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DYNAMIC PROPERTIES OF REINFORCED SAND
By Scott E. Shewbridge1 and Jorge B. Sousa,2
Associate Members, ASCE
ABSTRACT: A series of dynamic shear tests was performed on large hollow cylindrical reinforced and unreinforced sand samples. Steel rods were used to reinforce the specimens uniaxially and biaxially. A newly developed test apparatus was used to perform cyclic torsional shear strain control tests with strain amplitudes ranging from 1 X 10~2% to 7% and frequencies varying between 0.5 Hz and 10 Hz at different levels of confining pressure. Test results indicate that confining pressure and strain level affect the dynamic shear modulus and damping of unreinforced and reinforced sands. Dynamic shear modulus and internal damping were independent of frequency (up to 10 Hz) for both reinforced and unreinforced samples. Reinforcements have no effect on the complex shear modulus of the material up to a strain amplitude of 5%. At large strains (>7%) uniaxial reinforcements inhibited the formation of helical shear bands, which had been observed on the unreinforced specimen. However, vertical shear bands parallel to and equidistant from the reinforcements were observed. Biaxial reinforcements prevented the formation of all rupture surfaces, thus greatly increasing the ductility of the material. The results indicate that the effect of the reinforcements on the dynamic properties of the reinforced soil composite can be ignored when analyzing the dynamic response at small strains, but should be considered when evaluating the dynamic response at larger strains.
INTRODUCTION
Understanding the dynamic behavior of soils is essential in areas where seismic or man-made dynamic loading occurs. Modulus, damping, frequency response, and ductility are important factors to consider when studying dynamic response. In the last 20 years, the reinforcement of soil by high tensile strength materials has become a widespread technique in earthwork construction. Metal strips, plastic grids, and fabrics have all been used to increase the tensile strength, shear resistance, and stiffness of soils (Mitchell 1981). Generally, analysis of these systems has been limited to static loading conditions, as dynamic material parameters have been unavailable, and the mechanics of material interaction under dynamic conditions was unknown.
To analyze the dynamic response of a structure, a common methodology is the finite element method. The material behavior models can be of a simple linear-elastic type or, as in more recent years, can incorporate complex elastic and plastic material models. These analyses can be fairly simple when the geometries of soils and structures are not complicated. However, reinforced soil-structure interactions can be very difficult to model and may involve the mixing of behavior models. This can lead to numerical difficulties and can be indicative of strain compatibility problems occurring at the interfaces between two different material types. One method of approaching this problem is to use interface elements. However, the development of in-
lSr. Engr., Wahler Assoc, 1023 Corporation Wy., Palo Alto, CA 94303; Formerly Grad. Res. Asst., Dept. of Civ. Engrg., Univ. of California, Berkeley, CA 94720.
2Asst. Res. Engr., Inst, of Transp. Studies, Univ. of California, Berkeley, CA. Note. Discussion open until February 1, 1992. To extend the closing date one
month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on May 11, 1989. This paper is part of the Journal of Geotechnical Engineering, Vol. 117, No. 9, September, 1991. ©ASCE, ISSN 0733-9410/91/0009-1402/$1.00 + $.15 per page. Paper No. 26174.
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terface elements between soils and other materials is not a simple matter and is not always successful.
Another method of avoiding these difficulties is to develop a behavior model for the soil and structure that does not separate the behavior of the component parts. This technique is a form of homogenization, wherein the individual components of the structure are combined to form a composite with a unique set of behavioral characteristics. In this paper, the results of dynamic tests on several soil reinforcement composites are used to derive composite material response parameters that can be used for the dynamic analysis of reinforced soil structures.
METHODOLOGY
Determining the dynamic properties of any soil is made difficult by the nature of the material. The dynamic modulus and damping are affected by the mean stress and strain level. Furthermore, it is expected that a reinforced soil specimen's behavior is affected by the size of the sample; a small sample will not allow for the measurement of strains, or for the development of stresses in the reinforcement. A device for testing the dynamic response of pavement materials has recently been developed by Sousa (1986). The sample used in this device is large enough to allow the fabrication of samples with reinforcements long enough to develop significant stresses. The device also allows for varying the mean stress, mean strain, and frequency.
