural Frequencies and Amplitudes Linear weightless spring approach (i) Vertical Vibration (0 nz
=
Az
=
.A
W
(ii) Torsional Vibration
~
m
.-,.
Fz 2 2 m (OOnz-00 )
Mmz .(On",-~cw J, and
=
A '"
Mz 2 2 Mmz,,(OOnw-00 )
(iii) Combined Rocking and Sliding Sliding and rocking are coupled modes of vibration. The natural frequencies are determined as follows: conx ~ ~c, mA
<.oncp --
Mmo ~C.IWL 1
2
2
2
2
2
2
) (OOnx+OOn,) -4r OOnl,2 = 2; OOnx +OOn+:t [(
and
2
2
OOnxOOn,
]
The amplitudes of vibration can be computed with the following equations: 2
A x -
2
(Ct AI.;+C.I- WL- Mm00 ) FJ:+(Ct AL) M v
Y
~(oo2) 2 (CtAL)Fx+(CtA-moo
A ~'lihere,
)My
~ (002)
Ax = Linear 4orizontal amplitu_de of the combined center of gravity Acp = Rotational amplitude in radians around the combined center of gravity. 2
L\«o)=mMm
.
ill..
2
-
2
,
2
,.
2
((J)nl-(J) )((J)n2-~ )
-
'-
I j
406
Soil Dynamics & Machine
Fou1Jdatio"s
The amplitude of the block should be determined at the bearing level of the foundation as a ... ..'-~."
~~~.,,~~:E-C';:,;z::..;~~;_.-,,-~..,_.
where
'.'
Av
= Az+ '2 A.
Ah
= Ax + h A.
Ah = Horizontal amplitude at bearing level h = Height of the bearing above the combined center of gravity of the system Av = Maximumvertical amplitude
Elastic half-space approach Equivalent radius, mass ratio, spring constants and damping factors are listed in Table 9.7 Table 9.7 : Values of Equivalent Radius, Mass Ratio, Spring Constants and Damping Factor Modeof vibration (1) Vertica!
Sliding
Rocking
Torsional
Mass (or inertia) ratio (3)
Equivalent radius (2)
roz =
r ox
,o r
°'V
f! f! 1t
(7
-
1t
0.425
m 3
z = Bz '0.2875
pro
- 81l)
m
x= -a- .f
Bx= 32(I -Il) pr; B
31t (ba')'14
=
(l-Il) -4
Bz=
Damping factor (4)
r
ba(a2+b2) 6 1t (
= $
3 (I - J.l)Mmo 5 8 pro
MmB=---t Y pro
0.15 x
Spring Constant (5)
k=-
4Gro 1-1l
kx =
32(1-J.l)Gro 7-8J.l
z
k =
= (1+ B).JB; 0.5 'V= 1+2B
\jI
3 8G ro
3 (1- J.l)
.!
16
,t
3
k'V=3"Gro
Natural frequencies and amplitude of vibrations (I) Vertical Vibration OOnz
=
~; .'
(it) Torsfonal Vibration
A, 00
~
n'V
=
,,' '~,
K'[{l-(:JrZ+~,:Jr
)~
~
~1
'V
M rn,!,
and
~
~
K.[{1_(ro:Jr~(2~.ro:Jr
L
--..
407
='oundations of Recipr-Ocating Machines
(iil) Coupled sliding and rocking vibration 0)
nx
0
'
= fKx V-;;
~
~
Mmo
O),,~ =
Damped natural frequencies are obtained as the roots of the following equation: 2 4
O
(ID~,+ ID;") 4 V..
2
00 na - 00 nd
[
{ ~X 00nx 00nd
4
IDnx"";
-
r
0
00
0
oo~ 00;,
+
To,
r
.0
}
2 00n,
2
(
)
00
] 2
~'
00nd cOnip
(
2
2
- 00 lid + 00nx - 00nd =0 r r [ )] Undamped natural frequencies can be obtained by using following Equations: +
~
2 OOnl,2 -
2 +
)+ ( 2
2
)-
2 2
+
2
2
2r [( OOnx OOn, - OOn, OOnx 4 r oon, oonx ] Damped amplitudes for motion occasioned by the applied moment, can be obtained as below: 0
.
x
.-
1/2
2
[
A = My
~ . ,-
0
]
(oo~) +(2~x 00n.t)2
Mm"
~(002) 112 2 2
2
A =My [( OOnx-OO $
)
M 111
2
-',0,
]
+(2~xoonx).. /),,(00 2 )
where /)"(0)2) is given by Eq. ..
4
~ (ol) =
(OO~,+OO~)
2
00 -00
r
(
[{,
4 ;.t
~~ 00 nx 00 nljl
r.
0
0
+
+x
2
- 00~x 00 ~~
.
r
}
,
00
'.
00
2
2
0...
~, 00 n,
":-(IDn+-ID)+
}
2 1/2
00,
r
2
2
(IDn,-ID)} ]
Damped amplitudes for motion occasioned by an applied force Fx acting at the center of gravity of the foundation may be obtained as below: 0'
,
1/2
..
and
~
Fx [ (-Mm 002+k, +L2Kx)2 +40)2(~~~K~Mmo+L2~x.JK7n)2
-m-~.m
.A
A
- Fx
(
4 ~xco
0
,.'~:t'~ffl"
6...
(0)2) '- .
2 L coIIX coIIX+ (
/)".
'.-
-
2 1/2 .
)
,", ..
-.' "
]
0
408
Soil Dynamies & Machine Foundations
9.9.9. Check for Adequate Foundation. The natural frequencies computed in.step 8 should be away from the resonance zone i.e. Cl)
-
co
< 0.5 or -
Cl)n
< 1.5
con
The amplitudes computed in step 8 should be less than t~e limiting amplitudes of the machine which are usually specially by the manufacturer of the machine.
ILL USTRA TIVE EXAMPLE
I
Example 9.1 A reciprocating machine is symmetrically mounted on a block of size 4.0 m x 3.0 m x 3.5 m high. The soil at the site is sandy in nature having cl>= 350and Ysat= kN/m3.The watertable lies at a depth of 3.0 m below the ground surface. The block is embedded in the ground by 2.0 m depth. The machine vibrating at a speed of 250 rpm generates Maximum vertical unbalanced force = 2.5 kN Torque about Z-axis = 4.0 kN-m Maximum horizontal unbalanced force = 2:0 kN at a height of 0.2 m above the top of the block. The machine weight is small in comparison to the weight of foundation. Limiting amplitude of the machine is 150 microns. A block resonance test was conducted at the site to evaluate the dynamic elastic constants. The data obtained from the test is the same as given in Example 4.2 (Chapter 4). Determine the natural frequencies and amplitudes by (a) Weightless spring m!=thod,and (b) Elastic half-space approach. Solution: 1. Machine data (Fig. 9.38) 2.5 Sin G.>t 2.0 Sin CiJt
Machincz
4.0 Sin c..>t :.~...
e.g..
h = 3.5 m
2.om
1
11--
Block
T
.,.
T
b =3.0m
L = 1.75m
a =".Om (a)Section
"
1
~
J-
-".,.-- -
-,..~-, -- -
a = 4.0 m (b) Plan
I"" 0" f. Y"", Fig. 9.3S':, Details oUoundatlon
..
l -} ..,!' ,,(. ~l .~1
, ,--- ,I
F oundatitJnsuf
409
R:ecip,octlting Machines
Operating speed of machine =500 rpm
"
'.
Weight of machine is small and can be neglected. Fz = 2.5 sin Cl)t kN Mz, = 4.0 sin rot kN ,m Fx = 2.Dsin rot kN acting at a height of 0.2 m above the top of block 21t ro = 250 rpm = 250 x 60 = 26.2 rad/s 2. Dynamic elastic constants Refer example 4.2. The soil data and the size of the actual block are same as in that example. The procedure of determining the values of dynamic elastic constants for analysing the block foundation is illustrated there. Therefore, following values of dynamic elastic constants may be adopted. Cu = 3.62 x 104kN/m3 G = 1.10 x 104 kN/m2
E = 2.98 x 104kN/m2 f.1= 0.35 C 3.62 C = ---!!.= x 104 = 1 81 x 104 kN/m3 t
2
.
2
= 3.46 Ct = 3.46 x 1.81 x 104 = 6.26 x 104 kN/m3
C41
C", = 0.7S'Cu = 0.75 x 3.62 x 104= 2.71 x 104kN/m3
3. Foundationdata.
"
Let the block is casted in M20 concrete. The unit weight material is taken as 24.0 kN/mJ~ Weight of block = 24.0 x 4.0 m x 3.0 x 3.5
= 1008 kN 1008
2
m = 9.81 = 102.8 x 10 kg Area of foundationbase = 4.0 x 3.0 = 12.0 m2 In further analysis and design of foundation, the depth of embedment is neglected. 4. Linear weig,htles'sspring app'roach (a) Vertical Vibration
-;;;H
A- -
u
(Onz
=
3.62 x 104 x 12.0 = 65 rad/s
102.8
Fz (AZ>max
,
.r
,
,. .
= m(oo~ -002) .
\
'
~,-' .-:
2.5 ,
102~~J~?2
',."".:iI...'-'::
-.2~.22J'
=:=6.8 x. 10-6
".:'r~.~,l'S"""
,
.
m r. = 6.8 microns .'
. 410
SoU Dynamics & Machinl! Foundatio;
(b) Torsional vibration ab
2
2
.4 x 32
2
4
= -12 (a + b ). = - 12 = (4 + 3 ) = 25 m
J
z
.
M
=!!!.. (i + b2) = 102..8x 103 (42 + 32) = 214 x 103 Kg-m2 12 . . 12
(0
-
mz
nw -
--
M mz iC" J, -
- 56 .ras 2 d/
-
. 2.71 x214 104 x 25
(c) Coupled rocking and sliding
=
(0 nx
(On
re;;:=
V-;;-
1.81 x 104 x 12 = 46 rad/s
102.8
Mmo -- ~C.I-WL
I = ba3 = 3 x 43 = 16 m4 12 12. W = 1008 kN L = 1.75 m Mmo = Mm+ m L2 M ~!!!.. m 12 Therefore
Mmo OOn d.
(i
+ h2)
= 102.812x 102 (32 + 3.52) = 182 x
103 kg-m2
= 182 x 103+ 102.8 x 102 x 1.752 = 496.8 x 103kg-m2 = 6.26 x 104 x 16-1008 x 1.75 = 44 8 d/ . ra s
496.8 M 182 r = --!!L = = 0.366 Mmo 496.8 't'
2 (J) 11\,2
-
1
- -2r
[(
V
2
2 2 +" (J)n.\"+ (J)IIcp -
2 2
) ((J)ncp+ (J)nx) - 4 r
2
2
(J) ncp (J) n.\"
]
(462+44.82)2 -4 x 0.366 x 462 X44.8~ .I
(462+44.82):t = 2 x 0.366 1 . [
= 33.8 rad/s and oon2 = 100.6 rad/s
!J.(00 ) = m Mm
= = My
102.8
2 (J)nl
2
2
.'.".'
2
..
( - (J) )(Cl)n2 - ) X
Cl)
182 (33.82 - 26.22) (100.62 - 26.22)
102.8 x 182
X
] ,cot I
= ~ 3283] 0.732 [4123:t . OOnl 2
4~
456 x.9434
= 8.05
= 2 (0.2 + 1.75) =~.~,'kN'ni":,
X
1010 kg2 m2
, "..;~.I
------------
lundations
of Reciprocating
=
A
411
Machines
(C411- WL+C-rAL2 -Mmro2) Fx+(C-r AL) My
x -
=
~ (ro2) (6.26 x 104 x 16-1008
x 1.75+1.81 x 104 x 12x1.752_182 8.05 x 1010
x 26.22) x 2.0
4
1.81x 10 x 12 x 1.75x (3.9) + 8.05 x 1010
= 3080158+ 1482390 = 56.6 x 10-6 m 8.05 x 1010
A$ =
=
2 ) My
(C-rAL) Fx + (C-rA-mro ~ (ro2) (1.81x10
4
x12x1.75)x2.0+(1.81x10 8.05 x 1010
4
x12-102.8x26.2
= 760200+ 571876 = i6.5 x 10-6 rad . 8.05 x 1010 Hence,
a -6 4.0 -6 A = A + - A = 6 8 x 10 + x 165 x 10 v z 2 ~. 2' 0'
= 39.8 x 10-6 m = 39.8
microns
1/
,
= Ax + h' A$ = 56.6 x 10-6 + .1.75 x 16:5 x 10-6 = 85.47 x 10-6 m = 85.47 microns
Ah
5. Elastic half-space approach (a) Vertical vibration
=
r 0
fA =
V;
ru. = 1.95m
V-;
Average effective unit weight of soil = 20+ 10 = 15 kN/m 3 2 1008
-'
B = 1-~.m3 4
Z
=
K Z
pro
= 1-0.35 4
W
2-.
( 9.81 )
= 1.47
x 1.953
4Gro = 4 x 1.10 x 104 x 1.95 = 13.2 x 104 kN/m 1-~ 1-0.35 0.425 ' 0.425
~z. =-
t.
x
'
":= .
B z °0-..J.195 . :JB: ,t '
= 0.304
2
)3.9
412
Soil Dynamics & Machine Foulldations
CO nz .~Kz
-
m =~
13.2 x 104 = 35.8 rad/s 101.8
Fz
A,
k{{l-( :Jf
~
2.5
13.2
x 1o~[{I -
= 29.4
x 10-6 m
~
r
+(:~,~:J
rr
G::~
+ (2 x 0.304 - ~~:~
r
r
(b) Torsional vibration 1/4
r = 0
M mz
\14
ab(a2+b2) 67t
[
= 214
]
3
4x3(42+32)
=
[
2
x 10 K g -m
67t
= 1.9973 m
]
B = Mmz = 21~ x 9.81 = 4.40
p,;
\jI
15x 1.99735
16
~
3
16
.
4
3
4
= "3 G,o ="3 x 1.1x 10 x 1.9973 = 46.75 x 10 kNmlrad
X,"= T
co -nljf
0.5
0.5 = 1 2 4 4 = 0.051 1+ 2 BIjf + x .
~
-- Ijf M mz
~
46.75 x 104 -- 46 .7 rad/ s 214
9.9.10. Coupled sliding and rocking vibration Sliding
l ~
r = - = 7t
0
K = x
32(I-Jl)Gr
x3
7t 0
7-8Jl
= 1.95m =
4
32(1-0.35)x1.lxl0
x1.95
7-8 x 0.35
= 10.62 x 104kN/m B = 7-8Jl x 32(I-Jl)
.-m
= 7-8xO.35
pr;
32(1-0.35)
x
1008 15x1.953
= 1.83 ~ ~x
- 0.2875-: 0.2875,- 7B: - .J1.83 - 9.212
, '1 t';
..' ~" I
r;'
oundations
of Reciprocating
413
Ma.clJines
.
fk:: = /10.62 x 104 = 32.1 rad/s
-
mnx - I V-;;
(
ab
-
Rocking
V
102.8
3 1/4
ro - \. 3~
)
1/4
4.0x33
(
~
]x)
~
1.84
ID
K = 8 G r; = 8 x 1.1X 104 x 1.843 -
3(1-/l)
cjI
3(1-0.35)
4
-28.1x10kNm/rad
BcjI= 3(1-/l) x Mmo = 3(1-0.35)x496.8x9.818 Pl~ 8 15x1.845 -3.75 0.15 ~=
'" n
0.15
(l+B<\»jB;
J MK,1110
= (1+3.75).J3.75 = 0.016 496.828.1 x 104 = 23.7 radJ, .
=
9.9.11. Coupled Vibration. Undamped natural frequencies in couples rocking and sliding are given by 1
2
2 IIX
=
2
2
h [(m
m111,2 =
2
2
2
2
IIX
mmp]
(mnx + m ) - 4 r m
) + mn
n
1 2 x 0.366 [ (32.12+23.72):t
(32.12+23.72)2 -4XO.366X32.12X23.72 "
]
=~ 0.732 [1592:t 1299] (On1
~
= 20 rad/s anda mn2= 62.8 rad/s
( (co;.:co~)
.
r
4~x~.:"' co,.}+ co;',co;. 2 1/2
+4
{
(
~. mnx m 002 -m2 .\
r
II<\>
00 m2 -002 )+ ~<\>mn<\> ( n.\ r
]
)}
2 ~ «(02)
26.24 -26.22
= [{
1592 - 4 x 0.212 x 0.016x 32.1 x 23,7 + 32.12 X 23.72
( 0.366
)
0.366
0.366
2 1/2
.
+4
x 26.2 (23.72- 26.22)+ 0.016 x0.366 23.7 x 26.2 (32.12 - 26.22) } 0.366 { 0.212 x 32.1 ] :",,';
[
}
'.
,;-
.
.-
= (471200-2966467+ 1581341)2+4 (-60772+9337)2
.
In
] = 919697
.
" l
414
SoU Dynamics & Machine
Foundations ~;; ~'
A
~
~
xl
+K. +Kx L,)'12+4"'(~, ~K. Mm.+L' ~x~)'rl2
{(-Mm'"
mMm
=
~({i) 2
2.0 102.8 x 182 x 919697
[ (-182
x 26.22 + 28.1 x 104"+10.62 x 104 x 1.752) 2 1I2
+ 4 x 26.22 (0.016~28.1
=
x 104 x 496.8+1.752
x 0.212~1O.62
0 2 1/2
2
= <1>1
]
2.0 (2.3165x 1011+0.1496 X 1011)1/2 102.8 X 182 919697
= 58 x 10-6 ID
A
x 104 x 102.8)
~'(
.
M
( L COnx COnx+4
~x CO
)
~(CO2 )
111
.\::
r
1/2
")
(
32.1 321-+4XO.212X26.22) 919697.
