Transactions, SMiRT-25 Charlotte, NC, USA, August 4-9, 2019
Division IV
Experimental Study on Vibration Characteristics of the soil Under Drop-Weight Impact Loading
(2) Evaluation and Simulation Analyses of Test Results
Takashi Nozawa1, Takashi Tsuruga2, Takayuki Koyanagi2, Yuichi Aoyama2,
Makoto Takahashi2, Hiroyuki Nouji3, Aya Tanaka4, Yoshinori Mihara4, and Masuhiro Beppu5
1 Kajima Corporation, Tokyo, Japan ([email protected])
2 Tokyo Electric Power Company Holdings, Tokyo, Japan
3 Hitachi-GE Nuclear Energy, Ltd., Hitachi, Japan
4 Kajima Corporation, Tokyo, Japan
5 Professor, National Defense Academy of Japan, Yokosuka, Japan
ABSTRACT
The objective of these series of studies is to evaluate the vibration characteristics of the soil under impact loadings. This paper shows the detailed evaluation and simulation analyses based on the test results obtained by the previous paper Part 1.
Regarding the detailed evaluation of the test results, the damping constant of the soil in the dominant shock-wave propagation region is estimated to be more than almost 10% based on the detailed evaluation by measurement records in the soil shown in Part 1, as well as the identification results by hammer impact tests before and after the drop-weight impact tests.
Moreover, simulation analyses based on the test results are performed using the three-dimensional finite element method (3D-FEM). As a result, simulation analyses with stiffness proportional damping show relatively better agreement with test results in the tendency for acceleration waveforms in the soil, and simulation analysis results where the damping constant of the soil is assumed to be 1% are conservatively evaluated compared to the test results. For further discussion, simulation analyses show that the elasto-plastic effect of the soil is relatively small in the first travelling wave immediately after the drop-weight impact where maximum soil acceleration occurs and the elasto-plastic effect of the soil is relatively large in subsequent waves after the first wave that causes a decrease in the amplitude of acceleration due to impact energy absorption from the elasto-plasticity of the soil. The latter result by simulation analyses with the elasto-plasticity of the soil shows good agreement with the test results.
INTRODUCTION
Present vibration evaluations for underground structures against missile impacts in Japan do not consider covering soil effects conservatively because there are no assessment methodologies for soil impact in NEI07-13 [Revision 8P] (2011) and insufficient findings on impact vibration characteristics of the soil.
Therefore, for a realistic impact design of underground structures that cover the soil effects, drop-weight impact tests for soil are performed to obtain impact vibration characteristics such as the damping effect.
This paper reports the detailed evaluation and simulation analyses based on the test results obtained by the previous paper Part 1. First, regarding the detailed evaluation of test results, the damping constant of the soil is estimated based on detailed evaluation of the measurement records in the soil obtained in Part 1, as well as the identification results from hammer impact tests before and after drop- weight impact tests. Furthermore, from the viewpoint of supporting the identification results of the soil damping constant mentioned above, the simulation analyses based on test results are performed using the three-dimensional finite element method (3D-FEM) to obtain analytical findings on shock-wave propagation in the soil.
EVALUATION
This chapter proposes the general formulation to evaluate the damping constants for homogenous isotropic materials such as uniform fine sand using measured records. Based on the formulation, identification results of the damping constant of the soil are obtained from detailed evaluation of the drop- weight impact tests as shown in the previous paper Part 1, as well as from the hammer impact tests before and after the drop-weight impact tests.
General Formulation to Evaluate the Damping Constants
Wave amplitude Ui at the distance xi away from an impact loading point is generally given in equation (1), considering geometric attenuation Gi and scattering attenuation Ti.
0exp exp
i i i i i
U U x i t kx G T (1) such that k : wave number, : angular frequency, : attenuation factor, t: time
After extracting only the term related to soil damping, equation (2) is obtained.
0exp
i i i i
U U x G T (2)
Considering wave propagation problems between the two measured points (i=1 and i=2), then equation (3) is given.
2 1 1
2 1
1 2 2
ln U G T
x x U G T
(3)
The supposed homogenous uniform soil and spherical wave given in equation (4), equation (5) is derived.
1 2
lnT lnT 0
1 1 1, 2 1 2
G x G x (4)
2 2
1 1
2 1
ln U x U x x x
(5)
On the other hand, with reference to Nobuoka, D. et al. (2012), damping constant h is generally given in equation (6).