Material and Specimen Preparation The sand used for this study was dry, cohesionless, uniformly graded
Monterey sand #0 with a D50 of 0.5 mm, a minimum dry density of 1.36 Mg/m3 (85 lb/cu ft), and a maximum dry density of 1.68 Mg/m3 (105 lb/ cu ft). For this test series the samples were compacted to a relative density of 90%.
An isometric view of the hollow cylindrical specimen and the adjacent confining chamber is shown in Fig. 1. The hollow cylindrical specimen is 0.457 m high (18 in.), has a 0.229-m external diameter (9-in.), and a 0.025-m wall thickness (1 in.). The specimen is contained by a membrane and capping base rings made of hard anodized aluminum. Confinement for the specimens is provided by applying a vacuum between the membrane and the sand through porous stones on the bottom aluminum ring.
To prepare hollow cylindrical specimens, a unit consisting of inner and outer molds is required (Fig. 2). The outer mold is an aluminum cylinder that has been divided in half and is held together by bolts. The inner aluminum mold is inwardly collapsible and held together by an internal wedge. This mold can be collapsed by removing the wedge, thus allowing removal of the mold without imposing stress on the specimens. The two cylindrical molds are fastened to the bottom chamber plate with the bottom specimen ring and membrane in place. Vacuum grooves are provided on the inside wall of the outer mold and on the outside wall of the inner mold to keep the 0.51-mm (0.020-in.) thick latex membrane in position when fabricating fine-grained soil and sand specimens.
Each specimen was fabricated by pouring the sand into the mold, which was placed on a vibrating table. When the specimen was nearly completed,
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ELECTRICAL CONNECTORS
LOAD CELL
TOP SPECIMEN RING
TOP CHAMBER PLATE
CONNECTOR FOR PRESSURIZED AIR
— O-RING
ll
II II
II II II II II
l-i 1 1 1
1 1
CLEAR CHAMBER
- — SPECIMEN
BOTTOM CHAMBER PLATE
BOTTOM SPECIMEN RING
FIG. 1. Isometric View of Pressure Chamber, Specimen and Load Cell
the top ring was set in place, and additional sand was poured through holes in the ring so that the desired density could be achieved even between the ring lugs. This procedure was executed while the mold was vibrating.
Upon completion, the membrane was rolled over the top ring and sealed with O-rings. A vacuum was then applied to the specimen, and the mold was removed. Visual inspection indicated that this method produced a specimen of uniform shape and density (see Fig. 3).
In the reinforced samples, the reinforcements used were 3.175-mm (0.125-in.) diameter steel rods. For the first reinforced sample (uniaxially reinforced), eight 0.457-m (18-in.) rods were placed equidistant in the sample, parallel to the long axis of the specimen. For the second reinforced sample (biaxially reinforced), five 0.203-m (8-in.) diameter hoops were spaced equidistant along the length of the sample in addition to the eight rods. The reinforcements were placed in the sample after the sample mold was filled with a 0.102-m (4-in.) layer of sand, and were supported by this layer as the remaining sand was poured. Three samples were constructed: unrein-forced, unaxially reinforced, and biaxially reinforced (Fig. 4).
Load Frame The test loading frame, termed herein the dynamic loading system (DLS),
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FIG. 2. Molds for Fabricating Hollow Cylindrical Samples
was designed to test hollow cylindrical specimens in both torsional and axial loading. The device attempts to simulate the three-dimensional stress states that occur in materials in situ when subjected to static and dynamic loads. The torsional and axial loads can be applied either independently or simultaneously, thus enabling the rotation of the principal stress axes to follow predefined stress paths. Loads can be applied at frequencies up to 30 Hz. For a more complete discussion of the apparatus, see Sousa and Monismith (1987).
The DLS is designed so that the servorams can be mounted on a heavily reinforced frame beneath which the specimen (and/or chamber) is placed. Vertical loads are applied by a vertically mounted 44.5-kN (10-kip) MTS servoram with a 0.152-m (6-in.) stroke. Torsional loads are applied by a horizontally mounted 8.9-kN (2-kip) MTS servoram with a 0.152-m (6-in.) stroke. The latter, when connected to a 0.178-m (7-in.) torque arm provides up to a 1.58-N-m (14-kip-in.) torsional moment to the center shaft (Fig. 5).