= 2.0X1.75 182
l = 26.9
)
x 10-6
fad
'
1/2
A
-
My [ co~'(+(2~xconx)2 -
x2 - Mm
]
;'" .;
~ (co2) 2
= 3.9 [ (32.1) +(2xO.212x32.1) 182
= 0.81
,,'",:iJ
2 1/2 ]
:;;
t11
919697
x 10-6
';::;t
;~
ID
t~
1/2
2
~;J
'(\"
"j
- My [ (co~x-co2) +(2~xconxco)2 ]
A(\I2-
-
Mill
2
~ (co ) 1/2
2
3.9 [ (32.12_26.22) 182 = 8.3 x 10-6 fad
+(2xO.212x32.1x26.2)2
]
919697
Ax = A.d + Ax2
= 58 x 10-6+ 0:81 x 10-6= 58.81 x 10-6in A4I= A4II+ A4I2= 26.9 x 10-6+ 8.3 X 10-6= 35.2 X 10-6fad
ft~11 ;"
i~
,
,
'~",.1 iij;';i
-
..
III
~..~" 415 .
{}undations .of Reciprocating- Machines
Hence
.
a
= A + -2 An.= 29.4 x
A
4~
Z
y
= 99.8 Ah
-6
x 10
-6 4.0 -6 10 + - x 35.2 x 10
2
m
= Ax + h' A, = 58.81 x 10-6+ 1.75 x 35.2 x 10-6 = 120.41 x 10-6 m
It may be noted the there are significant differences in the magnitudes of natural frequencies and mplitudes computed by the two approaches. It may be due to the reason that the value of shear modulus J is computed from the block resonance test data using the relation between Cu' E and G. Actually in lastic half-space theory, it is desirable that the value of G is obtained from wave propagation test. Author's xperience indicates that the value of G obtained from wave propagation test is much higher than com'uted from block resonance test data. Use of appropriate value of G may bring the results of the two pproaches closer. :xample 9.2 )etermine the natural freque~cies and amplitudes of motion of the foundation (Example 9.1) taking into tccount the embedment effect and apparent soil mass. , ,olution : ~
(i) Assume that the values of dynamic elastic constants at a depth of 2.0 m are 20% higher than the values at the surface of ground. Therefore CuD = 1.2 x .3.62 x 104 = 4.35 x 104kN/m3 Ct D = 1.2 x 1.81 x 104 = 2.18 x 104 kN/m3 , CD
= 1.2
CIjID
= 1.2 x 2.71 x 104 = 3.25 x 104 kN/m3
x
6.26
x
104=7.si
x
104 kN/m3
The average values of dynamic elastic constants will be C,./aV = 4.0 x 104 kN/m3 Ctav
-- 2.0
x
104 kN/m3
Ccpav= 6.89 x 104 kN/m3 Cljlav = 3.0 x 104 kN/m3 D = 2.0 m
(ii) Vertical vibration Kze
= CuD x A
+ 2 Ctav (bD + aD)
= .4.35 x 104 x 12 + 2 x 2.0 x 104 (3 x 2 + 4 x 2) = .108.2 x 104 kN/m ..
24 x 3.5 he
=
.
15
= 5.6 m
ah
1.0x5.6
S = -:- =
3.0
a
= 1.867; r=
4
b = 3" = 1.334
416
Soil Dynamics & Machine Foundations
,
,
From Fig. 9.36 a for S
"C
'
= 1.867 and r = 1.334;-!!!. r = 1.45
or
Cm = 1.45 x 1.334 = 1.93
Therefore
m =-
,
'Y b3
S
ag
C = m
15 x 33 1.0x9.81 4"-
(m+ms)
=
-
= 75.8 rad/s
108.2 x 10ffinze= VI02.8+79.7 Fe A == 2 ze
3
x 1.93 = 79.7 x 10 Kg
2
(O)nze-O)
)
2.5 (102.8+79.7)(75.82_26:22) =2.70 x 10-6m
(iil) Coupled vibration Kxe
= C'tD . A
+ 2, Cuav bD + 2 Ctav aD
= 2.18 x 104 x 12 + 2 x 4 x 104x 3 x 2 + 2 x 2 x 104x 4 x 2
= 106.16 x 104 kN/m From Fig. 9.36 b, for S = 1.867 and r = 1.334,
=
M
'Y
b5
CX
m.'Cs 12ga
~r
15x3
=
i
= 17.72 kNms
= 0.43
2
5
0.43 x 1.334
12x9.81xl
Mmos= Mmxs+ ms x L2 = 17.72 + 79.7 x 1.752 = 261.80 kN ms2 Cl)
=
nxe
~
xe
m+m s
C K~ = Cd.D 1- WL + ~
.b
,av 24 3
= /106.16x 104 = 76.2 rad/s ~102.8+ 79.7 "
Db (16 D3 - 12 hD2) + 2 C$av x 10+ Ctav x_a 2 3
2
'
I=ba
=3x4 =16m4 12 11 W = 24 x 3 'x 41<.3.5 = 1008 kN 3 4 2} , I = aD = ~ ='.10.67 m4 0 3 3 '-, -~ ;~ ;f J :J-j
w,~'!!' i."
417
Foundations of RecfprocGtingMachilies.. 6 89 K$e
. 04
= 7.51 x 104 x 16 - 1008 x 1.75 + .
x24 I.
.
x 3 (16 x 23 - 12 x 3.5 x 22) .
2 + 2 x 6.89 x 10.67 + 2.0 x 104 x 2 x 3 x 4
2
= 120.16 x 104-0.1764
x 104-34.45
-
x 104+ 147 x 104+48
x 104
= 280.53 x 104 kN/m 4
(J)mj> e
280.53 x 10 182 + 261.80
K$e
=
+ Mmos
Mmo
= 79.5 rad/s Kxx = CtD x A + 2 Cuav bD + 2 Ctav aD = 2.18 x 104 x 12 + 2 x 4 x 104 x 3 x 2 + 2 x 2 x 104 x 4 x 2
= 90.16 x -104kN/m 2
Kx$
= CqJavb (D - 2DL) = 6.89 x 104 x 3 (22 =-231.81
x 104kN
3
2
I. = D a + a Db y 12 4
K$$
= C
CtD AL 2 x 2 x 1.75) - 2.18 x 104 x 12 x 1.75
=
~ 12
3
2
+ 4 x 2 x 3 = 28.67 m4 4
CtD AL 2 - WL + 2 C1jfOl'Iy + Ctav b
~ a2
+ ~ C
x 104 x 16 + 2.18 x 104 x 12 x 1.752 - 1008 x 1.75 + 2 x 3 x 104 x 28.67
+ 2 x 104 x 3 x 22x 42 + ~ x 6.89 x 104 [1.753 + (2.0 - 1.75)3] = (120.16 + 80.11 - 0.17 + 172.02 + 96 +24.69) x 104
= 512.81
Õ±òô
x 104kNm
¢
óÅÝôÜ ßÔõî C"", bD
~
-[2.18
x
104
x
12
( L- ~ )+2 C"" ( L- ~) aD]
x 1.75+2 x 4 x 104 x 3" 2 (J.75-~
= -[45.78+51.84+34.56] 104 = -132.18 . "
~
x 104 kN
)+2 x 2 x 104 (1.75-~)4
x 2]
'(.
418
SoU Dynamics & Machine Foundations
{KcMI-(Mm +Mmxs) Cil} Fx AXl=
2 2 0) }{ KcMI-(Mm +Mmxs) 0) } -Kx4l'
{ Kxx -(m+ms)
=
.1'
K~
{512.81X 104_(182+17.72) 26.22} 2.0
---
{90.16 x 104 - (102.8 + 79.7) 26.22} {512.81 ~<104,-(182 + 17.72) 26.22 }
[
-(-231.81 X 104) (-132.18
]
X 104)
4
998.18 x 10
998.18x10
= 77.64 x 104 x 498.3 x 104- 30639.32 x 108 = 8048.69
4
x 108
-6
= 12.40 x 10 m A
-K
= cpl
{Kxx -(m+ms).0)2}
~
.F
. x
{Kcpcp -(Mm + Mmxs) 0)2}-Kxcp' K~
= 132.18 X 104 X 2.0 8048.69 X 108
= 3.2 Ax2
x
10-6 fad - Kx . M.v
=
~
2
{ Kxx
- (m + ms) 0) } { Kcpcp - (M m + Mmxs )
231.81x104x3.9
=
.
8048.69 x 108
11.23 x 10
Kxcp . K~x
c:
m
,: ;J :,' ""',I
0)2} My 2
2 { K,n,-(m+ms)O)
-
-6
=
{Kxx -(m+ms)
Acp2 =
2 0) }
} { Kcpcp-(Mm+Mxs)O)
4
,
} -Kxcp K~x
2
- [90.16 x 10 -(102.8+79.7) x 26.2 ]x3.9 8048.69 x 108
Ax
= 3.7
x 10-6 fad
= Axl
+ Ax2
= (12.40 + 11.23) x 10-6 = 23.63 x 10-6 m
Acp = A4I1+ Acp2= (3.2 + 3.7) x 10-6 = 6.9 x 10-6fad
Displacement of the top of the block
= Ax + (h - L) AqI = 23.63 x 10-6 + (3.5 = 35.70 x 10-6 m
- 1.75) x 6.9 x 10-6
'~; it ~. r, ,.," [, ': ~ 1,'1 fJ 'j
",
J;:
419
Foundations of Reciprocating Machines
REFERENCE Anandakrishnan, M. and Krishnaswamy, N. R. (1973a), "Response of embedded footings to vertical vibrations", J. Soil Mech. Found. Div. , Am. Soc. Civ. Engg; 99, pp. 863-883. , '
Arnold R.N. , Bycroft, G.N. and Warburton, G. B. (1955), "Forced vibrations of a body on an infinite elastic solid", Trans. ASME, 77, pp. 391-401. Balkrishna, R. H. A. (1961). "The design of machine foundations related to the bulb of pressure", Proc. Int. Conf. Soil Mech. Found.' Eng., 5th, Paris, Vo\. 1, pp. 563-568. Baranov, V.A. (1967), "On the calculation of excited vibrations of an embedded foundation (in Russian)". Vopr. Dyn. Prochn. . 14 pp. 195-209. darkan. D.D. (1962). "Dynamics of base and foundations", McGraw-Hill, New York. Bereduge, Y.O. , and Novak, M. (1972). "Coupled horizontal and rocking vibration of embedded footings", Can. Geotech. J. , 9(4), pp. 477-497. ,
,
Beredugo, Y.O. (197 I), "Vibrations of embedded symmetric footings", Ph. D. Thesis, University of Westem Ontario, London, Canada. ', Bowles. J. E. 91982), "Foundation analysis and design" , McGraw-Hill, New York. Bycroft, G.N. 91956), "Forced vibrations ofa rigid circular plate on a semi-infinite elastic space and on an elastic stratum", Phi10s.Trans. R. Soc. London, Ser. A, 248, pp. 327-268. Chae. Y. S. (1971), "Dynamic behaviour of embedded foundation-soil system", Highw. Res. Rec., 323, pp. 49-59. ,. Fry, Z. B. 91973), "Development and evaluation Sta., Tech. Rep. No. 3-632.
of soil bearing capacity, /, '
Foundation
of Structures",
Waterways
Exp.
Gupta. B. N. (1972), "Effect of foundation embedment on the dynamic behaviour of the foundation-soil system", Geotechnique, 22 (I), pp. 129-137. '
Hall, J. R. (1967), "Coupled rocking and sliding oscillations of rigid circular footing", Proc. Int Symp. Wave Propag. Dyn. Prop. Earth Mater, Albuquerque, NM, pp. 139-148. Hsieh, T. K. 91962), "Foundation vibrations", Proc. Inst. Civ. Eng., 22, pp. 211-226. Lamb.H. (1904), "On the propagation of tremors over the surface of an elastic solid", Philos. Trans. R. Soc. London, Ser. A 203, pp. 1-72. Lysmer,J. and Richart, F. E. , Jr. (1966), "Dynamic response of footing to vertical loading", J. Soil Mech. Found. Div., Am. Soc. Civ. Eng., 92 (SM-I), pp. 65-91. Novak, M. (1970), "Prediction of footing vibrations", J. Soil Mech. Found. Div. , Am. Soc. Civ. Eng., 96 (SM-3), pp. 836-861. ,
'
,
Novak, M., (1985), "Experimeflts with shallow and deep foundations". Proc. Symp. Vib. Prob\. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., pp. 1-26. Novak, M.. and Beredugo, Y.C. (1971), "Effect of embedment on footing vibration", Proc. Can. Conf. Earthquake Eng. ,1st, Vancouver, pp. 1I 1-125. ,
Pauw,A. (1953), "A dynamic analogy for foundation soils system", ASTM Spec. Tech. Pub!., STP, pp. 3-34. Reissner,E.(1936),"StationareAxialymmeterischeduTcheineSchuttelndeMaSseErregteSchwingungeneine Homogenen ,Elastichen Halbraumes", Ing. Arch. 7(6), pp. 38I-39?,"" ',', Rc'ssner, E. (1937), "Freie underzwungene Torsionschwing-ungen des Elastiche~ Halbraumes", Ing. -Arch, 8 (4). i~,<,..,.!:;.; ...,C,": "",'" ,.(",.L.I>,..',~' , pp.
229-245.
'.':, ,
,
\.< !
"','.{'
,
I
".
,..FfAoJ'E:<:l!...\ S
} '' . .
, ":~ , .
.420
Soil Dynalllics &. Machine
Foundations
Reissner~.E.andSagoci, H. E.(1944),~'Forcedtorsional oscillations.ofan.elastichalfspace", 1. Appl. Phys., 15, pp. 652- 662. Richart, F. E. Jr. (1962), "Foundation vibrations", Trans. Am. Soc. Civ. Eng., 127, Part, I, pp. 863-898. Richart, F. E.)r. ,and'Whitman, R. V. ('1967), "Comparison of footing vibrations t~sts with theory", J. Soil Mech. Found. Div., Am. Soc. Civ. Eng., 93 (SM ~6), pp. 143-168. Stokoe,K.H.II (1972). "Dynamicrespo~seofemb'~ddedfoundations",Ph. D. thesis presented,Universityof Michigan, Ann Arbor, - Michigan. . . Stokoe, K. H., 11,and Richart, F. E. Jr. (1974), "Dynamic response of embedded machine foundation",1. Geotech. Eng. Div., Am. Soc. Civ. Eng. , 100 (Gt-4), pp. 427-447 . Sung, T. Y. (1953a), "Vibrations in semi-infinite solids due to periodic sutface loading", Ph. D. Thesis, Harvard University, Cambridge, Massachusetts. Vijayvergiya, R. C. (1981). "Response of embedded foundations", Ph. D. Thesis, Univeristy of Roorkee, Roorkee, India. Whitman; R. C. and Richart, F. E., Jr. (1967) , "Design procedures for dynamically loaded foundations", J. Soil Mech. Found. Div., Am. Soc. Civ. Eng., 93 (SM - 6), pp. 169-193. .
PRACTICE
PROBLEMS
9.1 (0) List the basic differences in analysing a reciprocating machine foundation by the two clp proaches namely (i) Linear weightless spring-mass system, and (ii) Elastic half-space theot) (b) Derive the expressions of natural frequency and amplitude of a block foundation subjectel to vertical vibration. ,',) 9.2 Starting
from fundamentals, derive the expressions of natural frequencies and amplitud~~l'
block foundation subjected to a horizontal force Fx sin rot and a moment My sin rot at the COlT bined e.g. of machine and foundation. ,~i 9.3 A concrete block shown in Fig. 9.39 is to be used as a foundation for a reciproc~ting e~fi U operating at 500 rpm and mounted symmetrically with respect to foundation. The weight O' ,
~ '
.
.'.
.
.
.
..
.
.
engine is 10kN. It is ,likely that the operation of machine exerts the following:
. t~
Unbalanced vertical force = 1.8 sin rot kN .;~ Unbalanced torsional moment = 6 sin rotkN I "/~ 1;1't. The values of the dynamic elastic constants for the design of the foundation may be adop'! given below: . )ni.1. CII
= 4.5
x 104 kN/m3
G = 4.8 x 104 kN/m3 J.1.
= 0.34 3
17 kN/m 3 Yconc. = 24 kN/m amplitUdes~f the biock by (i)Lin~ar elasticispj . Determinethe namralfrequenciesand -' "~..; Ysoil~
:.approach and (il) B~stic half space theory.
.
-' - . -
.-,
.:'~
o"ca; c: ,
, , ",,_.
421
"Ol~~i:;ptoCtlii,.g ~ac{,,",a
Fila1IliItfdils "
- .,
.
, "
,
,
"T' -,2,(0 1ft.
-0
.0
,'" -
-0
,'-,
"
,,1
.,'
8.0 m
~
1..'Q.5m T-
Elevation
T
It..Om ,.
1
','
Plan Fig. 9..~9: Details of foundation
9.4 Design a suitable foundation for a horizontal compressor driven by an electric motor. The following data are available: , '
/
,
Weight of compressor, = 160 kN Weight of motor = 60 k;N , Speed of compressor = 250 rpm Horizontal unbalanced' force , =- 75 kN acting at a height of 1.0 m above top of pedestaL , The following tests were performed at the site of determine the values of dynamic elastic constants :
(a) A verticalvibrationtest was conductedon an M-I50 concreteblock 1.5 x 0.75 x 0.7 m high, using different eccentricities. The data obtained is given in Table 9.8. Table 9.8
...
SI.No.
Angle, Of setting of . .'. eccentric masses
1,,: Hz
I 2 3 4
35.0 ,32.0 .31.0 29.5
5
15 30 45 60 120
6
1~9
I
28.0 -,
-'
.,:27.0
Amplitude at resonance mm 0.06375 0.150 0.210 0.30 0.525 0.620
422.
S,iI DyntUllics &. Machine .
FollndtUWns ,., ,
(b) A cyclic-plate load test was.done on plate 300 mm ~.300 mm. The elastic settlement corresponding to a load intensity of 250 kN/m2 was 6.00 mm. . (c) A wave -"propagation test gave an average value of travel time of compression waves as 0.02 s, corresponding to a distance between geophones of 6 m. The water table at the site is 2.0 m below the proposed depth of the foundation 3.0 m. The soil at the site is sandy in nature.