2
sgn
2 1
h
f c
(6)
such that c: phase velocity, f: frequency
Therefore, based on the measured records of two points in the soil, the damping constant h of the soil is evaluated from equation (5) and equation (6).
Identification Result of Damping Constant of the soil from Hammer Impact Tests
For hammer impact tests before and after drop-weight impact tests, the primary wave velocity of the soil Vp and damping constant of the soil h are evaluated based on the formulation mentioned above. The 4- point measured records on the initial rise of the primary acceleration waveform are used for focusing on the depth direction at the center point of falling weights. Identification results of the damping constant of the soil from the hammer impact tests are shown in Table 1 and Table 2.
Regarding identification results of damping constant of the soil from the hammer impact tests and the drop-weight impact tests, the following are considered:
*) To consider the assumed moving distance of the accelerometers of the lower depth based on the penetration depth of the falling weights
*) To confirm the relationship between measured result from accelerometers and that from the high- speed video
*) To confirm the actual moving distance of the accelerometers of the lower depth after the drop- weight impact tests
Table 1: Identification results of damping constant of the soil etc. from hammer impact tests.
(Basic soil type) Evaluation
points Evaluation
parameters Before C1 Before C2 Before C3 Before C4 After C4 0.1 to 0.3 m
depth
Vp (m/s) 142.5 160.0 191.5 177.5 210.0
h (%) 16.2 23.9 20.0 11.9 16.9
0.3 to 0.7 m depth
Vp (m/s) 214.9 226.0 230.0 230.0 238.3
h (%) 2.6 4.2 4.4 2.8 1.6
0.7 to 1.3 m depth
Vp (m/s) 231.3 226.0 226.0 234.1 231.3
h (%) 3.9 2.4 2.9 1.8 2.1
Table 2: Identification results of damping constant of the soil etc. from hammer impact tests.
(Varied soil type) Evaluation
points Evaluation
parameter Before C5 Before C6 Before C7 Before C8 ― 0.1 to 0.3m
depth
Vp (m/s) 152.4 164.1 168.3 186.4 ―
h (%) 24.0 25.6 27.1 14.5 ―
0.3 to 0.7m depth
Vp (m/s) 187.2 208.1 208.1 204.8 ―
h (%) 6.6 7.4 11.0 7.9 ―
0.7 to 1.3m depth
Vp (m/s) 269.3 258.7 273.1 265.7 ―
h (%) 8.4 5.0 5.9 7.0 ―
Identification Result of Damping Constant of the soil from Drop-weight Impact Tests
For drop-weight impact tests shown in the previous paper Part 1, the primary wave velocity of the soil Vp and damping constant of the soil h are evaluated similar to the evaluation procedure for the hammer impact tests. The identification results for the damping constant of the soil from drop-weight impact tests are shown in Table 3 and Table 4. Moreover, overall plots along the soil depth of the damping constant of the soil for both the hammer impact tests and the drop-weight impact tests are shown in Figure 1. As a result, while the damping constant of the soil in the dominant shock-wave propagation region with relatively large acceleration is estimated to be almost 10% to 30%, the damping constant of the soil in the deeper region with relatively small acceleration is estimated to be smaller value of about 2% to 10%.
Table 3: Identification results of damping constant of the soil etc. from drop-weight impact tests.
(Basic soil type) Evaluation
points Evaluation
parameters C1 C2 C3 C4
0.1 to 0.3 m depth
Vp (m/s) 145.6 160.0 147.3 146.6
h (%) 29.4 17.9 19.1 19.5
0.3 to 0.7 m depth
Vp (m/s) 214.9 222.2 222.2 222.2
h (%) 1.1 2.9 8.3 7.8
0.7 to 1.3 m depth
Vp (m/s) 228.6 223.4 220.9 220.9
h (%) 2.3 1.6 3.7 3.3
Table 4: Identification results of damping constant of the soil etc. from drop-weight impact tests.
(Varied soil type) Evaluation
points Evaluation
parameters C5 C6 C7 C8
0.1 to 0.3 m depth
Vp (m/s) 168.0 116.6 133.4 137.8
h (%) 19.3 17.0 17.2 14.8
0.3 to 0.7 m depth
Vp (m/s) 190.0 184.6 187.2 190.0
h (%) 3.1 5.4 6.4 8.9
0.7 to 1.3 m depth
Vp (m/s) 276.9 265.7 262.1 262.1
h (%) 3.4 3.3 5.6 6.2
Figure 1. Overall plots of damping constant of the soil.