A Rotolin linear and rotary ball bearing with a total linear travel of 25.4 mm (1 in.) allows the coupling of both loads through the same shaft. Decoupling of the vertical movement of the torque arm from the horizontal movement of the horizontal servoram is made possible by two spherical bearings that limit the angle of the torque to ±15°.
A wheel-shaped load cell was constructed with eight spokes, each in the form of a rectangular prism. Each spoke is assumed to behave as a beam. Vertical and torsional loads can then be measured independently since strain gages are mounted on the centerlines of alternate faces of the beams.
Measurements of specimen deformations were made using linear variable differential transformers (LVDTs). The LVDTs used were in constant con-
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r .W • I *^.
I - . T"
%$
FIG. 3. Unreinforced Specimen before Testing
1 V (a) (b) (c)
FIG. 4. Configuration of Reinforcements: (a) None; (b) Uniaxial; (c) Biaxial
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FIG. 5. Dynamic Loading System
tact, due to spring-loaded gaging heads, with targets mounted on the top of the specimen.
Data Acquisition and Equipment Control The development of the DLS required that wave-form generation, data
acquisition, equipment control, and data processing be accomplished almost simultaneously. To successfully perform such a task, a microcomputer was used in conjunction with a data acquisition board and a signal conditioning interface. The diagram in Fig. 6 shows schematically how the elements of the apparatus are interconnected.
Software and Data Analysis In order to run the tests, the software developed by Sousa (1986) for the
dynamic loading system was used. The tests were conducted using strain-controlled sinusoidal waves; an analysis of the results was performed using the program ANALYSER. This program assumes that the material behaves at low strain levels as a linear viscoelastic material. To determine the dynamic properties of a material (complex modulus and internal damping), the testing sequence based on strain-controlled sinusoidal excitation was used.
If a sinusoidal force P = P0 cos (wf) is applied to a specimen composed of an ideal massless linear viscoelastic material, the deformation response X will be sinusoidal and at the same frequency, <o, but it will lag by a phase angle 8, as given by the following expression (Lazan 1968):
X = X0 cos (wf - 8) (1)
where X0 = maximum amplitude of the deformation; and P0 = maximum amplitude of the force.
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Vertical w\\\^~^\\ Servoram S.
Strain Gauges ̂ ^j
H/Dr's-i^ \ | 5
—I Axial and Torsional Load ryCells -J- 1 rt-d
T - i r
Specimen
ni ii'n 11 a in) i in
I - SJ Signal Conditioning Interface Box
DASH 16 ̂ J.
Horizontal Servoram
.Signal Conditioning ' Interface Board
IBM PC/AT-
Graphics. Display
Q s • Graphics Printer
FIG. 6. Data Acquisition and Control Scheme for Dynamic Loading System
At each instant of time t, the relationship between P and X is a function of Po/X0 and the phase angle. It can be shown that if
P = P0 exp (iu>t)
then
X = X0 exp [i(M - 8)]
P P0
X X0 exp («8)
P0
= k' + ik" = K:
X0
and
k? + kf\l/2 = \k*
(2)
(3)
(4)
(5)
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k" tan 8 = — (6)
K The variable k represents a ratio between force and displacement deter
mined either in axial or shear loading. If appropriate factors dependent on specimen geometry are multiplied by this ratio, a dynamic modulus for the material can be determined.
For engineering applications, it is convenient to express the dynamic shear modulus as
G* = G(l + 2p,i) (7)
where ps = a measure of internal damping under shear loading. It should also be noted that
tan 8 P* = — ( 8 )
Thus, the internal damping of an ideal massless viscoelastic material can be directly derived from the phase angle 8 (between the sinusoidal forces and the sinusoidal displacement at the top of the specimen) and the stiffness modulus (from the ratio of the amplitudes).
However, materials may also exhibit damping behavior with nonelliptical stress/strain loops. For the specimens in this study, it was found that at high strain levels, measurement of damping is greatly affected by the shape of the stress-strain hysteresis loop. If the loop was not well approximated by an ellipse, then the area method for determining damping proposed by Seed et al. (1970) was used. In addition, the computer program makes other adjustments, which incorporate corrections for the mass of the specimen and the mass of the load cell.