DD
. ~~ft
1 ::1 '.
~
,
"
,
'
.
,
FOUNDA TIONSOF IMPACT TYPE MACHINES
10.1 GENERAL Impact type machines produce transient dynamic loads of short duration. Hammers are most typical of impact machines. A hammer-fouridation-soil system consists of a frame, a falling weight known as 'tup', the anvil and the foundation block. Figure 8.6 shows a typical foundation for a hammer with its frame mounted on the anvil. In Fig. 10.1, a foundation for a hammer with its' frame mounted on the foundation is shown. ~,
-
Fig. 10.1: Typical arrangement ofa hammer foundation with A frame mounted on foundation <
,',
.. ",'~,'f\"
.. '"
'
' },
;'i~,:;~$~~~4i.~,,'i.t!t:;..!,:~j'.;: ", "',7;
';,';
~',)' ,c"
i':(, ,'" ~,..,'
~
~, 0'"
',Lo-'
:
re
.\
424
','
Dy,"""ks&
Soil
'Mllc"inif'
F()Il1I~
The foundation of a hammer generally consists of a reinforced concrete block. 'Q1efollowing arrangements are used depending upon the size of the hammer: (i) For small hammers, the anvil may be directly mounted on, the foundation block (Fi~. IO.2a). This system can be modeled as single ,degree of freedom system as sh°W!lin Fig. lO.2b. ','
,~,',;"
,;:'::~',.:.\,.!','~
/':
. ..
..-
~~",.:"::':\:':',:,..",'~" "
'"
":':"',
",'c ;
,-
J'~
'9T::
1
,,
Tz1
~
'FOUNDATION, BLOCK.RESTING
ANVIL AND FOUNDATION
ON SOIL
'BLOCK"" '}
'~
(b)
(a)
F;g. 10.2 : Anvil resting directly on the foundation block
(ii) In medium capacity hammer, a vibration isolation layer is placed between the anvil and tl foundation block (Fig. lO.3a). Usually the isolation Jayer is an elastic pad consisting of rubbe felt, cork, 01 timber adequately protected against water and oil. In case of high capacity han mers, special elements such as coil springs and dampers may be used in place of elastic pa( (Fig. lO.3b). The systems shown in Fig. lO.3a and lO.3b can be modeled as two degrees I freedom system as shown in Fig. lO.3c. f.
~I
,
,
WI
. . .-
~
FOUNDA1'1 ON BLOCK RESTING, ON SOIL
-,:1 ~f&4 ,"~
(a)
.... t
(b)
Fig. 10.3 : Anvil resting on elastic pad/spring absorbers (...Contd.)
~,
"oundations
425
of Impact Type Machines
dJ~P DAMPING1N ANvrL
l ABSORBER
122
I
SPRINGk2 ELASTIC PAD
FOUNDATION Tzt BLOCK SOIL SPRING kl < ~
<
DAMPING IN SOIL (c)
Fig. 10.3 : Anvil resting on elastic pad/spring abs'!)rbers
(Ui) For reducing the transmission of vibrations to the adjoining machines or structures, the foundation block may also be supported on elastic pads (Fig. lOAa) or on spring absorbers (Fig. lO.4b). In such a case, the foundation is placed in a reinfor<;edconcrete trough. The space between the foundation and side of trough is filled up with some soft 'materials or an air gap is left. The systems shown in Fig. lO.4a and lO.4b can be modeled as three degrees freedom system as shown in Fig. lO.4c. The stiffness of trough (Fig. lO.4a) is very high compared to that of the pad below the foundation block, the trough may be assumed to be rigidly supported on the soil (Novak, 1983), and therefore a two degree freedom model (Fig. lO.3c) may give sufficiently, accurate results for all practical purposes. FOUNDATION BLOCK
FOUNDATION BLOCK
TROUGH
ELASTIc. PAD BELOW FOUNDATION BLOCK
SPRING ABSORBER BELOW FOUNDATION
BLOCK-:: TROUGH
"
.~'
.'~,. . (a),<
..
i
" (b)
"
,!1 ' . , Fig. 10.4 : Foundation block on elastic pad/spring absorbers (...Contd.)
.......
,.
,(
,'r!'
. 426
Soil Dynamics & M.achine Foun.d,,:tions
~TUP I
. ANVIL I
TZ)
I
SPRINGkJ OF PAD
DAMPING IN PAD
-.
FOUNDATION BLOCK
BELOW ANVIL.
TZ2
SPRING k2OF PAD BELOWTHE FOUNDATOO BLOCK
TZ1 SOtL SPRING k1
DAMPING IN SOIL (c) Fig. 10.4 : Foundation block on elastic pad/spring absorbers
In hammer foundations, tup, anvil and foundation are geometrically so aligned that their centres fall on one vertical axis. This will ensure that the loads act on the anvil and foundation without any eccentricity. 10.2 DYNAMIC ANALYSIS 10.2.1. Two Degree Freedom System. In general the anvil, pad, foundation, and soil constitute a tWo degree system as shqwn in Fig. 10.5. This model is based on the following assumptions: . .....,;'. (i) The anvil, foundation block, frame, and tup are rigid bodies. (ii) The pad and the soil can be simulated by equivalent weightless, elastic springs. (iii) The damping of the elastic pad and soil is neglected. (iv) The time of impact is short compared to the period of natural vibrations of the system. (v) Embedment effects are neglected.'
bO.
The notations used in the model have the following meaning:
,~
:r
ml = Mass of foundation and frame if the latter is mounted on the foundation, as in Fig. 'lO~t ml
= Mass of anvil (with frame if the latter is mounted on the anvil as in Fig. 8.6. ., ":~1$f
K) = C'u . AI = Equivalent spring constant
of soil under. consideration
425
r<'oundations of Impact Type Machines
dJ~P Tz2
DAMPING1N ANvrL
l ABSORBER
I
SPRING k2 ELASTIC PAD
FOUNDATION PI BLOCK SOIL SPRING kl '~ .
DAMPING IN SOIL (c) Fig. 10.3 : Anvil resting on elastic pad/spring abs'!)rbers
(iii) For reducing the transmission of vibratIons to the adjoining machines or structures, the foundation block may also be supported on elastic pads (Fig. lO.4a) or on spring absorbers (Fig. 10.4b). In such a case, the foundation is placed in a reinforc;edconcrete trough. The space between the foundation and side of trough is filled up with some soft materials or an air gap is left. The systems shown in Fig. lO.4a and lOAb can be modeled as three degrees freedom system as shown in Fig. 10Ac. The stiffness of trough (Fig. lO.4a) is very high compared to that of the pad below the foundation block, the trough may be assumed to be rigidly supported on the soil (Novak, 1983), and therefore a two degree freedom model (Fig. 1O.3c)may give sufficiently. accurate results for all practical purposes. FOUNDA. TlON
BLOCK
.,..
FOUNDATION BLOCK
'..'
,.
ELASTIc.
PAD BELOW FOUNDATION BLOCK
.:. ,-.' :
'8.
I'
TROUG H
SPRING ABSORBER BELOW FOUNDATION
B LOCK,:~ ""
.,..(a)r', I.. . ,
,.,""
.
.,"-.' ., . .,. . ,. ,,(b) . Fig. 10.4: Foundation block on elastic pad/spring absorbers (...Contd.)
........
,.
-~
ill
427
undations of Impact Type Machines Cu' =
A Cu = Modified coefficient of elastic uniform compression, to t,,;keinto account impact condition which is different from periodic loading
t.. = Multiplying factor that governs the relationship between Cu and Cu' usually 1-2, for impact depending upon the soil type Cu =
Coefficient of elastic uniform compression
-
K2 = (E / b) x Az = Equivalentspring constantof the pad under the anvil
E = Young's modulus of the pad material b = Thickness of the pad
Al A2 z\ z2
= Area of foundation in contact with soil = Area of the pad = Displacement of foundation from the equilibrium position = Displacement of anvil from the equilibrium position
.
TuP--cb
ANVIL SPRING K2 FOUNQATION ,,"
m1
:~, SPRING~1 I
.
Fig. to.5 : Two-mass-spring
BLOCK
analogy for hammer foundation
IO.l.I.I. Natura/frequencies. The equations of motion in free vibration are mlz\+Klzl+Kz(Zt-zz)
...(10.1)
=0
...(10.2)
~ 2Z+ Kz (zz --zl) = 0 The solution of the above equation can be written as Let,
zl = A sin oont
...(10.3)
and,.
zz=Bsinco"t.
...(10.4)
where A and.B are arbitrary constants. Substituting the values ofzl and zz fro~ Eqs. (10.3) and (10.4) in Eqs. (10.1) and (10.2) respectively, we get
z
.!! = .Kz+ Kt -1nl con A KZ
...(IO.S)
..
428
Soil Dynamics & Machine Foundations.
and,
B
K2
A
K2 -"'2 con
- =
...(10.6)
2
Equating Eqs. (10.5) and (10.6), on simplification we get 4
OOn-
K2 (ml +"'2)
{
ffi:-
(
l+~ ml
KZ+
)( "'2
KI 2 Kl K2 -- 0 +-COn+ ml } ml "'2
ml "'2
Kl
ml+"'2 )
00:+K2x
KI
"'2
ml +"'2
(
1+"'2 =0 ml)
Let,
ffina= Circularnaturalfrequencyof the foundationof the anvilon the pad
.".
"'na
..
~
~~
(J)nl
...(IO.S) = Limiting natural frequency of the foundation and anvil on soil
OOnl
=
~
I
...(10.9)
ml+"'2
"'2 Ilm
= -;;I
.
Substituting the values of OOna'oonland Ilm in Eq. (10.7), we get 4 OOn -(1
2 2 +Ilm) (OOna+ffinl)
2 COn+(1+llm)
2 2 OOnlOOna =
...(10.10 a)
0
r 002
= .!.I ,
nl,2
2
2
+
2l~I+llm)(COna+OOnl)-
. 2
2
{(1+llm)(OOna+COnl)}
2
2
2
-4(1+llm)OO,IiOOna
...(10.10 b)
]
,,',I
The two natural frequencies of the hammer foundation may be determined by solving the above equations,
10.2.1.2. Amplitude of vibration. The generalsolutionof the Eqs. (10.1)and (10.2) is given by z, = AI sin ronlt +A2 cosronlt +AJ sin ron2t +A4 cosron2t ...(10.11)\ Z2
=
BI sin ronl t + B2 cos (0nl t + BJ sin (0n2 t + B4 cos (0n2 t
...(10.12)
where, AI' A2' AJ' A4' BI' Bz, B3' B4 are arbitrary constants. If system is vibrating at frequency oonl'then from Eq. (10.6)
~
,,1
-
A Similarly
when con = con2
~ -
K2 K2 -"'2 CO2 nl
='
(02 na 0)2 2 = al (Say) na - 0)nl
"
...( 10.13arJi")~ .
.
..
~t;;J ;"c .
...
2 (Ona
A - (02na _(O2=a2(Say) n2 ."i.", IJ',J;~4
rdatioIJsoflmpa.ct Type,Machi~es
429
,
It may beno~e~,that values ofal andaz are known from Eqs. (10.13 a) and (10.13 b) respectiv.ely. r.racting az from a I we get
. 2
Cl) nl
2
Z
(
al
- az =
(
Cl)
2
-
Cl)
Z
- Cl)nZ ]
Cl)na
Z
)
nZ Cl)na
Z
na
1 2
2
.
[ Cl)na Cl)n I
or
.
1
z.
.
a I - az = Cl)na
.
2'
...(10.14)
2
- Cl)nl )(Cl)na - Cl)n2 )
.Equation (10.12) can therefore be written as : .~
= al
...(10.1S)
AI sin Cl)nlt +al Az cos Cl)nlt +az AJ smCl)n2 t +az A4 COSCl)nZ t
boundary condition:
(i}At t = 0, ZI = Zz = 0 and It gives
Az + A4 = 0
...(10.16a)
a I Az + az A4 = 0 A Z '= A 4 = 0
...(10.l6b) ...(10.17) , ,
(ii) At t = 0, zl = 0; Zz= Va (Velocity of anvil) Zl
= AI
Cl)nl cos Cl)nl
or
t - Az ronl sin ronl t + AJ ronZ cos ronZt - A4 ronZ sin ronZ t
AI (Onl + AJ (OnZ =
or
0
Cl) A J = - A I- ni
...(10.18)
. Cl) nZ 2'z = al Al Conl cos ronl t - al Az Cl)nl sin Cl)nl t + az AJ Cl) nZco£~ nZt - al A4 Cl)nZsin Cl) nZt
= al
or
Va
or
A =
and
A = a 3 (al -a2) Cl)nZ
I
AI (Onl + az AJ (OnZ
Va
...(10.19)
(al -az) Cl)nl
-v I
Therefore, Putting the value of(a,-
Z
. Va
='
.
SIn ronl
Va sinCl)nzt
I (a\-az) Cl)nl (al -az) Cl)nZ az) from Eq. (10.14) in Eq. (10.21), we get
(
Cl)nl-CI)n2
Similarly,
. ...(10.21)
~
ZI = (CI)~a-CI)~I)(CI)~a-CI)~z) ZZ Z
)
Cl)na
sinCl)nlt - sin Cl)n2t V Cl) Cl) ] a .[ nl nZ
(CI)~a-CI)~2)SinCl)nlt
.~
(
2
Z
=. . .Cl)nl-(J)nZ. 1
)[ .
./(10.20) I
Cl) nl
...(10.22)
v
(CI)~a-CI)~I)Sinron2t CI)' nZ
a
] .,'
. ...(10.23)
Field observation (Barkan, 1962) of the amplitudes of the anvil and the foundation showed that the ibrations occured at the lower frequency only. Therefore, it may be assumed that the amplitude of moion for sin (Onlt= 0 «Onl > (OnZ)'
430
Soil Dynamics & Machine Follndations
Hence approximate expressions for maximum displacement will be as follows (sin Cl)n2t = I):
ZI
=
z2
=
(OO;a-00;1) (OO;a-00;2) . Va 2 2 2 OOnaOOnl-OOn2 OOn2
)
(
" .. r:
...(10.24}
(OO;a-00;1) Va 2
...(1 0.25}
2
(OOnl -OOn2 )OOn2
.
10.2.2. Single Degree Freedom System. Sometimes in the case of light hammers, no pad is used between anvil and foundation. rhe system can then be represented as single degree freedom system (Fig. 10.6). In this case the equation of vertical free vibrations of the foundation will be .
m i + Kz = 0
...(10.26)
where, z = Vertical displacement of centre of mass of foundation and anvil, measured from equilibrium position m = Total vibrating mass, K =C' u A 1 TUP--m
m1
SPRING K, ..'
Fig. 10.6: Single-mass-spring
analogy for hammer foundation
The natural frequency of the system will be given by
Cl) nz
=
~
The Eq. (10.26) is the equation of free vibrationsof the foundationwithoutdamping.ThegeJi~' solutionof this equationis :(~~. z = A sin COn t + B co~ Cl)n t The constants A and B, as usual, are dete~ined from initial conditions of motion. z = 0'a and z = V' ., At t = 0
...(l~t.,
,
'JH"ir ; ,i2
. . Foundations of rmpllCt Type Machines
~
431
"
Using these initial conditions., we get Va A = - and E=-O Q)n
Va
A =-
Therefore,
Z
.
.
sm 0> t
Q)n
...(10.29)
n
The maximum displacement will be Va
(Az)max
10.2.3. Determination
...(10.30)
= IDn
of Initial Velocity of Anvil, V if
For a single acting drop hammer, the initial velocity of the tup VTi at the time of impact is given by VTi
...(10.31)
= 1'\ ~2 g h
where, h = Drop of tuP in meters g = Acceleration due to gravity, mls2 1'\= Efficiency of drop (It lies between 0.45 to 0.80. An average value equal to 0.65 may be
adopted) For double
.
- acting hammers, operated by pneumatic
or steam pressure, VTi is given by
VTi = 1'\.12 gh (Wl + pAc)
...(10.32)
.
Wl
where
W1
= Gross weight of the dropping parts, includ~ngupper half of the die in kN.
P = Pneumatic (i.e. steam or air) pressure in kN/m2 Ac
= Net area of-.cylinder in m2.
The initial velocity of anvil just after the tup' s. impact can be detennined by using the law the of
conservation of momentum. Since the anvil is stationary:
.
W
Momentum of tup and anvil before impact = ---1 VTi . g and Wl
...(10.33 a)
W2
Momentum of tup and anvil after impact = - VTa + Va g g where W2= weight of anvil (plus frame if it mounted on the anvil) .
Va' = Velocity of anvil after impact
VTa ='Velocity oftup after impact Therefore,
----
Wl v. - W, W2 VTa + Va T' -
g'g.
g
L
432
SoU Dy,,1IIIIics
~ Machine. FoundatiDns
According to Newton's law, the coefficient of elastic restitution, e, is given by Relative velocity after impact e = Relative-.velocity before impact Va - VTa
e=
or
...(10.35)
VTi
The value of e depends upon the material of the bodies involved in'iinpact. Theoretically value' of e lies between 0 and 1. In forge hammer, usually the value of e does not exceed 0.5 (Barkan, 1962). Since a larger e gives larger amplitudes of motion, the value of e equal to 0.5 is adopted in designing hammer foundation. On solving Eqs. (10,34) and (10.35) we get. ' l+e 'Y ' V ...(10.36) W TI a
1+-1. WI
10.2.4. Stress in the Pad. Maximum compressive stress in the elastic pad below the anvil depends upon the relative displacements of anvil and the foundation block. The worst case of compression in the pad developes when the anvil moves downward, and at same instant of time, the foundation block moves upward. The maximum compressive stress in the pad is thus expressed by zl + z2 (Jp
= K2
A' 2
. , (zl' z2 In absolute values)
...(10.3?)