(Left: Basic soil type, Right: Varied soil type) SIMULATION ANALYSES OF TEST RESULTS
In this section, from the viewpoint of supporting the identification results of the soil damping constant as mentioned in the previous section, simulation analyses based on the test results are performed using the three-dimensional finite element method (3D-FEM) to obtain analytical findings on shock-wave
propagation in the soil. The objectives of the simulation analyses are to confirm the validation process to conservatively evaluate the vibration characteristics of acceleration in the soil by focusing on only the relatively small nonlinear region in the deeper soil except the severe nonlinear region, such as penetration the phenomena around the impact point.
Numerical Model
The following are the concepts for the numerical model for the simulation analyses of drop-weight impact tests.
* The element size is supposed to be 0.02 m and sufficient enough to simulate shock-wave propagation considering wave frequencies up to 2 kHz.
0
0.5
1
1.5
0 10 20 30
C1 C2 C3 C4
Hammer test before C1 Hammer test before C2 Hammer test before C3 Hammer test before C4
depth from the ground (m)
h (%)
0
0.5
1
1.5
0 10 20 30
C5 C6 C7 C8
Hammer test before C5 Hammer test before C6 Hammer test before C7 Hammer test before C8
depth from the ground (m)
h (%)
* The dimensions of numerical model are 1.5 m x 1.0 m in plan and 1.5 m in depth, similar to those of the soil specimens. Also, the numerical model is supposed to be uniform soil and 1/4 symmetric.
* Considering the reservoir of the soil, the boundary conditions of the numerical model are applied at the fixed bottom and the fixed sides in the out-of-plane direction. But the difference in the side boundary conditions does not affect the simulation results according to a sensitivity study.
* Regarding the impact effect due to falling weight, the time series of the displacement obtained by high-speed video is applied uniform in the impact region.
Following the above concept, the numerical model for simulation analyses is shown in Figure 2.
Figure 2. Numerical model for simulation analyses.
Material Property of the Soil Specimen
The basic soil type specimen is the target for simulation analyses with or without the elasto-plasticity of the soil. Elastic material properties of the soil are based on the results of the hammer impact tests before the drop-weight impact tests. Material constitutive characteristics for the soil are subjected to the Mohr- Coulomb plastic model. the material properties of the soil specimens for the simulation analyses are shown in Table 5.
Table 5: Material properties of the soil specimens for simulation analyses (Basic soil type) Soil density
(t/m3)
Young’s modulus
（kN/m2）
Poisson’s ratio
（-）
Cohesive strength
（kN/m2）
Internal friction angle (degree)
1.44 5.76×104 0.4 0 15
Damping Characteristics
The material damping characteristics for the simulation analyses are considered with three types of material damping. One is the Rayleigh damping type as the damping constant of 0.1% at 1 Hz and 2 kHz.
Second is the Rayleigh damping type as the damping constant of 1.0% at 1 Hz and 2 kHz. Third is the stiffness proportional damping type as the damping constant of 1.0% at 48.8 Hz of 1st mode of the soil specimen.
Simulation Cases and Numerical Code
Test case C1 is the target for the simulation analyses. It is performed for basic soil types and the smallest dropping height of 0.1 m that is the most appropriate to confirm the validation process to conservatively evaluate the vibration characteristics of acceleration in the soil by focusing on only the relatively small nonlinear region for test case C1 except the severe nonlinear region caused by the larger dropping height.
Simulation cases are shown in Table 6 that are totally four cases varied with three types of material damping and the effect of the elasto-plasticity of the soil. Commercial finite element code Abaqus/Standard 6.14-3 is used as a numerical code for the simulation analyses.
Table 6: Simulation cases.
Simulation
case Test
case Dropping
height Type of
material damping Damping
constant Elasto-plasticity of the soil
S1 C1 0.1 m Rayleigh damping 0.1% w/o
S2 C1 0.1 m Rayleigh damping 1% w/o
S3 C1 0.1 m Stiffness proportional damping 1% w/o
S4 C1 0.1 m Stiffness proportional damping 1% w/
Simulation Results
Focusing on the measured records at the center point of the falling weight, comparisons of the
acceleration waveforms of the soil in the depth direction are shown in Figure 3. To quantitatively evaluate the effect of the soil damping regarding shock-wave propagation in the soil due to impact loadings, the comparison of the amplitude ratio of maximum acceleration in the depth direction is shown in Table 7 normalized with the value at a depth of 0.1m from the ground. Furthermore, for the simulation case S4 with the elasto-plasticity of the soil, snapshot contours of the effective plastic strain of the soil are shown in Figure 4 and Figure 5. In these figures, snapshot times are selected according to the following reasons such that 0.3 ms is the time when maximum acceleration is observed at a depth of 0.1 m, 1 ms is the time when maximum acceleration is observed at a depth of 0.3 m, 2 ms is the time when maximum
acceleration is observed at a depth of 0.7 m and 50 ms is the time when maximum deformation is observed on the ground surface.
