In order to study the dynamic behavior of a shaft in soil during earthquakes,

dynamic centrifuge tests were carried out. Two types of models were taken up; (1) level ground

of sand; and (2) a hollow shaft of two different depth embedded into the level ground. An

effective boundary treatment method was presented so that the semi-infinite ground condition

can be satisfied approximately. The effects of input level and shaft geometry on the response

of shaft-soil system were discussed

Hayashi, H.

Honda, M.

Ohbo, N.

Scott, R. F.

Zheng, J.

Caltech Authors - Main

Centrifuge 94, Leung, Lee & Tan (eds)© 1994 Balkema, Rotterdam, ISBN 90 5410 352 3 
Dynamic response of a shaft in dry sands 
M. Honda, N. Ohbo, H. Hayashi & J. Zheng 
Kajima Technical Research Institute, Tokyo, Japan 
R.F.Scott 
California Institute of Technology, Pasadena, Calif., USA 
ABSTRACT: In order to study the dynamic behavior of a shaft in soil during earthquakes, 
dynamic centrifuge tests were carried out. Two types of models were taken up;(l)level ground 
of sand; and (2) a hollow shaft of two different depth embedded into the level ground. An 
effective boundary treatment method was presented so that the semi-infinite ground condition 
can be satisfied approximately. The effects of input level and shaft geometry on the response 
of shaft-soil system were discussed. 
1 INTRODUCTION 
The behavior of underground structures 
during earthquakes can be simulated on a 
centrifuge much more realistically 
compared with conventional model test. In 
view of the above, the seismic behavior of 
hollow shaft embedded in a dry sand 
deposit was modeled experimentally on a 
centrifuge at 1/50 scale in order to 
investigate the dynamic behavior of shaft-
soil system during earthquakes. Main 
attention were given to the following two 
aspects: ( 1) to propose a boundary 
treatment method; (2) to study the effects 
of input motion level and shaft geometry 
on the response of shaft-soil system. 
2 PRELIMINARY CENTRIFUGE TEST 
ON THE BOUNDARY TREATMENT 
One of the problems in physical modeling 
of dynamic phenomena is that the model is 
bounded by the container walls, and the 
artificial boundaries at the end and side 
walls can significantly affect the response 
of ground model. Thus it is important to 
reduce both the end wall effects and the 
friction between side wall and the ground 
model<!). In this section. a method to treat 
the boundary at the container walls is 
proposed. The verification of the proposed 
boundary treatment method is also 
presented through both dynamic 
centrifuge tests and the 1-D numerical 
analysis(2). 
2.1 Experiment arrangement 
Silicone rubber was placed between sands 
and end wall, as shown in Fig.I, so that the 
shear deformation of the ground model 
would not be greatly affected by the end 
wall. The stiffness of silicone rubber is 
determined by preliminary numerical 
analysis. The Young's modulus of silicon 
rubber is 2.4kgf/cm2, and the damping ratio 
is 4.8%(at the first peak). These values 
3cm 44cm 3cm 
TT ,, 
Container 
1 
'==T=O=Y=O=U=R=A===:'...l ]18cm ~ sand 
./ > 
excitation direction 
Fig. I Outline of preliminary test 
257 
TOYOURA sand 
_ll 
Container 
l 195mm l 
Fig.2 A section of container 
Table. I Preliminary test condition 
height of relative density 
maximum acceleration ground mode I of ground mode I 
h Or of input motion 
with boundary 18. 4cm 95.2% 1. Sg 
treatment 
i w 1 :hou t boundary 17. 4cm 97.9% 1. Bg 
i ~reatmentj 
were obtained from lg excitation tests of 
the silicone rubber. Rubber membrane was 
set up between sands and side walls with 
grease pasted along the interface between 
them, as shown in Fig. 2. The rubber 
membrane is supposed to move with the 
ground model, and the grease reduces the 
friction between rubber membrane and side 
walls. 
To verify the proposed boundary 
treatment method, two types of the 
boundary conditions(with treatment and 
without treatment) were considered. Fig.I 
shows the cut view of the experiment 
model. Table l shows the model condition 
and the maximum acceleration of input 
motion for two test cases with different 
boundary conditions. All tests were carried 
out under 50g centrifugal condition. The 
input signal for the shaker was intended to 
simulate the 1940 El Centro NS wave. The 