10.2.S. Stresses in the Soil. Stresses transmitted to the soil q through the combined static dynamic loads are expressed by q = 10.3 DESIGN PROCEDURE
Wt +W2 +zl Kt
...(10.38)
Al
FOR A HAMMER FOUNDATION
The design of a hammer foundation may be carried out in following.steps: 10.3.1. Machine Data. The following information about the hammer is required for the design: (a) Type and weight of striking part of hammer; (b) Dimensions of base area of anvil and its weight; (c) Maximum stroke or fall of hammer, mean effective pressure on piston and effective area of piston; . (d) Arrangement and size of anchor bolts; and (e) Permissible amplitudes of the anvil motion and the foundation on block. If this information is not available, the amplitudes of motion given in Table 8.2 may be considered as limiting values.
10.3.2.Soil Data. The followinginformationabout the sub-surfacesoil shouldbe known:
:
(a) Soil profile: For drop hammers of up to 10 kN Capacity, soil investigations should generally
be
done to a depth of 6 m. For heavier impact machines, it is preferable to investigate soil con~~. tions to a depth of 12 m or to a hard stratum. If piles are used, the investigation should ~ conducted to a suitable depth, ~
I
1
~~-
433
'oundations of Impact Type Machines
(b) Soil investigation to ascertain allowable soil pressure and to determine the dynamic properties of the soil specifically the value of Cu' (c) 'The relative position of the water table below ground at different time of the year. 0.3.3. Trial Size of the Foundation. (a) Weight and area: The weight of the foundation for a hammer and the size of its area in contact with the soil should be selected in such a way that (i) the static pressure on the soil does not exceed the reduced allowable soil pressure, and (ii) the foundation does not bounce on the soiL These conditions may be written as ...(10.38 a)
PsI ~ a qa , and Nhere,
PSI
a = Reduction'factor
(= 0.8)
q a = Allowable soil pressure Az
...(10.38 b)
Az < Ap = Static pressure intensity
-
= Amplitudeof motion
Ap = Permissible value of amplitude For Eq. (10.38a) W
/'.
..(10.39) 'Considering an average value of Ap as I mm. = (10-3 m), and assuming the system as single degree freedom system, the Eq. (10.38b) can be written as . A . ~ a qa
(I +e) W VI 0
~C' u W A .g
where,
' I
3
...(IOAO)
< IO-
Wo = Weight oftup, kN W = Weight of foundation, anvil and frame, kN
Al = Base area of foundation in contact with soil, m2 VIi = Initial velocity of tup, mls g = Acceleration due to gravity, mls2 C' u = Coefficient of elastic uniform compression for hammer foundation, kN/m3 Substituting the value ofW from Eq. (10.39) into Eq. (10.40), we get (1+e)WoVTi A. ~
g ~Cuaqa '
32 x 10 m ." ':
...(10.41)
',<,'
'""",
'"
;i~< ,~~df .'1t~. "'r;l'
434
Soil Dynamics & Machine
Foundatio~~-"
Substituting the value of AI fromEq. (10.39) into (10.40), Weget (1+e) Wo VIi
W = Let where,
~ "C~ g
~
"
-3
x 10 kN
...(10.42)
'I,:
r.;~
W=W\+W2 W \ = Weight of foundation
l/
= Weight of anvil
W2
't
~,; Q.'i~
The Eq. (10.42) can be written as WI
(l+e) VIi
Vi" =
~
0
"Cu
~ g
-3 W2 x 10 -Vi" 0
...(10.43)
From, Eq. (10.42), total weight of the anvil and foundation can be obtained. Knowing the weight ot anvil, using Eq. (10.43) weight of the foundation can be worked out. On the basis of experience (Barkan, 1962), the weight of the anvil is kept generally 20 times the weight of the tup. Further it is recommended" that the weight of the foundation block should be at least 3 to 5 times that of the anvil. (b) Depth: The depth of the foundation block shall be so designed that the block is safe both in punching shear and bending. For the calculations the inertia forces developed shall also be included. However, the following minimum thickness of foundation block below the anvil sha,If be provided: ~ Mass of Tup kN
Thickness (Depth) of foundation Block, Min (m)
Up to 10
1.0
10 to 20
1.25
20 to 40
1.75
40 to 60
2.25
Over 60
'2.50 ..'
10.3.4. Selecting the dynamic elastic Constant C 'u. The procedure of obtaining Cu has already bei4:
discussed in Chapter 4 for relevant strain level. The value of C'Umay be taken as A C'u where A vari~ between 1 and 2. The value of Cu may also be obtained from the following relation: if'. 4Gr .~. C = -1 0 ...(10.44,) u -~ G = Shear modulus where, ~.
.<
..
r = Equivalentradius = 0
J.1
J AnI
= Poisson' s ratio
10.3.5, Natural frequencies, Compute ,
O>na
1& = v-;;;;
, I!'
.
,I
'
D
435
(}undations. 0/ Impact Type Machines
and
oon/
~ V~
K2
= b . A2
"
E in w~ich,
,
'
,
E = Yourig's'modulus of pad material b = ThicIaiess of the pad A2 = Area of the pad K 1 =C' u A=A.C u A "
,0
..'
,",",
'
"-.,,,
Natural frequencies of the combined system are given by: 2
'"
1
2 (()nl,2
= '2
2
[ (l+~m)
2
2
[
) (OOna+OOnl:t
2
(OOna+OOnl)]
(l+~m)
"
2
2
(rona,wnl )]
-4(1+~m)
10.3.6. Velocity of Dropping Parts to Anvil. Compute the velocity VTiof the tup before impact VTi
11 ~2g
~
(WIW + pA)h
.-
in which, W 1 = Gross weight of dropping parts p = Steam or air pressure .. "
,.
Ac = Area of the piston h = Drop of the tup 11 = Efficiency of drop, usually 0.65 Compute the velocity of the anvil Vaafter impact by l+e ' Va =. W , VT I
1+ ---1Wl
in which,
e = Coefficient of elastic restitution. The value of e may be adopted as 0.6.
10.3.7.Motion Amplitudes of the Foundation and Anvil. Compute the maximum foundation and anvil amplitudes with following equations
(ro~a Z1
=
- (O~I) ( (O~a-
2 OOna
(Onl
2,."-
0.,
-
(
(co
2
na,
2
- OOn2 )OOn2 2
CO 'I
n
)
'
V
Z2 ~ (CO~I: ro~2)cDn2I),;, where oon2 is the smaller natural frequency.
...
00~2)
""0"""""" . .. ' '~~-~'-_...
VI)
436
SoU Dynamics et Machine Foundations
10.3.8. Dynamic Stress in Pad (jp' Compute dynamic stress in ~e pad by (J P
= Kz(zJ+zz) Az
Computed values of natural frequencies should satisfy the criteria for the frequency of operation of the hammer. Also, motion amplitudes should be smaller than permissible values, and the stress in the" elastic pad should be smaller than the permissible stress of the pad material. 5
!ILLUSTRATIVE EXAMPLES'
Example 10.1 A 15 kN forging hammer is proposed to install in an industrial Complex. The hammer has the following specifications: Weight of tup without die
= 11 kN
Maximum tup stroke
Supply steam pressure
= 800 mm = 40 kN = 0.12 mz = 600.kN/mz
Weight of anvil block
= 300 kN
Total weight of hammer
= 400 kN = 1.8m x 1.8m = 1.5 mm = 1.0 mm
Weight of the upper half of the die Area of piston
Bearingarea of anvil Permissible vibration amplitude for anvil Permissible amplitude for foundation
It is proposed to use a pine wood pad of thickness 0.5 m below the anvil. The modulus ,of elastici' of pad"material is 5 x 105 kN/mz, and allowable compressive stress in pad is 3500 kN/mz. 'I'~ A vertical resonance test was conducted on a 1.5 ID x 0.75 m x 0.70 m high concrete block at f proposed depth of fo~ndation. The data obtained are given below: S. No.
8 (Deg)
/"Z (Nz)
Amplitude at resonance (mic1'f!'Y
I. 2.
36 72
41 40
13 24
3.
108
34
32'~
4.
144
31
40
.'
.~
iJ
t .,
I
A
'ine soil at the site is sandy in nalnre and water table lies at a depth of 3,0 m below ground
~
Allowable soil pressure is 225 kN/mz. Design a suitable foundation for the hammer.
Ji;
437
Ff!undation';-of Impact'type Machines ,
Solution:
,
(i) Trial dimensions of foundation.. Let the weight of the block is kept about 5 times the weight of anvil.' The details of the suggeste~ foundation are shown in Fig. 10.7.
.
Weight of foundation
= 24
x 7 x 5 x2
= 1680kN
(ii) Evaluation of Cu Area of test block = 1.5 x 0.75 = 1.125 m2 Weight of test block
= (1.125
x 0.75) x 24
= 18.9 kN.
Weight of oscillator and motor = 1.0 kN , (Assumed) ,-',':' ',.'j ,,-<,' .,,'
L.
";
.
438
SoU Dynamics & Machine Foundations' Total weight of test block, oscillator and motor = 18.9 + 1.0. = 19.9 kN 2
CII =
41t2
2
'
,
x 19.9 x fnz =71.1 Afn~ m -- 41t1.125 x 9.81
fn~ kN / m3
. Amplitude at resonance Stram level = 'Width of block Putting the given values of fnz and amplitude at resonance, CIIvalues and corresponding strain levels are computed and listed in cols. 2 and 3 of Table 10..1respectively. Table 10.1 : Values of Cu and strain levels -'
S. No.
CuI x 104 kN/m3
Strain Level x 10-4
Cu2 x 104 kN/m3
1.
11.95
0.173
2.96
2.
11.38
0.320
2.82
3.
8.22
0.427
2.04
4.
6.83
0.533'
1.69
Correction for confining pressure and area:
The mean effective confining pressure croI at a depth orone-half of the widthof block is givenby 0'01
= cr v-(1+2K
3
-where,
O'v
crvl
) 0
= crvl + crv2
= Effective overburden pressure at the depth under consideration cc
crv2 = Increase in vertical pressure due to the weight of block Assuming that the top 2.0. m soil has a moist unit weight of 18 kN/m3, and the next 1.0.m soil i..e upto water table is saturated, then "
-
0.70.
crvI = 18 x 2.0.+ 20.2-
;--
4q
"
= 43 kN/m2
2mn~m2+n2+1
.m2+n2+2
v2 - -41t m2 +n 2 +1+m 2 n 2 m2 +n 2 +1 + srn
[
-L m = -2 z
-B n ""',"2
-1.5 =- 2
0..70.
,'-".
. -} 2mn~m2+n2+1
m2 +n 2 +1+m 2 n 2
]
= 2.14 ., ,.!
2 0..75 -
.
= .£.z = ,0..70. -L = 1.0.7'
'Cl ::..
'
q ='24 x 0..70.= 16.8 teN/ni2, '
"
'",'
,,',
. . 0'
. , ",:, '..~
,.!,'..,'"
-,~."~
439
Ilndations of Impact Type M~~hines
Substituting the above values of rn, nand q in the expression of av2, we get al2 = 13.44 kN/m2 Gv = 43 + 13.44 = 56.44 kN/m2 Assuming
<1> =
35°, Ko = 1 - sin 35° = 0.426 1+2XO.426
-
= 56.44 [
aol
3
2
] = 34.84 kN/m
For the actual foundation av!
= 18 x 2.0 + 20 x 1.0 + (20 - 10) x 1.5 = 71 kN/m2
For the actual foundation 7.0 2 m = 5.0 = 1.4 2 5.0 - 2 _ 10 n - 5.0 2
.
q = 24 x 2.0 = 48 kN/m3 Substnuting the above values of rn, nand q in the expres~ion of crv2, we get av2 cry
= 37.2 kN/m2 = 71.0 + 37.2 = 108.2 kN/m2
G02 = 108.2[1+2 x30.4266] = 66.8 kN/m2 0.5 0' 02
Cu2 = CuI
[
0'01
]
= 66.80 [ 34.84 ]
~
0.5
.
[ A2 ] °.5
1.5x 0.75
[ 7.0 x 5.0 ]
°.5 -
- 0.248
The values of Cu of the actual foundation for different strain levels are listed in cot 4 of Table 10.1. 1.0 4 Stain level in actual foundation = 5.0 x 1000 = 2 x 10The strain level in actual foundation is higher than the strain level observed in the tests. Seeing the variation of Cu with respect to strain level, the value of Cu equal to 1.3 x 104 kN/m3 may be adopted in design. .
\ . --,
440
',.' "'t~,
Soil DYllamics & Machine Foulldaii~~
(iii) Computations of KI' K2' ml and m2
C' = A' C 11
11
= 2.0 x 1.3 x 104 = 2.6 x 104kN/m3(A = 2.0, Assumed) K I = C'!I ,A I = 2.6 x 104x 7 x 5 = 91 x 104kN/m E
K, = -b .A 2 = ~
5 x 105 1.8x 1.8 = 324 0.5
x
104kN/m
Weight of the foundation block = (7 x 5 x 2 x 1.8 x 1.8 x 0.8) x 24 = 1618 kN 1618 = 165 x 103 K g 9.81
m =I
Weight of the anvil and frame = 400 kN 400 m2
=
'
9.81 =40.8 x 10"'Kg
(iv) Natural frequencies of soil-foundation system (J)1/1 =
~
l
324 x 10 = J79411 = 281 rad/s 40.8
= v-;;;;
11
=
= .J4421 = 66.4 rad/s
4
r&= (J)/la
91 x 104
-
- ~ nIl +"'2 '165+40.8
~ = 40.8 = 0 247 1/l\
165
.
Natural frequencies are given by : 2
,
.
1/2
(J.)~1.2 = ~ [(1 + Ilm) ((J)~a+ (J)~l)J=t~ [(1 + Ilm) (O)~a + (J)~l)] -4 (1 + 11/11) (J)~a.(J)~l} { 1
In
=: "2[(l + 0.247) (79411 +4421)] (J) /11.2
::t
~{[(1+ 0.247)(79411
+ 4421)]2 -4 (1 + 0.247)79411
x 4421}
= 316 rad/s, 66 rad/s
(v) Velocity of dropping parts
VTi
~
~ ,{W, ~.A,
11 = 0.70 (assumed)
)2gh ; W\
=
15 kN, p = 600 kN/m2, Ac = 0.12 m2, h = 0.8 "
J
..;1
-\I~ VTi
"
= 0.70 ,iC5+ 60105 x 0.12) x 2 x9.81 x 0.8 = 6.68 mls .~ 'h4 .:;i ~I
441
rtions of Impact Type Machines
l+e
.
Va = .
-
w' VTi 1+-1.. W1
1+ 0.5 .6 68 400' 1+15
(e = 0.5, assumed)
= 0.362 m/s i) Amplitudes of vibration ZI
=
(ro:a-ro:l) (ro:a-ro:2) 2
2
2
(rona)(ronl -ron2 )ron2
.Vu
(79411-100167) (79411-4371) = (79411)(100167-4371)(4371) .(0.362)
= - 1.695 x 10-4 m = - 0.1695 mm «1.0 mm, safe) Z2 =
(ro:a-ro:l)
2 -ro\n (ronl
,Va
) ro/12
(79411-100167) x 0.362 = (100167-4371)(4371)
..
= - 1.7944 x 10-5 m = - 0.0179 mm [< 1.5 mm, safe]
vii) Dynamic stress in pad k2 (zl +z2) (J2
=
A2
. ' (zl' z2 In absolute values)
= 324 X 104 (0.1695+0.0179)
10-3
1.8x 1.8
= 187.4 kN/m2
(\
[< 350 kN/m2 , safe]
442
Soil Dynamics & Machine Fouml'!!i!:l
ÎÛÚÛÎÛÒÝÛÍ Barkan, 0.0. (1962), "Dynamics of bases and foundations", McGraw-HiIl, New York. Novak, M. (1983), "foundations for shock producing machines", Can, Geotech. J. 20 (1), pp. 141-158.
PRACTICE
-
PROBLEMS
10.1 Discuss with neat sketches the various possible arrangements ofa hammer foundation to minimist the vibrations. 10.2 Consideringa two-degree-freedommodel, derivethe expressionof amplitudesof anviland foundatio of a hammer. 10.3 A 20 kN forging hammer is proposed to install in an industrial Complex. The hammer has th following specifications: Weight oftup without die Maximum tup stroke Weight of the upper half of the die Area of piston Supply steam pressure Weight of anvil block Total weight of anvil and frame
= 12 kN = 900 mm = 50 kN = 0.15 m2 = 700 kN/m = 400 kN = 500 kN = 2.1 m x 2.1 m
Bearingarea of anvil
Permissible vibration amplitude for anvil = 1.0 mm Permissible amplitude for foundation = 0.8 mm It is proposed to use a pine wood pad of thickness 0.5 m below the anvil. The modulus of elasticity pad material is 6 x 10 kN/m2, and allowable compressive stress in pad is 4000 kN/m2. A vertical resonance test was conducted on a 1.5 m x 0.75 m x 0.70 m high concrete block at : proposed depth of foundation. The data obtained are given below:
S.No.
e (Deg)
f nz(Hz)
1. 2. 3. 4.
36 72 108 144
40 38 35 29
Amplitude at resonance (micron 14 26 33 41
The soil at the site is sandy in nature and water table lies at a depth of 2.0 m below ground surf Allowable soil pressure = 200 kN/m2. Design a suitable foundation.
C]
.~ i, ..Ji.