As a result, simulation analyses with stiffness proportional damping show relatively better agreement with test results in the tendency for the acceleration waveform in the soil and the simulation analyses where the damping constant of the soil is assumed to be 1% are conservatively evaluated compared to the test results. For further discussion, simulation analyses show that the elasto-plastic effect of the soil is relatively small in the first travelling wave immediately after the drop-weight impact where maximum soil acceleration occurs and the elasto-plastic effect of the soil is relatively large in the subsequent waves after the first wave that causes a decrease in the amplitude of acceleration due to impact energy absorption by the elasto-plasticity of the soil. The latter result by simulation analyses with the elasto-plasticity of the soil shows good agreement with the test result.
(a) S1
(b) S2
(c) S3
(d) S4
Figure 3. Comparison of acceleration waveforms of the soil in depth direction.
-10000 -5000 0 5000 10000
0 0.002 0.004 0.006 0.008 0.01
0.1m depth 0.3m depth 0.7m depth 1.3m depth
accerelation (m/s2 )
time (sec)
-10000 -5000 0 5000 10000
0 0.002 0.004 0.006 0.008 0.01
accerelation (m/s2 )
time (sec)
-4000 -3000 -2000 -1000 0 1000
0 0.002 0.004 0.006 0.008 0.01
accerelation (m/s2 )
time (sec)
-4000 -3000 -2000 -1000 0 1000
0 0.002 0.004 0.006 0.008 0.01
accerelation (m/s2 )
time (sec)
Table 7: Comparison of amplitude ratio of maximum acceleration in depth direction.
(Normalized with the value at 0.1m depth from the ground) Depth
from the ground
Simulation case Test case
S1 S2 S3 S4 C1
0.1 m 1.00 1.00 1.00 1.00 1.00
0.3 m 0.83 0.73 0.25 0.25 0.01
0.7 m 0.43 0.30 0.05 0.04 0.01
1.3 m 0.33 0.11 0.02 0.01 < 0.01
(a) time 0.3 ms (b) time 1.0 ms
Figure 4. Snapshot contours of effective plastic strain of the soil #1 (Max 30%).
(a) time 2 ms (b) time 50 ms
Figure 5. Snapshot contours of effective plastic strain of the soil #2 (Max 30%).
CONCLUSION
The objective of these series of studies is to evaluate the vibration characteristics of the soil under impact loadings. This paper shows the detailed evaluation and simulation analyses based on the test results obtained by the previous paper Part 1.
Regarding the detailed evaluation of test results, damping constant of the soil in the dominant shock-wave propagation region is estimated to be more than almost 10% based on the detail evaluation of the measurement records in the soil obtained in Part 1 and the identification results from the hammer impact tests before and after the drop-weight impact tests.
Moreover, the simulation analyses based on the test results are performed using the three- dimensional finite element method (3D-FEM). As a result, simulation analyses with stiffness proportional damping show relatively better agreement with test results for the tendency toward the acceleration waveform in the soil and the simulation analyses where the damping constant of the soil is assumed to be 1% are conservatively evaluated compared to the test results. For further discussion, the simulation analyses show that the elasto-plastic effect of the soil is relatively small in the first travelling wave immediately after the drop-weight impact where the maximum soil acceleration occurs and the elasto- plastic effect of the soil is relatively large in the subsequent waves after the first wave that causes a decrease in the amplitude of acceleration due to impact energy absorption from the elasto-plasticity of the soil. The latter result by simulation analyses with the elasto-plasticity of the soil shows good agreement with the test result.
ACKNOWLEDGEMENT
This study was conducted as a part of the joint project under the eleven electric power companies in Japan.
The authors hereby express gratitude to the organizations and all who supported the project.
REFERENCES
NEI 07-13 [Revision 8P], "Methodology for Performing Aircraft Impact Assessments for New Plant Designs", April 2011
Nobuoka, D., Azuma, H. and Oba, M., "Evaluation Technique of the Ground Attenuation Property by Using a PS-logging Method ", BUTSURI-TANSA, Geophysical Exploration, Vol. 65, No.1 & 2, pp.79-90, 2012 (in Japanese)