TOYOURA standard sand was pluviated 
through air directly into the container to be 
the model ground. Fig. 3 and Table 2 show 
the physical properties of TOYOURA 
standard sand. 
g 100 
BO 
60 
00 
c: 
en 
en 
"' a 
if 
ll 
0. 01 0. 1 
Grain diameter(mm) 
Fig.3 Grain size distribution of 
TOYOURA standard sand 
Table.2 Physical properties of 
TOYOURA standard sand 
~ 
E 
··········· without boundary treatment 
---- with boundary treatment 
-- 1-D simulation 
30~~-~-~-~~ 
:a 20 
e 
< 
0o 2 4 6 8 10 
' Frequency(Hz) 
Fig.4 Effects of boundary condition 
on transfer function 
2.2 Test results 
Fig.4 for the preliminary test shows the 
transfer functions for two cases of different 
boundary condition computed from the 
ratio of Fourier spectra of response at the 
surface to those at the bottom. The 
frequency shown in these figures is at 
prototype scale. The result of 1-D non-
linear simulation using lumped mass model 
is also shown in the same figure. The input 
motion in 1-D simulation is the acceleration 
observed at the bottom. 
258 
40 llO 100 
'Al.2 A4·~A3 
• a 
A9 Al2• AlO 
16 All, 
Container 
I Plan Vie• I 
40 
• AB 
Shaft model 
(H/D=5) 
I Side View I 
40 
Al2 All, AID 
A4 I llr.A""'3 .....,.,,.fl 
~ Al~ A5 , TOYOURA 
ii Al4 A6 I Sand 
~ AlS .\7 i 
Container 
direction of vibration 
Silicone 
(H/D=5) 
\ Shaft model \ 
H/D=5 H/D=l. 5 
Gt=2 2 @=2 
~ 60 --..j jE- 60 --I 
A4 All AIO A3 
Sl S2 
AS El E2 
E3 E4 
S3 S4 
A4 All AlO A3 
Sl S2 ~ A5~~j~ 
E3 A6 E4 
E5 E6 ;:: 
-
S5 S6 aA7 
E7 E8 
= ;:: 
A2 
o A : Accelerometer (Horizontal) 
• A : Accelerometer (Vertical) 
r- E : Pressure Transducer 
• S : Strain gauge 
Fig.S Outline of experiment model 
In comparison between the case with and 
without boundary treatment(silicone 
rubber and rubber membrane), large 
difference could be observed around 7Hz 
in the amplitude of the transfer function. In 
the case without treatment, there is not a 
clear peak. On the contrary, the transfer 
function in the case of boundary treatment 
is very similar to that computed by 1-D 
simulation, which demonstrates the value 
of the proposed boundary treatment 
method. 
3 CENTRIFUGE TESTS OF A SHAFT 
INDRYSANDS 
The dynamic centrifuge tests with three 
sorts of arrangements(only ground model, 
ground model with long shaft and ground 
model with short shaft) were carried out. In 
this section, the effects of the shaft 
condition and input amplitude level on the 
response of acceleration are discussed. In 
addition, the influence of shaft geometry 
on the dynamic earth pressure and bending 
Table.3 Test condition 
shaft inout m. 1nout height of relative density of 
condi tlon wave a<:cel. W model (co) ground oode I Or 00 
case 1 
without 1. 2 33. 0 82 
case 2 9. 2 33. 0 83 
case 3 short El 1.1 32. 7 87 
case 4 (H/0=1. 5l Centro 9. B 32.6 BB 
:::: : long 1.2 33. 0 83 (H/0=5l 10. 0 32. 9 85 
case 7 shart SinelOOHz 4. B 32. 9 BB 
case B GVO=l. 5l Sine2fi0Hz SB 32. 9 B9 
case 9 long SlnelOOHz 4. 5 32. B BB 
case10 QV0=5) Si ne2ff!Hz 7. 3 32. B 87 
moment of the shaft are studied by 
analyzing the test results. 
3.1 Experiment arrangement 
Fig.5 shows the outline of the experiment 
model. Test cases with different shaft 
conditions are shown in Table 3. The 
parameter for the shaft geometry is the 
length/diameter ratio(H/D) with the 
diameter and wall thickness being kept the 
259 
.g 8 
~ 6 
! 4 
·········· without shaft : easel 
----short shaft (H/D=l.5) : case3 
-- long shaft (H/D=S) : cases 
100 200 300 400 
Frequency(Hz) 
Fig.6 Transfer function at A4 in the 
case with small amplitude input 
.g 
.~ 
-a. 
e 
<!'.'. 
········ ··· without shaft : case2 
---- short shaft (H/D=l.5) : case4 
-- long shaft (H/D=S) : case6 
g..--~-~--..--~ 
100 200 300 400 
Frequency(Hz) 
Fig. 7 Transfer function at A4 in the 
case with large amplitude input 
same. Two types of shaft conditions, short 
shaft(H/D=l .5) and long shaft(H/D=5), are 
considered. The shaft models are made of 
aluminum. Table 3 also shows the density 
of ground model and the maximum 
acceleration of input motion. All tests were 
carried out under 50g centrifugal 
condition. The model ground, composed of 
the dry TOYOURA standard sand, was 
made by air-pluviation. 
El Centro 1940 NS record was used as the 
input signal for shaker. The shaking time 
was shortened to 1/50 of the original 