FOUNDATIONS OF ROTARY MACHINES
.1 GENERAL le unprecedented burden cast by the importance of oil can be relieved only by exploiting indigenous lergy resources efficiently. Major power energy resources, in long term power plan, incorporate an ,timal mix of thermal, hydel and nuclear generation. Power intensity is relatively high in our country le to various reasons including the substantial substitution among the forms of energy in the various lportant sectors along with the accelerated programme in rural electrification and assured power supy for agricultural sector etc. -Theaim of planning to generate power higher than the demand by at least ) percent, calls for a coordinated development of the power supply inqustry. As per the present ratio, the ermal sector caters 54 percent; hydro sector 43 percent and 3 percent catered by nuclear sector. The 15 :ar National Power Plan from 1980 onwards envisages the installation of additional generating capacity t'almost 100,000 MW in the thermal, hydro and nuclear sectors taken together as against the existing :neration capacity of 31,000 MW. Large capacity thermal power station at coal pit heads called Super hermal Power stations (2000 MW capacity or more) because of their size and sophisticated technology, ill account for as much as 50,000 MW to be commissioned during the next five year plan. The turbogenerator unit is most expensive, vital and important part in a thermal power plant. The perating speeds of trubogenerators may range from 3000 rpm to 10000 rpm. Auxiliary equipments such s condensers, heat exchanger, pipe lines, air vents and ducts for electric wiring are essential features of turbogenerator installation. Frame foundations are commonly used for turbogenerators with four :asons : (i) (ii) (ui) (iv)
auxiallary equipment can be arranged more conveniently, the inspection of and access to all parts of the machine become more convenient, less liable to cracking due to settlement and temperature changes, and more economical due to the saving in material and freedom to add more members to stiffen if
needed.
.
The frame foundation is the assemblage of columns, longitudinal and transverse beams. The trans:erse beams may be often eccentric with respect to the column centre lines and generally have varying :ross-section due to several opening in the top deck and haunches at the junction with columns. The sometric view of a typical frame foundation is shown in Fig. 11.1. "I' . . In a power - plant, the long term satisfactory performance of the turbogenerators is affected by their r'oundations,pence there is vital need to adequately design these foundations for all possible combinations )f static and dynamic loads. Interaction with the mechanical engine~r is also required for any adjustment 10the layout of machinery and auxiliary fittings.
...
~.;;;t.
n!l""cU~
/
444
Soil Dynamics & Machine Foundatiol'
""'>'
Longitudinal beam
Top deck ~
"
Column's , ;.,.
'~
-,-,
.. .1 ,'.' 1
Fig. 11.1 : Typical frame foundation for a turbo-generator' "
11.2 SPECIAL CONSIDERATIONS For better perfonnance of aT. G. foundation, following points may be kept in view:
.<
-, (i) The entire foundation should be separated from the main building in order to isolate the tfaIf~re of vibrations from the top deck of the foundation to the building floor of the machine'-roo~~). clear gap should be provided all around. . ,(~
(ii) Other footings placed near to the machine foundation should be checked for non-uniform str~~. impo')cdby adjacent footing. The Pressure-bulbs under the adjacent footings should not inie1fi
significantlywith each other.
'
\wa:
~
445
lundations of Rotary Machines
(iii) All the junctions of beams and columns of the foundation should be provided with adequate haunches in order to increase the general rigidity of the frame foundation. (iv) The cross-sectional height of the cantilever elements at the embedment point should not be less than 60 to 75 percent of its span,- being susceptible to excessive local vibrations. . (v) The transverse beams should have their axes vertically below the bearings to avoid torsion. For the same reason the axes of columns and transverse beams should lie in the same vertical plane. (vi) The upper platform should be as rigid as possible in its plane. (vii) Permissible pressure on soil may be reduced by' 20 percent to account for the vibration of the foundation slab. This slab has much smaller amplitudes of vibration than the upper platfom1. -
.-
-
-
-
(viii) The lower foundation shib should be sufficiently rigid to resist non-uniform settlement and heavy enough to lower the common centre of gravity of the machine and foundation. It is therefore: made thicker than required by static computations. For 25 MW machine its thickness is 2m and increases with the power of the machine to a maximum of 4m. Its weight should not be less than the weight of the machine plus the weight of the foundation excluding the base slab and condensers. (ix) Special reinforcement detailing as laid down in the code IS-2974 Pt III should be followed. (x) Special care in construction is called for to avoid cracking of concrete. The foundation slab should be completed in one continuous pouring. In this case the joint between the two concretes, preferably at one-third column height, is specially treated to ensure 100 percent bond. (xi) Piles may be provided to meet the bearing capacity requirement but then the consideration of sub grade effect is essential. (xii) As far as possible the foundation should be dimensioned such that the centre of gravity of the foundation with the machine should be in vertical alignment with that of the base area in contact with. the soil. . --(xiii) The ground-water table should be as low as possible and deeper by at least one-fourth of the width of foundation below the.base plane. This limits the vibrations propagation, ground-water being a good conductor to wave transmission.
(xiv) Soil-profile and characteristics of soil upto at least thrice the width of the turbine foundation or . till hard stratum is reached or upto pile depth, if piles are provided, should be investigated. 11.3 DESIGN CRITERIA The design of a T. G. foundation is based on the following design criteria: (i) From the point of view of vibration, the natural frequencies of foundation system should preferably be at a variance of at least 30 percent from the operating speed of the machine as well as critical speeds of the rotor. Thus resommce is avoided. An uncertainty of 10 to 20 percent may be assumed in the computed natural frequencies. However, it may not be necessary to avoid resonance in higher modes, if the resulting resonant amplitude is relatively insignificant. It is preferable to maintain a frequency separation of 50 percent. (ii) The amplitudes of vibration should be within permissible limits. Values of permissible amplitudes are given in Table 8.2..,. , (
.
-=:
~r
':}J!
1.1
, .,
>
--,-,.
(
c
. -_.._-
446
Soil Dynamics & Machine Found.
11.4 LOADS ON A T. G. FOUNDATION The loads acting on a turbogenerator are as given below: 11.4.1. Dead Loads (DL). These include the self weight of the foundation and dead 'Yeight of the chine. 11.4.2. Operation Loads (OL). These loads are supplied by the manufacturer of the machine anc elude frictional forces, power torque, thermal elongation forces, vaccum in the condenser, piping fo etc. The load due to vaccum in condenser, ifnot supplied by the manufacturer, can be obtained fron following equation: ...(1
Pc = A (pa - Pc) where.
Pc = Condenser vaccum load A = Cross-sectional area of the connecting tie between the condenser and turbine
Pa = Atmosphericpressure Pc = Vac cum pressure The value of (p a - Pc) may be taken as 100 kN/m2.
TA
TR A
-8-
Shaf-
t] H.P.
Gcznczrator Tu r bin czs Fig. 11.2 : Torque due to normal operation oh multistage turbine-generator
unit
The magnitude of the torque depends upon the operational speed and power output capacity ( turbines. For a T. G. unit having multistage turbine (Fig. 11.2), the torque may be calculated as be - 105 PA kN TA N m - 105(PB - PA) kN
TB -
N
...( 1
m
- 105(Pc - PB) kN Tc N m 105P T = c kNm g
where,
TA
= Torque
N
due to high-pressure (H. P.) turbine in kNm
TB = Torque due to intermediate-pressure
(LP.) turbine in kNm
...( 1 ...( I ...(1
-447
Foundations of Rotary Machines
Tc = Torque due to low-pressure (L. P.) turbine in kNm Tg= Torque due to generator in kNm PA' Pa and Pc = Power transferred by couplings A, B and C respectively in KW N = Operating speed in rpm 11.4.3. Normal Machine Unbalanced Load (NUL). As mentioned in sec. 8.2, rotary machines are balanced before erection. However, in actual operation some unbalance always exists. The unbalance is specified as the distance between the axis of the shaft and mass centre of gravity of rotor, and is known as effective eccentricity. The magnitude of unbalanced forces can be obtained using Eqs. (8.8) and (8.9).
_l-
(0)
(b) Fig. 11.3: Unbalanced forces due to rotary machines
For the case of rotors shown in Fig. 11.3a, the resultant unbalanced forces due to the two masses at any time cancel out, but there is a resulting moment M given by 2 M=mero./ ...(11.3) e where,
/ = Distance between the mass centre of gravities of rotors
The components of the moment M in vertical and horizontal directions are given by Mv=meero2/sinrot 2 MH =meero /cosrot
...(11.4 a) ...(11.4 b)
When masses have an orientation as shown in Fig. 11.4b, the machine operation will give rise to both an unbalanced force and a moment. The unbalanced force is given by 2 F = 2 me e ro
...(11.5)
The unbalanced moment can be computedusing Eq. (11.3), For more than two rotors on a common shaft, combined unbalanced forces and moments can be computed in similar manner.
.
448
Soil Dynamics & Machilre Foundat
11.4.4. Temperature loads in the foundation (TLF). The effect of differential thermal expansion a! shrinkage should be considered in the design of frame foundations. In the absence of the exact data. differential temperature of 200 may be assumed between the upper and lower slabs. Besides, a different: temperature of 200 may be: assu\TIedbetween the inner and outer faces of the upper slab. The upper s1;. should be treated as a horizontal closed frame and analysed for the induced moments due to differenti temperature. To account for the shrinkage of the upper slab relative to the-base slab, a temperature fall of 100C 150C may be assumed.. . . 11.4.5. Short circuit forces (SCF). Short circuit condition imposes moment on the turbogenerat foundation. A fault of this type occur when any two of the three generator phase terminals are shorte The shock, which is in the form of couple known as "shortcircuit moment", tends to break the stator ( the foundation, and this imposes vertical loads on the longitudinal beam supporting the generator state If accurate information is not available from the manufacturer, the short circuit moment (Msc) m. be taken emipiricallyas four times the rated capacity (in MW) of turbogenerator unit. Major (1980) has suggested the following fomlula for estimating the short circuit moment: Msc = 10 r Wr kNm Where,
...( 11.
W,. = Capacity ofT. G. Unit in MW r = Radius of the rotor in m
11.4.6. Loss of blade unbalance (LBL) or bearing failure load (BFL). One of the buckets or blades the turbine rotor may break during the operation of turbo-generator unit. It will increase the unbalanc force. This additional unbalanced forced wi1l depend on the weight of the bucket,. the distance of centre of gravity from the axis of rotation and operational speed.. .. . 11.4.7. Seismic load (EQL). The horizontal seismic force is considered both in logitudinal and transve: directions separately. It may be computed from the following equation (IS 1893-1984) : Fs
where,
Fs
= ah
I
~C
S W ...(11.7)
= Horizontal seismic force
ah = Seismic zone coefficient I = Importance factor
~ = Soil-
foundation
factor
C = Numerical base shear coefficient S = Numerical site structure response coefficient W
= Verticalload
due to weight of all permanent components.
When earthquake forces are considered in design, the permissible stresses in materials and the lowable soil pressure may be increased as per IS 1893-1984.
11.4.8. Construction loads (CL). Constructionloads occur only when the machineis being erected. such they are not to be consider~das acting simultaneouslywith dynamic loadswhich occur only dur the operation of the machine. The construction loads are generally taken as uniformly distributed l( varying from 10 kN/m2 to 30 kN/m2depending on the size ofT. G. unit. . .
449
r.°un~a~io.nso! R~tary M~chines
The design of a T. G. unit should be checked for the following load combinations: (a) Operati?n,: ~~ndition DL+OL+N1J:r..:+JL[ (b) Short circuit condition
..:
DL + OL + NUL + TLF + SCF} .,'...,
.
(c) Loss of blade conditionlbearing failure'doridition DL + OL + TLF + LBLIBFL
.
(d) Seismic condition DL + OL + NUL + TLF + EQL 11.5 METHODS OF ANALYSIS AND DESIGN In the case of a frame foundation, it is necessary to check the frequencies and amplitudes of vibration and also to design the members of frame from structural considerations. The methods for carrying out dynamic analysis may be divided into two categories: (a) Two-.dimentional analysis (b) Three-dimensional analysis The two-dimensional analysis is based on the following assumptions: (i) The difference between the deformations of individual frame coluinns is insignificalll. (ii) The deformation of the longitudinal and transverse beams is almost identical. (iii) The torsional resistance of the longitudinal beams is insignificant in relation to the deformation
of the transverse beams.
. -..
..
.
(iv) The vertical vibrations of the frames can be determined for each frame individually. (v) The weight transmitted from the longitudinal beam can be considered as a load supported by the column head, even in case where the transverse beam is eccentrically placed with respect to the centre line of the column. (vi) Both the columns and beams can be replaced by weightless elements with the masses lumped at a few points by equating the kinetic energies of the actual and the idealised systems. (vii) The effect of elasticity .of subsoil is neglected, it being relatively much flexible. . (viii) When considering horizontal displacement the upper slab is regarded as a rigid plate in its own
plane.
.
The two dimensional analysis may be carried out by the following methods: 1. Resonance method (Rausch, 1959) 2. Amplitude method (Barkan, 1962)
.
3. Combined method (Major, 1980) In subsequent sections, salient features of the above methods are given.
I
i
"--'--'..
-"
.-
450
Soil Dynamics & Machine Foundations
ïïòêÎÛÍÑÒßÒÝÛ
METHOD
In this method, the frame foundation is idealized as a single-degree freedom system, and consideration is given only to natural frequencies of the system in relation to the operating speed of the machine. The amplitudes of vibration are not computed in this method. 11.6.1. Vertical Frequency. For obtaining vertical frequency, each transverse frame that consists of two columns and a beam perpendicular to main shaft of the machine, is considered separately (Fig. IlAa). Fz sin cut Fz sin c..>t
t
Wz
Wz
Wl ID
I.-
lo
I
~
~
-2a
I
ILl
I
1
I
lr
m= ql j-W,+2' 9
I
h, ho
Zb
,-
Column
assumed fixed
Gekrmns
.>71
1- - -7 /"
J~
!
- 1I
Y- - - - - - - - - - -_J Base slab
(b)
I
L
(a) 11.4: (a) Typical transverse frame; (b) Idealised model
The loads acting on this frame are (I) Dead load of the machine and bearing, W I (ii) Load transferred (iii) Uniformly (Iv) Unbalanced
to the columns by longitudinal
distributed
beams, W 2
load due to self weight of cross beam, q per unit length
vertical force due to machine operation,
Fz sin w t
451
Foundations of Rotary Machines
The frame is modelled as mass-spring system as shown in Fig. II.4b. The stiffness of equivalent spring (K) is computed as the combined stiffness of the beam and columns acting together. It is given by W K =...( 11.8) ::
851
where,
W = Total load on the frame
or
W
= W\
+ 2W2 + q ./...(11.9)
1 = Effective °51
span
= Total vertical deflection at the centre of the beam due to bending action of beam and axial compression
in columns.
o
= °1
51
where,
...(11.10)
+ °2 + °3 +°4
°1 = Vertical deflection of beam due to load WI °2 = Vertical deflection of beam due to the distributed load q °3 = Vertical deflection of the beam due to shear 84 = Axial compression in column
The magnitudes of 81' 82, °3 and 84 can be obtained using following expressions: Wj ,3 2 K + 1 81 = 96 E Ib K + 2
...(11.11)
q 14 5K + 2 °2 = 384 Elb' K+2
...(11.12)
3
1
ql
83 ="5 E Ab ( WI+2 84
=
~
EA e
(
...(11.13)
)
W + WI +ql 2 2
)
Ih h K =-.le 1
where,
...(11.14)
...(11.15)
Ab = Cross-sectional area of beam Ac = Cross-sectional area of column Ib
= Moment of inertia of beam about the axis of bending
le = Moment of inertia of column E = Young's modulus of concrete K = Relative stiffness factor 1 = Effective span of frame h = Effective height of frame Values of 1and h are obtained as below: '=1
0
-2ab
h =h 0 -2aa
...(11.16) ...(11.17)
452
Soil Dynamics & Machine Foundatiolls
where,
10
= Centre to centre distance between columns (Fig. 11.4 a)
ho = Height of the column from the top of the base slab to the centre of the frame beam (Fig. 11.4 a) a = One-half of the depth of the beam for a frame without haunches (Fig. 11.4 a) or the distance as shown in Fig. 11.5 for a frame with haunches b = One-half of the column width for a frame without haunches (Fig. 11.4 a) or ~he distance as shown in Fig. 11.5 for a frame with haunches. Knowing the values of ho' 10and b, excan be obtained from Fig. 11.6.
0.40
0.30 0<
®óóó I I
I I
0.20 0
0.10
-1 b~
00
0.04
0.08
0.12 b/ lo
Fig. 11.5 : Values of a and b for a frame with haunches
Fi. 11.6 : CLversus b/lo
The natural frequency of a transverse frame in vertical vibrations is given by
(j)
...(11.18)
nz = ~KzW. g
Average vertical natural frequency of the T. G. Foundation is taken as: COn:1+cun:2+...+con;n (j) n;a
=
n
...(11.19)
where, (J.)I/;!'(J.)1/;2'" = Vertical frequencies of individual transverse frames
.. ..
!.~-
Foundations
453
of Rotary Machines
The average value of vertical amplitude of T. G. foundation may be computed as IF:
A:a
(I Ko),'ll-(
...(11.20)
oo:Jf+( ~:o:)'
A:a = Average vertical amplitude of T. G. foundation
L F~ = Total vertical imbalance force 2::Kz = Sum of the stiffness of the individual frames
~ = Damping For under-tuned
ratio
foundation,
i.e. (() < (() lI~a ~, (()n = (()n~a should be used in Eq. (11.20). Then
2::F:
A za
...(11.20 a)
(2::KJ (2~)
11.6.2. Horizontal Vibrations. In a T. G. frame formulation, the deck slab undergoes horizontal vibration in the direction perpendicular to the main shaft of the machine. The spring stiffness is provided by the columns due to their bending action, and for any transverse frame it is given by
6K+l
K = 12E le x where,
If
113 ( 3K+2
...(11.21)
)
Kt = Lateral stiffness of an individual transverse frame
L Kt
= Sum of the lateral stiffness of all the transverse frames
WT = Total weight of deck slab and machine Then the natural frequency of the T. G. frame foundation is given by
"'"xa
~
...(11.22)
p:~~)g
The average horizontal amplitude of the foundation may be computed as follows: A
L~t
=
...( 11.23) 2
'" For under-tuned
foundation,
(LK,),/ I- ( ~ nxa ) 2 + ~nxa J [ ] ( Cl)
2
Cl)
i.e. (() < (()n.m'
(J)11= (J)I/.HIshould be used in Eq. (11.23).
Thus
LFt
ß©ãøÔÕÖøî¢÷
...(11.23
(l)
454
Soil Dynamics & Machille
FoUIldatiolls
As mentioned earlier, in this method only the possibility of resonance is checked i.e. the natural frequencies computed from Eqs. (11.19) and ( 11.22) should differ by atleast 30 percent from the ope rati:1g speed of the machine. The Eqs. (11.20), (11.20a) and (11.23) for determining amplitudes are gi\'en to be used further in combined method. Resonance method based on idealising each transverse frame to single mass-spring system is an oversimplification of a complex problem. Therefore the values of natural frequencies computed by this method are very approximate. 11.7 AMPLITUDE
METHOD
In this method also, the vibration analysis is carried out for each transverse frame independently. However, the frame has been idealised as a two-degree-freedom system (Fig. 11.7). The main criterion for design is that the amplitudes due to forced vibrations are within permissible limits (Barkan, 1962).
Z2
t Fz sin
1-
wt
m,
\ m, /1
TI K,
z
\
-----"::::::;-mz
KZ
\ K, 1-
\
I
\ \ \
mz
12
c-tzz
I
I I
columns
:.r::
:"l
Z1
z
m,
/
/
(a)
Section
of
cross
frame
(b)
Mathemetical
model
Fig. 11.7: (a) Vertical vibration ora cross frame as a two-degree-of-freec.lom system; (b) !\Iass-spring model
11.7.1. Vertical Vibration. For the vertical frequency a two-degrce-spring-mass Fig. 11.7 b is adopted. Mass m I lumped over the columns is given by
= WI + W2+0.33W3 +0.25W4
m I
g
system shown in
...( 11.24)
,\ ,ii
IIIJ
455
iations of Rotary Machines
Aass mz acting at the centre of the cross beam is given by m = Z e.
Wz + 0.45W4 g
W 1 = Dead load of the machine and bearing W 2 = Load transferred to be columns by longitudinal beams W 3 = Weight of two columns constituting the transverse frame W4 = Weight of the transverse beam
The stiffness Kl of both the columns of a transverse frame is given by 2EAc KI = h The stiffness Kz of the frame beam is given by 1 K =-
z
-re,
/ (1+2K)
.
...(11.26)
...(11.27)
°SI
3/
...(11.28)
= 96 E rb (2 + K) + 8G Ab G = Shear modulus of beam material
OS!
E = Young's modulus of the material of columns Ac
= Cross-sectional area of a column
h = Effective height of the column I = Effective span of the beam AiJ
= Cross-sectional
area of the beam
rb = Moment of inertia of the beam K is defined by Eq. 11.15. The system shown in Fig. 11.7 b is identical to the system shown in Fig 1 18, and therefore can be alysed by the procedure explained in Art. 2.8. The equations of motion in free vibration will be: ...( 11.29)
ml ZI + KI ZI - Kz (Zz - ZI) = 0
...( 11.30)
mz Zz + Kz (Zz - ZI) = 0 The solution of above equations are: Z \ = A I sin 0)III Zz =A z sinO) III
...( 11.31 )
...(11.32)
Substituting Eqs. (11.31) and (11.32) into eqs. (11.29) and (11.30), on simplification. we get 4 (I)/I-(I+~l)
.z
l
2
(1)/111-0)12
zz
)+(I+P)(J)IlIIO)IlIZ
...( 11.3 3)
=0
Kl
here,
0) III 1 -
~ 0) Il 1Z
...( 11.35)
= V--;;;;
Ilm =
.u-_.'
...( 11.34)
/Ill + /Il2
nlz ml
...( 11.36)
~56
Soil Dy"amics & Machi"e Fo""dations
The two natural frequencies of the system can be obtained by solving Eq. (11.33). In forced vibration. the equations of motion will be: /1/] ZI+KI
...(11.37)
ZI-K2(Z2-ZI)=0
= F;
1112Z2 + K2 (Z2 - ZI) The solution
Substituting
of the above equations
Eqs. (11.39) All
AZ2
can be presented
...( 11.39)
Z2 = An
.. .(11.40)
and (11.40)
sin OOt
into Eqs. (11.37)
and (11.38), 2 00,,12. F;
=
=
as
ZI = AZI sin OO{
1111
and
...(11.38)
sin OO{
[
(
- (1 + ~ ,J
004
00 ~ 1 I + CO~ 12
) CO2 +
...(11.41)
(1 + ~ Ill) 00 ~ I 1 CO ~ I 2
2 2 [ (l+~IIl)OOllll+~IIlOOIl12-CO
[
and then solving them we get
]
2
] .F; 2 2 2 22 OOIlII+OOIl12 00 +(I+~/Il)OOIlIIOOIl12
-+
)
(
1Il2 00 -(1+~m)
]
...(11.42)
11.7.2. Horizontal vibration. For analysing the frame foundation in horizontal vibration as t\VOdegree-freedom problem, the upper and lower foundation slabs are assumed to be infinitely rigid. The columns are taken to act as leaf-springs. The stiffness of a leaf spring is considered equal to the lateral stiffness of the individual transverse frame. dl
~
IQ
I
r
j
d k3
d k1
dm1
11
I
II I
I
d
It
dm3~
m2 /. .
.
C
KX2
T X
G2
A1
01
m1
.
-L A3~--
- ---
AZ
Due to horizontal displacement Fi~. 11.1' : Sprin~-l1Ia"
81 m3
m2
l!1odl'l for combincd
-----
T
\ 1
Due to translation and rotation horizontal
and rotational
"'1'
... 8 z
óóóùè 3
vibrations
of the deck slab
-
457
Idations of Rotary Machines
Figure 11.8 shows a typical mathematical model for a two bays frame foundation. The equivalent s ( mi) lumped over the spring i (representing frame i) is given by: m, = m ,+ m b ' + 0.33 m . + m ' ...(llAJ) I ml I Cl gl
re.
1I111/i
IIIbi
= Mass of machine resting on cross-beam of ith frame
= Mass of cross-beam of i'h frame
lIIei = Mass of columns of i'h frame l1lu",I
= Mass transfered from longitudinal girders on either side
Using Eq. (11.43), values of ml' 1112and 1".13 can be obtained. K,i represents the lateral stiffness of ith transverse frame. Thus Kxl' Kx2 and Kd can be evaluated. In Fig. 11.8, points G1 and G2 represent the cent:e of masses (i.e. ml' m2 and 11l3) and centre of fness (i.e. Kxl' K,2 and Kd) respectively. Line A1Bl shows the initial position of deck slab. The final placed position of the deck slab is represented by the line A3B3 . The deck slab rotates about the mass ltre G I' dkl' dk2 and dk3 are the distances of different masses flom point G2. The distances of different ,sses from point G 1 are shown as d 11/I ' d mL and dm 3 . e rep resents the distance between G I and G L ~
~
.
The equations of motion for the system shown in Fig. 11.8 will be: (Iml).1+ (I/1/i 1ere,
dl~lI)
\it +
I[Kti
I[ K'i
...(11.44)
(x+dmi 'V)] = F, sin cD(
...(IIA5)
(x + d,lli'V) dllli] = M. sin cD(
Fx = Horizontal unbalanced force ivL = Unbalanced moment
Denoting lI/
= 'Ill/,I = Total mass
...(II.4())
")
Mm:
='I 11li dl~ll = Polar mass moment of inertia of all the masses about the vertical axis through G I ...{11.47) IKti(x+dll/i'V)
=xIK.ti+(IK.tidmi)'V ...( 11.48)
= x Kx + Kx . e 'V = Kx (x + e 'V) K, represents the total lateral stiffness.
I K, i
(x + dm i 'V) dill i
=
(I Kti
dilli) X + ( I Kti dl~1i) \jI
= K, . e. x + [ L Kti( e2 + d; i)] \jI = K, . e .\'+ K r
/
\jI +
(L K, id; i )
= K, (x+e\jl) e+ KIjI. \jI
\jf
...( 11.49)
11
458
Soil Dynamics & Machine Foundations K", represents the equivalent torsional spring stiffness for the frame columns and is given by ., ...(11.50) K", = IK.d dki
It may be noted that IK
XI .
dk I = 0 '
Making the substitutions from Eqs, (11.46) to (11.50) in Eqs. (11.44) and 911,45), we get m .~ + K .\,\ + K X . e \jI = F .r sin W{
= Mz sin
Mmz\jl + K\ex + (K, e2 + KII')\jI
..,(11.51) ...( 11.52)
W{
Equations (11.51) and (11.52) are similar to the Eqs. (9.51) and (9.52). Proceeding exactly in the same way as discussed earlier, the solutions of Eqs. (11.51) and (11.52) can be obtained from the following equation: 4 CDn -
where
( a
22 CDnx
+ CD/I IjI
)
222
CDn + CDnx
...(11.53)
. CD nljl = 0
fK. Wnx= v-; W nII'
=
...(11.54)
~
...(11.55)
e r
...(11.56)
II'
Mm= 2
a=l+Z
,..(11.57)
m r = ~Mm= The amplitude of vibration in translation and rotation are given by e22
-
A = [
2
+
,2 CD/IX ]
2 - CD F\ ~
CD1lII'
x
]
Mc - CD2 . --=---
m
n\
Mm=
...( 11.58)
L1(CD2) 2
e
2
F X
?CD/I\r. m
-
2
-
2
CD
(conx
M~
)---=M IIIz ...( 11.59)
AIjI
where
2
=
( ) = (J)
L\ co
L1(co2)
4
(
2
2
2
2
2
- a (J)n x + co n 'v ) CD + co n x con IV
...( 11.60)
The net amplitude Air is given by \vhere,
...(11.61) Air = At + y AIjI v '= Distance of the point at which the amplitude is being calculated from the centre 01 gravity of the system
..,
Juniations of Rotary Machines 1.8 COMBINED 1
459
METHOD
fact, the resonance and amplitude methods are complimentary. Since the amplitude method is based
n a system of two degree of freedom, it is obviously an improvement over the resonance method. Howvel', the. fact that in under tuned foundation the increase in amplitude of vibration during acceleration nd deacceleration stages has been ignored in this method. In combined method which is also known as xtended resonance method, the possibilities of resonance and excessive amplitudes both during steady ibration arid acceleration or deacceleration stages are investigated. The analysis may be carried out in he following steps: (i) The natural frequencies in vertical and horizontal modes of vibration as computed from Eqs. (11.33) and (11.53) by amplitude method are compared with the operating speed. The natural frequency in any mode of vibration should be atleast 30 percent away from the operating speed. (ii) Amplitude.s of vibration computed from Eqs. (11.41), (11.42) and (11.61) should be within permissible limits. (iii) Amplitudes of vibration computed from Eqs. (11.20 a) and (11.23 a) should also be compared with permissible amplitudes to take care the possibility of excessive amplitudes during acceleration and deacceleration stages. A suitable value of damping ratio may be adopted to use in these equations.
11.9 THREE DIMENSIONAL ANALYSIS For turbogenerator foundations of more than 100 MW capacity, a three-dimensional space frame model is preferred for analysis. The modelling should take into account the basic characteristics of the system, that is, mass, stiffness and damping. Special attention is required while idealising the points of excitation. Nodes are specified to all bearing points, beam-column junctions, mid-points and quarter points of beams and columns and where the rhember cross- sections change significantly. Generally "thenumber of nodes specified on any member should be sufficient to calculate all the modes having frequencies less than or equal to the operating speed. Lumped-mass approach is used having lumped masses at the node points. The machine shall be modelled to lump its mass together with the mass of the foundation. Equivalent sectional properties of beams and columns are used. The computation of equivalent mass moment of inertia of the frame members pose some difficulty since these depends upon the deflection shape in each mode. These may be discretised in the first step and considered data in an iterative manner if desired. The columns may be assumed to be fixed at the base, disregarding the base mat. A typical space frame model is shown in Fig. 11.9. The dynamic analysis of the frame foundation requires the calculation of Eigen values of the system. The problem can be handled in a systematic manner in the matrix notation. The structure is idealised into a skeleton system which retains the properties of the original structure. The stiffness matrix of the structure as a whole is assembled from the stiffness matrices of individual members. The resulting equations are then solved for the time periods and amplitudes.
...
460
Soil Dynamics
& Machine
Fou/1datio/1s
\
~"
/"7
Fig. 1\.9: Space frame model of the found.ation shown in Fig. 1 \.1
ÎÛÚÛÎÛÒÝÛÍ 11.1 BARKAN, D. D. (1962), "Dynamics of bases and foundations," McGraw-Hill Book Co Inc., Ne\\ York. 11.2 IS 2<174(Pt. [[1-1992), "Foundations for rotary-type macines (Medium and high frequency)". I ,.~ \IAJOR, A. (I %2), Vib:'atlon analysis and design of foundations for machines and turbines, AkademiJI Klado. Budapest, Collet's Hoidlngs Limited. London. 11.4 RAUSCII, E (I 9S(J). ":Y1Jchlnen fundamente und andere dynamisch beanspruchte Baukonstruclionen,"
VDI
VcrlJg. Dusscldorf.
DD
.:,
11
VIBRATION ISOLATION AND SCREENING
2.1 GENERAL n machine foundations, following two types of the problems may arise: (i) Machines directly mounted on foundation block (Fig. 12.1 a) may cause objectionable tions.
vibra-
(ii) Machine foundation suffers excessive amplitudes due to the vibrations transmitted from the neighbouring machines (Fig. 12.1 b).
.~.
Machine
!F
=
Fo s;n
Machine
"'.
Z=ZoSin(Jt Foundation
(0)
Foundation
Excessive vibrations .. du
(b)
Excessive amplitu de due to vibrations transmitted from adjacent source
Fig. 12.1 : Machine directly mounted on foundation
The first problem may be tackled by isolating the machine from the foundation through a suitably designed mounting system (Fig. 12.2) such that the transmitted force is reduced which in turn will reduce the amplitude. This type of isolation is termed as force isolation. This type of arrangement will also help in absorbing the vibrations transmitted from adjacent machines. The system used for this pur:p'ose is termed as motion isolation. For heavier machines, the isolating system may be placed bet\\'~~n 'the foundation block and concrete slab as shown in Fig. 12.3. Here the machines are rigidly bolted to the \:Oundation block which is isol1ted from the concrete slab through the mounting system. The mounting system is an elastic layer which may be in the form of rubber pad. timber pad, cork pad or metal springs. These have been already discussed in Sec. 2.5.
~62
Soil Dynamics & Machine Foundatioll.
Machi ne
.,
'I
Isolator
Foundation
=
Fig. 12.2: An isolator placed between machine and foundation
Machine
Founda'tion
block
Isolator
'i/i;
å óº
¢
ßÁòþ ùþ
'A.
A
.' 6
I" .~
'.
ôßù
t::>"-.l:i'.~~'~-'O""'I\'"
. -,
Concrete A.-.,
/},':
." slab'."
.:..s...".'.;::':
. ' .'
. -..'to.:".'
=
::. .~:~:..~
Fig. 12.3 : An isolator placed between foundation block and concrete slab
The systems shown in Figs, 12.2 and 12.3 can be represented by a simple mathematical model sho\' In Fig. 12.4. In this m represents the mass of machine (Fig. 12.2) or mass of machine plus foundati, block (Fig.12.3). The mounting system (i.e. the elastic layer) is characterised by a linear spring witlspring constant K and dashpot with damping constant C. This mathematical representation involves 0 basic assumption that the underlying soil or rock possess infinite rigidity. This system is identical to t ane shown in Fig, 2.17 (or Fig. 2,19), and the detailed analysis has already been presented in Sec. = considering both force isolation and motion isolation separately.
.
~
.
.
';1',,:,,:;.{
.. . Vibratioll
/so/(ltion
463
(llld Screening
m
Fig. t 2.4 : Mathematical model
A more realistic model will be that in which the soil or rock is considered as an elastic medium. This will make the system as a two- degree-freedom problem. the solutions of which are presented in the next section. The second problem in which the vibrations are transmitted from the neighbouring machines can be solved by controlling the vibrating energy reaching the desired location. This is referred to as Vibration screening. Effective screening of vibration may be achieved by proper interception. scattering. and diffraction of surface waves by using barriers such as trenches. sheet-pile walls, and piles. If the screening devices are provided near the source of vibration, then it is termed as active screening or active isolation. In case screening devIces are used by providing barriers at a point remote from the source of disturbance but near a site where vibration has to be reduced, it is termed as passive screening or passive isolation. Both the methods of screening the vibrations have been discussed subsequently. 12.2 FORCE ISOLATION Since the underlying soil or rock supporting the foundation block (or base slab) does not possess infinite rigidity, the foundation soil should be represented by a spring and not solely by a rigid sllpport as W:iS done in Sec. 8.2. Then the mathematical model becomes as shown in Fig. 12.5. The various ~termsused are explall1ed below: III I
= Mass of foundation block or mass of base slab
11l1 =
Mass of machine if isolator is introduced between machine and foundation block or mass of machine plus mass of foundation block if isolator is placed between foundation block and base slab.
K I = Stiffness of the soil K1 = Stiffness of the isolator
'
111
~," .'t_~
..64
Soil Dynar.lics & Mac/line Folllltlatiol/s
T Zz
tF=FOSinGJt
T Z1
rn,
~kl
////////
///////
Fig. 12.5: Two degrees offreedom model
If the machine is subjected to a harmonic force (Fa sin wt), the equation of motion will be: 111121 -rKI2,-K2(22-21)
=0
11/222 + K2 (22
The solution ofEqs.
-
...( 12.1 )
= Fa
21)
sin ((){
...( 12.2)
(J2.1) and (12.2) gives
(~
nIl n~ )Sinrol
21 =
.