one in accordance with similitude. The sine 
acceleration waves with frequency of 
1 OOHz and 250Hz were also used as input 
motions in separate tests. The first natural 
frequency of the ground model is 
225Hz(see Fig.6). 
Fig. 5 shows the location of the sensors. In 
order to observe the acceleration response 
of the ground model and shaft, 
accelerometers were set up at various 
locations. To measure the horizontal earth 
pressure acting on the shaft, pressure 
transducers were instrumented at the outer 
without shaft : easel 
long shaft (H/0=5) : cases 
100 200 300 400 
Frequency( Hz) 
Fig.8 Transfer function at Al 2 in the 
case with small amplitude input 
without shaft : case2 
long shaft (H/0=5) : case6 
QJ 8 
"O 
.s 
6 :.:l Q.. 
~· 4 
2 
00 100 200 300 400 
Frequency(Hz) 
Fig. 9 Transfer function at Al 2 in the 
case with large amplitude input 
face of shaft. Eight and four pressure 
transducers were used at long and short 
shafts, respectively. In order to observe the 
strain in axial direction of the shaft during 
excitation, pairs of strain gauges were 
bounded at both ends of a diameter to the 
inner face of shafts. Eight and six gauges 
were used at long and short shafts, 
respectively. 
3.2 Test results and analysis 
Responses of shaft and ground are first 
investigated through the analysis of 
transfer function. In order to study the 
effects of the input amplitude level, the 
transfer functions subjected to input 
motions of small and large amplitudes are 
compared. Fig.6 and Fig.7 show the 
transfer functions at the A4(see Fig.S)for 
the case with short shaft, long shaft and 
without shaft under low and high 
excitation levels. All the transfer functions 
are computed from the ratio of Fourier 
spectra of acceleration response at A4 to 
those at the bottom. It can be seen that, the 
260 
predominant period becomes longer and 
the amplification ratio decreases with the 
input level. As this tendency does not 
depend on the shaft condition, it is 
believed to be due to soil non-linearity. 
To study the effects of the shaft 
conditions, the transfer functions for the 
cases with different shaft conditions are 
compared. It can be seen from Fig.6 and 
Fig.7 that the transfer functions of the 
shaft( at the surface, point A4) have smaller 
amplitudes compared with that of the 
ground at the same position at the non-
shaft case. Fig.6 and Fig.7 show that the 
transfer functions for the case with long 
shaft have smaller amplitude compared 
with that for the case with short shaft. 
These tendencies don't depend on the 
input level. Hence, the shaft, which is much 
stiffer than the ground, restrains the shear 
deformation of the ground model, and the 
long shaft has a stronger effect than the 
short one. Fig.8 and Fig.9 show the 
transfer functions at the point Al2 for the 
cases with and without shaft, under low 
and high excitation levels. Transfer 
functions of the ground point Al2, which 
is l OOmm away from the shaft center, are 
quite similar in all cases. This tendency 
doesn't depend on the input level. 
Therefore, it could be concluded that the 
extent of the ground to which the shaft 
could restrain is less than l .SD(D:diameter 
of the shaft) from the shaft center. 
The distribution of relative displacement 
amplitude(peak to peak) between shaft and 
ground, which could be calculated from 
accelerations, is also studied. Fig.10 shows 
the results for the cases under sine wave 
excitation. The relative displacement is 
very small in the case of lOOHz wave 
excitation regardless of the shaft condition. 
This excitation frequency is lower than fo 
(the natural frequency of the ground 
model), which is around 225Hz (see Fig.6). 
Therefore the shaft and ground move 
together when the excitation frequency is 
lower than fo. However, in the case of 
250Hz wave excitation, which is close to 
fo, relative displacement in the region near 
the ground surface is large, and the relative 
displacement in the case with long shaft is 
larger than that in the case with short shaft. 
Thus, when the excitation frequency is 
close to the fo, the response at ground 
surface is largely amplified, resulting in 
relative displacement amplitude 
~ £ 
·--· .g 10 lOOHz 
(case7) 
0--0 
250Hz 
(case8) 
short shaft 
(H/D=L5) 
~ ·----= lOOHz 
"' (case9) 
long shaft 
(H/D=5) 
0-0 
250Hz 
(caselO) 
Fig.10 Distribution of relative 
displacement amplitude 
maximum bending moment amplitude 
x 103 (kgf·cm) 
0 1 2 3 4 
x 103 (kgf·cm) 
1 2 3 4 
!~r-
£ ~t- hno- ~r ol.nf 
.8 9 <-- base of shaft 
•---• 100Hz (casc7) 
0---0 250Hz 
(casc8) 
short shaft 
(HID=l.5) 
20 
25 
•---etOOHz 
(casc9) 
0--0250Hz 
(casclO) 
30 <-- base of shaft 
long shaft 
(H/D=5) 
Fig.11 Distribution of dynamic 
bending moment amplitude 
significant different response between 
shaft and ground. As the shaft is longer, the 
difference between responses of ground 
and shaft tends to be larger. 
The bending moments of shaft during 
shaking are calculated using the record of 
the strain gauges. Fig 11 shows the 
distributions of dynamic bending moment 
amplitude (peak to peak) during sine wave 
excitation. In the case of short shaft, 
bending moment is very small, which 
suggests that the short shaft vibrates as a 
rigid body during shaking. In the case of 
long shaft, on the other hand, as the depth 
of shaft increases, the bending moment 
becomes larger, and bending moments at 
each depth change all in phase. Therefore, 
the vibration mode of the long shaft could 
be considered as the deformation of a 
261 
dynamic earth pressure amplitude 
(kgf/cm 2) (kgf/cm2 ) 
1.0 0.5 0 0 0.5 1.0 1.0 0.5 0 0 0.5 1.0 
0 0 ~ 
10 10 ~ 
..d 
"' 
20 20 
short shaft long shaft 
(H/D=l.5) (H/D=5) 
30 30 
depth (cm) depth (cm) 
•----• lOOHz (case7) •----• lOOHz (case9) 
0--0 250Hz (case8) 0--0 250Hz (caselO) 
Fig.12 Distribution of dynamic 
earth pressure amplitude 
cantilever under distributed load around 
tip. 
The distribution of dynamic earth pressure 
amplitude(peak to peak), observed in the 
experiment under sine wave excitation, is 
shown in Fig.12. In the case with long 
shaft under 250Hz sine wave excitation, 
the dynamic earth pressure acting on the 
top and middle part of the shaft have larger 
amplitudes. This distribution tendency 
results from the distribution of the relative 
displacement between shaft and ground. 
At the depth near the ground surface, the 
relative displacement between ground and 
shaft becomes significant, which results in 
large earth pressure. To balance the large 
pressure at the top, large reaction force is 
needed at around the middle of the shaft 
where the soil is stiffer. It can be also seen 
that the distribution pattern of dynamic 
earth pressure amplitude varies as shaft 
condition is changed. Therefore, the 
distribution of dynamic earth pressure 
depends on the shaft vibration mode and 
the distribution pattern of the relative 
displacement, which is determined by the 
shaft geometry. 
4 CONCLUSIONS 
In order to reveal the earthquake response 
characteristics of a shaft in dry sand, 
dynamic centrifuge tests have been 
successfully carried out. Following are the 
main conclusions drawn from this study: 
I )The proposed boundary treatment 
method is shown to be very promising 
262 
through both dynamic centrifuge test and 
the 1-D numerical analysis; 
2)The acceleration response of shaft-
ground system is significantly affected by 
the amplitude level of input motion; 
3)Shaft condition has large effect on the 
transfer function at the shaft, but little 
effect on that at the ground surface; 
4)Short shaft vibrates in almost the same 
way as ground. However, in the case of 
long shaft, large response difference 
between the shaft and ground is observed 
near the ground surface; 
5)Short shaft vibrates as a rigid body. Long 
shaft deforms during shaking, and its 
vibration mode is very similar to the 
deformation mode of a cantilever under 
distributed load around tip; 
6)The distribution of dynamic earth 
pressure depends on tlte shaft vibration 
mode, which is determined by the shaft 
geometry. 
REFERENCES 
1 )Campbell D. J ., Cheney J. A. & Kutter 
BL. 1991. Boundary effects in dynamic 
centrifuge model tests. Centrifuge 91 
:441-448 
2)Hayashi H., Honda H., Yamada T. & 
Tatuoka F. 1992. Modeling of nonlinear 
stress strain relations of sands for dynamic 
response analysis. lOtlt World conference 
on Earthquake Engineering : 19-24 


Dynamic response of a shaft in dry sands

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Dynamic response of a shaft in dry sands

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