W4 -
K2 + (Kl + K2)
[ 11~
]
1nl
(KI+K~) 2
[
z-
-0)-
~
1111n~
0)4_ Kz+(Kl+KZ) [~ 1nl
.,
. Fa
...(12.3)
w2 + Kl K2
~
nIl nIl
. S111 0) (
]
.F
0)2+KlK2
]
...(12.4) a
nIl 1112
The principal natural frequencies of the system shown in Fig. 12.5 can be obtained by solving the following frequency equation: .; c!) I;
I K~ -
If
L III~ 4(,r) /1
I and
2
(Kl + K~) 1111
-
]
K, Kz
W 1/ + Ill, 1112
CD1/-~ re p resent
...( 12.5)
=0
the roots of the above e quation, Eq . (12.3) can be written as K~ ~
2~'
l IIII III~- )
I
where
tl(0)2)
.F
tl (0/ )
2 4 Kz tl(w ) - (J) - -+ [~
or
. S1l1 0) I
= (0)2
...( 12.6)
0
(KI+Kz)
-0)~1)(0)2
1nl
-0)~2)
(J)2 +-KIKz
]
...(12.7
ll)
1nl n~
...(12.7 b)
IiJ
465
Vibration Isolation and Screening
Force transferred to the foundation block or base slab F{ -= K} Zj Kj K2 -
l
sin (I)f
11/} / 11-,
-
-
)
F
~(c!)2)
...( 12.8 j
. a
The transmissibility of the system will be T
Ft
-F -
...(12.9)
Fa sin OJf Kj K2
or,
1/1}/I~ ') Ll((!)~)
TF =
...( 12.9 a)
The transmissibility TF depends on the system parameters given by Eq. (12.9). ror il1ustration, a special C3se will be examined in which KI - K2 - --p 11/, /110
2
/1/')
Denoting.
...(12.10)
.
...( 12.11)
-=- = mass ratIO = 11 11/,
The natural frequency ignored, is given by
of machine
11/
foundation
system ((Oil::)in which the isolating K)
-
')
-
KI
spring (k2) IS
')
~=~
(J)~:: -= 1/11+ /I~
1+ n~ rn,j
...(12.12)
1+llm
Using Eqs. (12.10), (12.11) and (12.12), the Eq. (12.9) can be written as 1 ...(1~2.13 a)
T, = l ~~- {(I+~ )+(~'" +~f f'r II ~-{(I+~ [
01'."
, -= rI .
1 ;')
.,
., I.-
(j)-.
"(1+~;,
1 11m. ó¥
(
11~I
õî Ø¢³õó¼
Figure 12.6 shows the plots ofT I" with ~
)-(~'" +~},f't}
"
~
l
(!) 11::'
)
r
., (0':'
} 1 l(1+~m)W;
{( I+~;' H~",+~I'
J"}
1
ratio for two values of mass ratio 11 . It is evident from m
this figure that the frequency ratio ((0 I (Oil::)must exceed a particular value (depending on the magnitude of 1111/) before the transmissibility fal1s below unity. The particular values of ((0I (OJ,)for which the transmissibility equals unity are given in Fig. 12.7.
..
.
~
466
Soil Dynamics & Machille 'Foulldatiolls
10 8 6 4
l1..
.
>-
2
'-+'--
.D ,11\
,-11\ E 11\
c
1.0
0.8
0 ....
+-
0.6
c:.I u
.... 0 l1..
).Jm=1.0
0-4
,.urn= 0.5 -
--
}J.m = 0.1 .------Z
0.2
{,Jnz = kIf (rnl +rnZ) kllrnl
=
\
kZ Irn 2
\
0.1 0
0-..5
1-.0
Fr~quency
1.5
\
\
3.0
2.0
ratio,
GJ/GJnz
Fig. t 2.6: T F versus ro/ronzfor different values of mass ratio /Jm
A desIgner is more concerned in examining whether the isolating system reduces the amplitudes of the machine and foundation. III case no isolating system is used the maximum amplitude of machine foundation is given by Fa A. = ...(12.14) 2 ) (/1/ 1 + Ill)) CD . - (I)/1-
(
)
)
In most of the machines. written as
the force Fa is frequency
dependent
(2meeun.
The Eq. (12.14) can be
)
2 me emA; =
(
2
(ml + 11/2) m n;
or where,
-
A. -
2 me e
2 11/1(1+~II/) (al -1) CD
-
a I =-2b... (J)
2
- m
...(12.15
a)
...(12.15
h)
)
...(12.16)
467
VibratiOll lsolatio" a"d Scree"i"g
2.0 I
1.5
W =j<2+)Jm )(l+}Jm) GJnz
I
E =<. 1.0
0.5
0
3.0
1.0
0
4.0
GJ G.)nz
Fig. 12.7 : Mass ratio versus m/mOl
If CDlla represents the natural frequency of mass m2 resting on isolating spring, then CD
-
~ -
2
...(12.17)
IIlz
IlU
The values of maximum amplitudes of two degree freedom system obtained from Eqs. (12.3) and (12.4) for frequency dependent force will be CD~a(2 me e (i)
A=1 = ml
[
2
4 CD -(I+l1m)
2
2
(CDlla-CDII= )CD +(I+l1m)CDlla
2
2
]
...(12.18 a)
CDII=
2
or
A =1 =
(2 me e) . (/2
/Ill
where
Similarly.
[
J 2 J J 1- ( 1+ I1m) (/j + (/2 - (/j (/2
(
)]
UJ/la (/1
=
...( 12.19)
(I)
A=2 = [(l+I1I11)(/~+l1m{/i-l](2mee) m2
~
...(12.18b)
[
l-(I+l1m)
~.-- -~. ,-
2
2
((/( +(/2-(/1
2
...(12.20)
2
(/2
)]
468
Soil DYllamics & Machille Follndations
The Eq. (12.18b) indicates that the amplitude of vibration of foundation will be small if a2 is small.
For this oonashould be small. This can be achieved by appropriate selection of the mass above the isolator spring ("'2) and stiffness of isolator spring (K2)' The efficiency of isolation system is defined as A:I
11 =- A-
...(12.21)
Value of A: can be- ~omputed from Eq. (12.15a). The value of A:I may be taken equal to the permissible amplitude. The value of 11 can also expressed as given below by dividing Eq. (12.18 b) by Eq. (12.15 b). Thus 11 =
ai (1 + ~m) (a~ -1) 2
[l-(1+~m)
2
(a) +a2 -a]
...(
2"
12.22)
a; )]
Solving Eqs. (12.21) and (12.22) one can obtain the value of a2' For this a2' appropriate values oflll2 and K2 are selected. It will ensnre the amplitude of vibration to be within pf'mlissible limit ...( 12.23)
Total force on the isolator, Fa = Kz . A:2 Fa should be less than the allowable capacity of the Isolator in compression. 12.3 MOTION ISOLATION
Let us examine a case when a sensitive equipment of mass 11/I ISplaced on a foundation block of mass 1112' The spring K, represents the foundation soil and spring K2 is an isolating spring which is placed bet\\een the masses Ill, and 1112' in order to minimise the transmission of vibrations from the ground to the equipment. If the ground is subjected to a periodic displacement given by Zo sin w l. the equations of motIOn \vill be: nzl ZI + KI Z\
-
K2 (Z2 - ZI)
11/2Z2 +K2(l2-Z,)
= KI
...( 12.24)
Zo sin W f
...(12.25)
=0
The vafues of maximum amplitudes of motion are given by 2
~-~ ml m2
A_I = KI lo -
004 -
[
m1
...(12.26)
(KI + K2) +- K2 W2 +- Kl K2 m, 1112] 11I,n/2
~ 111) Ill.,
A :2 = K 1 Z 0 W
-
K I+J K )
4
- [(
Ill,
-
KJ
2)J
K K
...( 12.27)
+ III~] W + Ill, 111;
The displacement transmissibility of the machine (Tn) is defined as the ratio of displacement amplitude of mass 1f/2to the displacement amplitude of the rigid support. Then KIK2 A:2
111,1112
T --0- Zo - w4_
(KI+K2)+K2
[
~
111)
...( 12.28) ())2+K)K2
1112Ã
111)11/2
I
469
Vibration Isolation alld Screenillg
The Eq. (12.28) is identical to Eq. (12.9), and therefore the results shown in Figs. 12.6 and 12.7 hold good in this case also.. Equation (12.28) can also be written as: A :2 l -
To -
0
2 2 a( a2 (1 + Ilm) l-(l+llnr)
If A-1 -- is taken as permissible amplitude. and l
0
A:2i lQ is known. For this value. Cl2can be determined spring i.e. K2'
12.4 SCREENING
OF VIBRATIONS
(
2
2
al +°2 -al
2
...( 12.29)
.,
a2
)
is the applied dynamic displacement. then the ratio which in turn will give the stiffness of the isolator
BY USE OF OPEN TRENCHES
12.4.1. Active Screening. In this case the screening of vibrations is done near the source of vibration. Figure 12.8 shows a circular trench of radius R and depth H which surrounds the machine foundation that is the source of disturbance. The design of trench barriers is based on some field observations. Barkan ( 1962) mentioned that the reduction in vibration amplitudes occurs only when the trench dimensions are sufficiently large compared with the wave length of the surface waves generated by the source of dIsturbance. Dolling (1966) studied the effect of size and shape of the trench on its ability to screen the vibratIOns.
F (t)
Amplitude of s u da c e displaclZmlZnt
Footing L'
Circular open t re n c h ot radius R an d depth H
I
I
~
H
1
R-1
Fig. 12.8: Vibration screening using a circular trench surrounding
the source
of vibration-Active screening (Woods. t 968)
The first comprehensive study of screening vibratIOns by use of open trenches was made by Woods and Richart ( 1967) and \Voods ( 1968). They conducted field tests by creating vertical vibrallons with a small \"ibrator resting on a small pad at a prepared site. The vibrator could create a maximum force of ~O0:. The soil conditions at the site \vere as shown in Fig. 12.9. The water table was below 14.3 m depth. The depth H of trenches was varied from 150 mm to 600 mm, the radius R of annular trench varied from 150 mm to 300 mm, and the angular dimension e was varied from 900 to 3600 around the source of vibration. Frequencies of 200 to 350 Hz were used in the tests. Using velocity transducers. the amplitudes
-
. 0;"
,"
"'
i;',C",'.
."0,'. ',c.,
470
Soil Dynamics & Machine FolllldatiOlls
of vertical ground motion were measured at selected points throughout the test site before installation of the trench and after installation of the trench. Woods (1968) has introduced a term amplitude reduction .factor which is defined as ARF = Amplitude reduction factor - Amplitude of vertical vibration with trench - Amplitude of vertical vibration without trench
T
1.2m
uniform
Wn
= 7
silty 0/0,
Vc
3.1 m
~
~
Wn =
=
fine
sand
e = 0.61) 287 rnls
td
sandy
silt
230/0)
Cl = 0.68)
(SM) = 16.4.6 k N /
m3
(ML) 'id = 15.77 kN/m3
Vc=534.m/s
Fig. 12.9: Soil stratum at the test site
Some of the resuits of field tests conducted by Woods (1968) are shown in Fig. 12.10 in the form of ARF contour diagrams. The dimensions of the trench are expressed in non-dimensional forms by dividing Hand R by the wave length ARof Rayleigh waves. ARis obtained by determining the number of waves (/1L occuring at
in Table 12.1.
distance x from the source (AR= x/1I).Wavelengths ARfor different frequenciesare given
Table 12.1: Wavelength and wave velocity for the Rayleigh Wave at the test site (Woods, 1968) Frequency Hz
AR mm
m/s
lOO
687
137
250
513
128
300
421
126
350
336
117
VR
'".:,.
-~.,,:i,:>},\
.
(,"
.;'if'..":.
.~.
471
Vibratio" Isolatio" a"d Scree"i"g
Boundary
of screened zone
/..
-
1,25
~>1.25
D
H/"'R 1.452
1.25-0.50
RO/AR 0.726
D
1.25-0,50
f:::~::JO,50-0'25
F::,::':':lo,50-0,25
1::::~::)lo,25-0,125
1(':,\/:1 0,25 -0.125
:-,:,':,-:- < 0 ' 125 t~:;:.~
,',', '," p',-::-:,:,) < 0 ' 125
H/AR 0,596
Ro/AR 0,596
(b)
(a)
Fig, 12,10: Amplitude reduction factor contour diagrams for active screening (Woods, 1968)
The field tests of Woods (1968) thus correspond to
A
r-R
H
= 0.222 - 0.910 and -;= 0,222 - 1.82 AR'
For satisfactory screening of vibrations, Woods (1968) recommended that ARF shouid be less than or equal to 0,25, The conclusions made on the basis of this study to keep ARF ~ 0,25 are:
(i) For full circle trenches (8 = 360°) , a minimum value of H/AR = 0.6 is required. The zone screened in this case extended to a distance of atleast 10 wavelengths (I OAR)from the source of disturbance (ii) For partial circle trenches (900 < 8 < 360°), the screened zone was defined as an area outside the trench extending to at least 10 wave lengths (10 AR)from the source and bounded on the sides by
...
,~
"""
.t72
Soil Dynamics & Machine
Fouudations
radial lines from the centre of source through points 45° from ends of trench. In this case also. a minimum value of H/AR = 0.6 is required. (iii) Partial circle trenches with 8 < 90°, effective screening of vibration is not achieved. (i,') Trench width is not an important parameter. 12A.2. Passive Screening, Woods (1968) has also performed field tests to study the effectiveness of open trenches in passive screening (Fig. 12.11), A typical layout of these tests consisting of two vibration e:\citers (used one at a time for the tests), 75 transducer locations. and a trench is shown in Fig, 12 12. The sizes of trenches ranged from 100 mm x 300 mm x 300 mm deep to 2440 mm x 3050 mm x 1220 mm deep, Frequencies of excitation varied from 200 to 350 Hz.
Amp! itude 0 f s u rfa c e vibration
Source of disturbance .'
. .
..
,
Amplitude of surface vibration ..
,
Equipmrznt to be protected
.
~
.' ,
,
, ,
. '.
~
..
.
"
-.
. .' t.
r
, . :'
-.
R
.
"
::.
,. . Open t rrznch Fi~. 1:!.I] : "ihration
~ ';
H
"
.'
1
screening using a straight trench. Passive screening (Woods. I96S)
The values of Hij'I{varied from 0.444 to 3.64 and R/AR from 2.22 to 9.10. It \vas assumed in these tests that the zones screened by the trench would be symmetrical about the 0° line. Figure 12.13 shows the ARf contour diagram for one of these tests. For satisfactory screening. Woods (1968) recommended that the ARF should be less than or equal to 0,25 in a semi-CIrcular zone of radius (1/2) L behind the trench. The conclusions made on the hasis of this field study to keep ARF ::: 0.25 are: (i) I li'I{ should be atleast 1.33 (id To maintain the same degree of screening, the least area of the trench in the vertical direction ((c,
1.11"" AI)'
should
be a<; follo\\'s:
-AI ,~ I'R
R -=2 'i at -=7 .) 'R
AI -, ,/'R
R = 7.0 .C6.0 at /' 'R
-.
0
..
m 473
Vibratioll Isolatioll alld Screenillg
(hi) Trench width had practically no influence on the effectiveness of screeI'.ing. Experimentai investigations of Sridharan et. al (1981) indicated that the open unfilled trenches are the most effective. However the open (unfilled) trenches may present instability problems necessitating trenches backfilled with sawdust, sand or bentonite slurry. The performance of open trench \vith sawdust was found better as compared with sand or bentonite slurry.
320 240
/
/
0
N
0
0
/
~
",
0
0
0,
/
I
0 ""'
0
0
O
0
I
0
0 0 L
~'
' , . ",,~" .,..""
-', ;{~:', ,
0
0
0
,/
.
"
\,
" /
.\,
0
0
l.Sm
./
/'t 0
75 pi c k u P benches
0
r>.
..
0 0
0
0
',0 0 ',('", ", , ' " "
0
/
0
~
0
<, " '<:(
0
0
0
0
O
/".
/
0
0
OOA
/
0
0
/0
0
0
0
0
0
0
0
0
/
0
80
0
6.1 m /
0
0
/
[°
0
0
0
16
v' \,
'."
K
Tr en' ch
barriers
,
,,"
'>
/
"v :?s B
//
0 0
/ l.Sm
/
vibration
exciter
footings
Fig. 12.12 : Plan \iew of the field site layout for passive screening (Woods. 1968)
474
Soil Dynamics & Mac/,ine Forllldatioll.'i
Centqr tin q
lliII1J ARF>1.25
0
ARF 1.25-0.5
[tJ
+
AA F 0.5-0.25
n
ARF
ó¬ó´ó
+
+
. Q
4.6m
Trench
0.25-0.125
~ ARF<0.125
m HI AR
=
2.38
LIAR
=
4.76
=
5.96 0.17
1.5
R I AA W/'AR
=
Fig. 12.\3: Amplitude reduction factor contour diagrams for !Jassive screening
12.5 PASSIVE SCREE1\I;\C
BY USE OF PILE BARRIERS
There may be situations in which Rayleigh wavelengths may be in the range of 40 to SOm. For such a casc. the open trench will be effective ifits depth range from 53 m to 66 m (i.e. 1.33 AR)' Open tren~hes (filled or unfilled) with such deep depths are not practical. For this reason, possible use of rows of piles as an energy barrier has been studied by Woods et al. (1974) and Liao and Sangrey ( 1978). Woods et a1. (1974) used the principle of holography and observed vibrations in a model half-space to evaluate the effect of void cylinderical obstacles on reduction of vibration amplitudes. A box of size 1000 mm x 1000 mm x 300 mm deep filled with fine sand constituted the model half-space (Fig. 12.14). In this figure. 0 is the diameter of the void cylinderical obstacle and SI! is the net space between two consecutive void holes through which energy can pass through the barrier. The numerical evaluation of barrier effectiveness was made by obtaining the average of the values of ARF obtained on several lines beyond the barrier in a section ::I:15° of both sides of an axis through the source of disturbance and perpendicular \0 the barrier. In all tests. H/AR and LIARwere kept as 1.4 and 2.5 respectively. The Isolation effectiveness is defined as
Effectiveness = I
-
ARF
...(12.31)
;"'ii[;
Vibration
Isolation
475
and Screeni1lg
F(!)
Fig. 12.14 : Definition of parameters
t
for cylindrical hole barriers (Woods et aI., 1974)
1,0
LL
a::
« I
11
0,5
."
111
"'-
111 t>I
C
I
t>I > .;: u
\\\ \ \
I
'O"
"-
,
.:
"',
~',
,
0:
"
,
.
"
"
'
0,:
"
':""
J
. '. . . . '0.: . '0:~, :', ': :' "'"
0.4
'0,. 0 ""
' :'0
J:-
'
,
0,15
t>I
....
....... W
. . l
" , 0,.
I
,,\\
"-
\\0.125
\\
0.21
"-
"-
\ 0.075
"
\
0 0
0.05
0,10
0.15
0.20
0.25
Snf"r Fig. 12.15: Isolation effectiveness as a function of hole diameter and spacing (Woods et al.. 1974)
0,30
""1iItJ'i
",","',,'
">,,',,'
-
,
.P6
Soil Dynamics & Machine Foundations
L Slllg the data of different tests, a non-dimensional plot of the isolation effectiveness versus S/A.R ratio for different vaues ofD/A.R was plotted as shown in Fig. 12.15. Woods et al. (1974) recommended that J row of void cylinderical holes may act as an isolation barrier if D 1
-
->-6
AR
I
SI! -
:.1nJ
...( 12.32)
...( 12.33)
4
Soil
type
Barrier
materials
107
f 10
Infinitely rigid pile
6
VI
-0 C
Steel
0
ZI
VI
N
10
.
5
E
Concrete
QJ
u c
0 -0
10
4
Gravel
Tim ber
0.
'-E > 0
0 ens e san d Hard clay
103 -
Si!t Loose sand Very soft clay
..c
crI
-
>0 Cl:
10
2
Plasti c foam
1 10 e--
.
Void boreho le
0 10 '--- ! Fi:!. 12.16 : Estimatcd \'aluc\ of Haylcigh wave impedance for ,'arious soils and p:le materials (Liao and Sangrcy. 1978)
I.lao and Sangn:y (1978) used an acoustic modcl employing sound waves in a fluid medium to ,lIuak thl' possibility of the use of row of piles as passive isolation barriers. They have studied the effect of dlal1l~.t..:r.spacing. and material properties of the soil pile system on the isolation effectiveness They ~'\
'~\'II1~'iLld~d
that:
!Z.'...,'~J
477
Vibration Isolation and Screelling
(i) The Eqs. (12.32) and (12.33) proposed by Woods et al (1974) are generally valid. (ii) Sn = 0.4 ARmay be the upper limit for a barrier to have some effectiveness. (iii) The effectiveness of the barrier is significantly affected by the material of the pile :md void holes. Acoustically soft piles (IR < 1) are more efficient than acoustically hard piles (IR > 1). IR is impedance ratio which is defined as IR = where.
Pp
Pp VRP Ps VRS
...( 12.34)
= Density of pile material
Ps = Density of soil medium VRP = Rayleigh wave velocity in pile material, \' RS = Rayleigh wave velocity in soil medium Figure 12.16 gives a general range of the Rayleigh wave impedance (p V R) for various soils and pile materials. .
(iv) Two rows of barriers are more effective than single row barriers,
jILLUSTRATIVE
EXAMPLES'
Example 12.1 Determine the stiffness of the isolator to be kept between a reciprocating machine and the foundation shown in Fig. 12.17 to bring the vibration amplitude to less than 0.02 mm. The weight of the machine is 18 kN, and it produces an unbalanced force of 4.0 kN when operated at a speed of 600 rpm. The dynamic shear modulus and Poisson's ratio of the soil are 2 x 104 kN/m2 and '0.35 respectively, Solution: 1.
Mass of the foundation
block =
24x4.0x3.0x1.5
9.81
')
= 44 kNs-/m
Mass of machine = 9~~1 = 1.83 kNs2/m Total mass of machine and foundation = 44 + 1.83 = 55.83 kNs2/111 Stiffness of the soil,
K==
r0 =
4 G ro 1- ~l
f;fA = ~4 1tx 3 = 1.95 m 4
K = 4x2xl0 =
1-0.35
x 1.95 =24
Natural frequency of the whole system without any isolation will be:
x
104 kN/m
478
Soil Dynamics & Machine Foundation
Machine Weight
18kN
=
T 1.5m
1
blo c k
Foundation
///,\,\
r
/\\
~
4.0 m Section
3.0m
~
~
4.0 m Plan Fig. 12.17: Machine-foundation
CD n;
~
-
--
m
system (Example 12.1)
4 24 X 10 1 55.83
= 65.0
rad/s
2 Ttx 550 CD= 60 = 57.6 rad!s (A ) ; max -
Fa 2
2 -
m(oo//;-oo)
4.0
2 ., 55.83(65.0 -57.6- )
=-07.9¨ 10-501=0.0790101
479
Vibratioll Isolatioll alld Screellillg
The amplitude is greater than the permissible amplitude i.e. 0.02 mm. Hece isolator is required between machine and foundation. 2. Let the isolator be having stiffness Kz. Adopting the two degrees freedom system as shown in Fig. 12.5. 2
11/2 =
1.83kNs Im 2 11/1 = 44 kNs Im 4 Kl = K: = 24 x 10 kN/m - = 65.0 rad/s 1/-
(0
0.02 = 0.253
AZI -
11= Az - 0.079
= 65.0 = 1.13
Cl)n:
a I = ---;;'nz
57.0
- 1.83 = 0.0416
/-1./11 = 1111-
44
From Eq. (12.22) 11 =
or
'"' -0.k53
or
a~
a~ (l+/-I.m)(a~-l) 2
[l-(1-llm)
2
(al +a2 -al
2
2
a2 )J
2 ., a2(1+0.0416)(1.13--1) 2
2
[1- (1 + 0.0416) (1.13 + a2 -1.13
2
2
a2 )J
= 0.2310 i.e. a2 = 0.481
(jJ
11i!...= 0.481. It gives (011(1 = 0.481 x 57.6 = 27.7 rad's
(j)
.. 3.
Ko = 1112 A
Z2
=
(jJ
2
1
Il{/
= 1.83 x 27.7- = 1.40
x
[(l+llm)a~+Il",a;-1][2mee(O2] 2 2 1 1 2 1112(0[ 1-(l+1l",) al +az -aj ai
(
=
3
10 kN/m
)
[(1+0.0416)x1.l32+0.0416xO.2310-1.-(4.0) ~ 1 1.83 x 57.6-
= - 8.5
1-(1+0.0416)
(1.13- +0.2310-1.13-
~ A
0.2310)
-~
x
Force in the isolator = Ko' An "'1.18kt'
10 m
= lAD x
103 x 8.5 x 10-~kN
110
480
Soil Dynamics
& Machine
Foundations
A suitable isolation system may be selected which has total stiffness of 1.40 x 103kN/m and allowable compressive 'load more than 1.18 kN.
Example 12.2 Determine the stiffness of the isolation system if it is placed between the foundation block and base slab as shown in Fig. 12.3. Use the data given in example 12.1. Solution: (i) Let the foundation block of size 4.0 m x 3.0 m x 1.0 m high is rigidly connected Isolators are placed between this block and base slab as shown in Fig. 12.18. Then
with machine.
Machine
T
Foundation
1.0 rn
block
i Isolator
T
O.Srn
Base
-L
~
slab
~
4.0 rn Fi~. 12.18: :\-Iachine-foundation
1//2
==
1.83 +
isolator system (Example 12.2)
-}x 44
= 31.2 kNs2/m
I 111( == 3"
J
x
44 = 14.7 kNs-/m 4
2
= 24 x 10 kNs fm (j)n:: = 65.0 rad/s Kl
11 = 0.253
From eq. (12.22),
al = 1.13 31.2 2 12 1111/ = t4.7 = . 2 2 a2 (1+2.12)(1.13 -1) 0.253 = 2 2 221.13 '+a2 -1.13 ([2)] [ 1-(1+2.12)
(
.-
WA.--'}..',..".~;:.'
bratioll Isolation
,"'ti'~:,,<.
,."
';",-,:,.
"~,<~Atj;!~";,;
"'.,:,' ">:!',., E
481
alld Screellillg
or
ai = 0.697 i.e. a2 = 0.835
Hence,
O)na = 0.835. It gives (J)na= 0.835 x 57.6 = 48 rad/s 0) 2
4
Ka = 11120)2 na =31.2 x 48 = 7.2 x 10 kN/m (ii)
[0+2.12)
- An =
.,
x 1.132 +2.12 x 0.697 -1]4.0
r
31.2 x 57.6- Ll-(1+2.12)
2
(1.13
2'
+0.697~1.13
x 0.697) ]
17.85
= 103514[-2.38] = - 7.24 x 10--5m Force in the isolator = Ka A.,=- ==7.2 x 104 x 7.24 x 10- 5 = 5.2 kN On comparing with the results of example 12.1, it can be concluded that the stiffness of the isolator and force on it depends significantly on the location of the isolator.
Example 12.3 It is planned to install a compressor having operating speed of 1000 rpm at a distance of 50m from a precision machine. Suggest a suitable open trench barrier to provide effective vibration isolation. The velocity of shear waves at the site was found as 140 mIs. Solution: Active screening 1000 f = 60 = 16.7 Hz
.
OperatIng frequency,
Rayleigh wave velocity VR may be taken approximately equal to shear wave velocity i.e. 140 m/so V Therefore,
Wave
length
AI{ ==
;
140 == 16.7 ==8.4 m
Depth of the trench for active screening is given by
H
==0.6 AI{
= 0.6 x 8.4 ==5.04 m
A partial ci~cle trench with e = 120° may be located at 4.0m distance from the source (Fig. 12.19 a), Passive screening Depth of the trench for passive isolation is given by .
.
H = 1.33 AR= 1.33 x 8.4 = 11.2 m
Let the trench is provided at a distance of 12 m from the precision machine.
For
R - (50-12) = 4.52 (O.K. as lies between 2 and 7). -AR 84. R = 4.52, AT = 25 4.52-2.0(6.0-2.5)=4.26 ')..2 . + 7.0-2.0 AR R
"1I
Soil Dynamics & Machille Foulldations 4.26 x 8.4 Length of trench =
2
11.2
= 26.8 m say 27 m
The layout of trench with respect to compressor
and precision machine is shown in Fig. 12.19b.
Precision machine
Precision
machine
T 12n
Tren ch
)
I 27m
{
I 50m I
38m. 0
120
Tre
../
,/
eññ£³
.
Co m pre ssor
Compre ssor (a)
Active
isolation
12.19.-Layout
-'-
(b)
Passive
isolation
of trench with respect to compressor and precision machine
~FERENCES "kan, D. D. (19621 "Dynamics of bases and foundations", McGraw Hill, New York." Iling, H. J. (1966), "Efficiency of trenches in isolating structures against vibration", Proc. Symp. Vib. Civ. Eng. Butterworth, London. .0, S, and Sangrey, D. A. (1978), "Use of piles as isolation barriers", J. Geotech. Engg. Div.. Am. Soc. Clv. Eng.. 104, (GT9), 1139-1152. dharan. A. , Nagendra, M. V. and Parthasarathy, T. (1981). "Isolation of machine foundations by barriers", 1nL Conf. Recent Ad\'. Geotech. Earthquake Eng., St. Louis, Vot. 1,279-282.
lads. R. D. (1968). "Screeningof surfacewaves in soils", J. Soil Mech.Found.Div. , Proc. Am. Soc. Ci\'. Eng., 94. (SM-4),951-979.
483
'atio" Iso/atio" a"d Scree"i"g
(1967), "Screening of elastic waves by trenches", Proc. Int. Symp. Wave Propag. Dyn. Prop. Earth Mater, Albuquerque., NM, 275-284.
)ds, R. D., and Richart, F. E. Jr.
lds, R. D., Barnett, N. E. and Sagessor, R. (1974), "Holography - A new tool for soil dynamics", 1. Geotech. Eng. Div., Am. Soc. Civ., lOO, (GT-I1),
1231-1247.
PRACTICE PROBLEMS !.1. Explain the difference between 'force isolation' and 'motion isolation'. Sketch a suitable system for 'force isolation'. Represent it by a mathematical model and then give the procedure of getting the stiffness of the isolator. 2. Starting from fundamentals, derive the expression for the efficiency of isolation system. 3. Explain the difference between "Active screening" and "Passive screening". Give the procedure of designing the open trench barrier in both the cases. 4. Give the salient features of passive screening by use of pile barriers. 5. Design a suitable isolation system for keeping the amplitude of the foundation of a reciprocating machine less than 0.025 mm. The weight of the machine is 25 kN and it produces a sinusoidally varying unbalanced force of 4kN in the vertical direction. The operating speed of the machine is
800 rpm. The dynamic shear modulus 'of the soil is 2.5 x 104kN/m2. Assume suitably any data not given. 6. A compressor having an operating speed of 1300 rpm was installed in an industrial unit. Later on it was planned to place a precision machine at a distance of 60 m from it. It was felt necessary to protect this precision machine from any damaging vibration caused by the compressor. Design open trench barrier to provide effective vibration screening for the cases of (a) active and (b) passive screening. The velocity of shear waves was found as 160 m/s.
DD
-- -----
SUBJECT INDEX F f
A
Culmann's construction:
f
modified, 191
f
Amplitude of motion, 15 Amplitude method, 454
Cyclic: mobility, 279
Antiliquefaction measures, 324
simple shear test, 118, 128, 286, 296 torsional shear test, 118, 131, 286
B
triaxial compression test, 118, 133
Bandwidth method, 57 Bearing capacity of footings, 238 factors, 241
D Damping:
critical, factor, ratio,
Blasting tests, field, 314 Block foundations: degree of freedom. 352 effect of shape on response, 394 embedded. 394 method of analysis, 353 modes of vibration, 352 rocking, vibrations, 354, 360, 383, 396 rocking and sliding vibrations, 354, 363, 389, 397 sliding vibrations, 354, 359, 379, 395 vertical vibrations, 354, 358, 370, 394 yawing vibrations, 354. 362, 386 Block-resonance test: horizontal, 154
Degrees of freedom, 14 Displacement analysis of retaining wall, 20 I in pure rotation, 210 in pure translation, 205 - using Richard Elms model, 201 using Saran et al model, 214 Dynamic bearing capacity, 238 dynamic analysis, 252 pseudo-static analysis, 238 Dynamic earth pressure, 187 effect of saturation, 187 effect of submergence, 190 effect of uniform surcharge, 189 for c- soils, 193 Mononobe-Okabe's theory. 200
vertical,151
c CodTicienr of clastic:
point of application, 200 pseudo-static methods, 238
nonllniform col11pn.:ssion,356 nonuniform shear, 357 uniform compreSSion,354 uniform shear. 356 CrItical distance, 96,98, 100
'
E Earthquake: epit:enter, 3 equivalent dynamic load, 6 focus, 3
J
'ct Index
485
M
:ensity, 4 19nitude, 5
Machines
constants, 68
types, 340
constants, 68 half-space method, 370 ded block foundation, 394
reciprocating, 340 impact, 344 rotary, 345 Machine foundations categories of, 340 criteria for satisfactory action, 347
sS :
permissible amplitudes of, 348
:aring capacity, 238 :neralised bearing capacity equation, 240 )rizontal displacement, 249 ttlement, 249 t, 249
Mononobe-Okabe's theory, 187 Motion amplitude of, 15. harmonic, 15 periodic, 15 isolation, 35
ncy: Ircing, 15 ltural, 15
N Natural frequency,
ysical prospecting, 93
15
p t machine foundations,
421
Periodic motion, 15
~sign procedure,
432
Permissible amplitudes, 348
v'namic analysis,
426
'Pt time, %,97,
100
Phase Jag, 16 Phase lead, 16
R spring method, 354
Resonance method, 450
'action, 279
Resonant column test, 118, 119
Ictors affecting, 323 eld tests for, 314
Rotating mass type excitation, 30
.om standard penetration tests, 319 litial, 279 Iboratory studies, 283 lechanism, 281 :andard curves and correlatIOns, 30 I .:rr11lnology,279 one of, 306
Rotary machines design criteria, 445 loads on, 446 three dimensional analysis of, 460 resonance method of analysis, 450 amplitude method of analysis, 454 combined method of analysis, 459
486
Soil Dynamics & Machine Foundations
s
fiee, 19,22,40,43
Sc\eening of waves, 469 Seismic:
isolation, 32 force, 32 motion, 35
coefficients, 9 cross-borehole survey, 147, 148 down-hole survey, 147, ISO forces, 9 zones, 9 refraction, 94, 147, 151 up-hole survey, 147,150
measuring instruments, 36 acceleratioI" pickup, 38,39 displacement pickup, 38 velocity pick up, 39 Vibrations of rod end conditions, 74 fixed-fixed, 75. 79 fixed-free, 78 free-free, 74, 76
Shake table tests, 286, 309 Shear modulus, 163,165,171
of finite length, 76 longitudinal, 76 torsional, 80
Soil mass participation in vibration, 400 Standard penetration test, 159
of infinite length, 70 longitudinal, 70 torsional, 72
Strain, 67
T Theory of vibrations,
w
13
Wave propagation
Time period, IS Transient tests
in elastic half space. 86 in elastic rods. 70, 72, 76, 80 in infinite medium, 81
u
Waves
Ultrasonic pulse test, 118,126
compressIon, 83, 108 head, 95
Lndamped free vibration, 19
Rayleigh, 9\ shear. 84, 108
v Vibration absorber, 48
z
forced, 25,42,47
Zone of liquefaction, 306
ODD
if-:"
. _'M-
_.-
~
,
'----""~"'-:--~"'--:-'""~'l ",
.\. ,- ',.
~ . :,
1
., ~
\
~
I
k~,
" '-
J
~L.._":~",-"
-,.~.~
:.c
,.:..."""",:"""""""","