Testing Procedures and Results In a series of tests, the influence of mean effective stress, frequency, and
strain level on the dynamic shear properties for an unreinforced and two reinforced sand samples were investigated. At each strain level (0.01%, 0.1%, and 1%), the samples were tested at different levels of effective mean stress and at four different frequencies (0.5 Hz, 1 Hz, 5 Hz, and 10 Hz) (see Table 1). The tests were performed under torsional strain control and axial stress control. The torsional actuator was controlled using feedback from an LVDT mounted on the top of the specimen. The vertical actuator was controlled using feedback from the vertical load cell, and a command value of zero thus maintains zero axial stress throughout the tests.
Fig. 7 shows the relationship between the dynamic shear modulus, \G*\, and the shear strain (%) for different mean confining stresses for all three samples. The material properties from each sample are in agreement with information reported in previous research by Seed et al. (1970), with the modulus increasing with mean stress and decreasing with increased strain. There is no distinct difference in behavior between the samples.
Fig. 8 shows the relationship between dynamic shear modulus and frequency for the unreinforced sample. The modulus is invariant with frequency. The reinforced samples also showed no variation with a change in frequency.
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TABLE 1. Summary of Sample Configurations and Test Conditions
(%) (1)
Frequency
(Hz)
(2)
Testl
(3)
Confining Pressures <kPa)
Test 2
(4)
Test 3
(5)
(a) No Reinforcement
0.01
0.01
0.01
0.01
0.1
0.1
0.1
0.1
1
1
1
1
>5
>5
>5
>5
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
88
88
88
88
88
88
88
88
88
88
88
88
88
88
88
88
68
68
68
68
68
68
68
68
68
68
68
68
— — — -
54
54
54
54
54
54
54
54
54
54
54
54
— — — -
(b) Uniaxial Reinforcement
0.01
0.01
0.01
0.01
0.1
0.1
0.1
0.1
1
1
1
1
>5
>5
>5
>5
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
74
74
74
74
74
74
74
74
74
74
74
74
74
74
74
74
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
54
34
34
34
34
34
34
34
34
34
34
34
34
34
34
34
34
(c) Biaxial Reinforcement
0.01
0.01
0.01
0.01
0.1
0.1
0.1
0.1
1
1
1
1
>5
>5
>5
>5
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
0.5
1
5
10
85
85
85
85
85
85
85
85
85
85
85
85
85
85
85
85
68
68
68
68
68
68
68
68
68
68
68
68
68
68
68
68
51
51
51
51
51
5!
51
51
51
51
51
51
51
51
51
51
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0.00 160
0.01 1
8
.Si 120-
"-S
1 SOTS o *!
cu 4 0 -
CO
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
Mean Confining Stress (Kpa)
0 - | > 1—r-i-rm-r 0.001 0.01 0.1 1
Shear Strain (%)
160
FIG. 7. Shear Modulus versus Shear Strain
^ 150-
.^
^ 1 0 0 -
o
s-
tf 50-
Co
0-0.00
1 1 1 1 1 1 1 ( < 1 1 1 1 ( ( <
Unreinforced Monterey sand ffm - 54 KPa ffm = 68 KPa ffm = 88 KPa
—.Strain ( » ) - .01
—.Strain («)= .01
•..Stroin ( * ) - .10
— — —.Strain (*)= .10
.Strain (»)= .10
^ S t r a i n (»j = \.0
2.00 4.00 6.00 8.00 10.00
Frequency (Hz) 12.00
FIG. 8. Shear Modulus versus Frequency
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^ 150
5-i
| 100
3 O
ti 50-
on
Strain Level
ior* r.,
w'3 ••''
10' 10'
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
r ^ f f T i -TT -n -60 0 20 40 60 80 100
Confining Pressure (KPa)
FIG. 9. Shear Modulus versus Confining Pressure
05 0 .001 ^20,-
o
a 15-
Co
a) .2 10-
S-O
Co CO cu 5-
0.01 0.1 1
Co
0-0.001
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
0.01 0.,
Shear Strain (%)
20
15
10
FIG. 10. Stress Normalized Shear Modulus versus Shear Strain
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Fig. 9 shows the dynamic shear modulus versus confining pressure for the three specimens at four strain levels. The modulus is approximately proportional to the square root of the confining stress. This is in agreement with the published results of Seed et al. (1970).
Fig. 10 shows stress-normalized dynamic shear modulus versus strain. The dynamic shear modulus is normalized by division of the square root of the confining stress. This normalization has collapsed the field of values to a narrow band. This normalization is in agreement with the correlations of Seed et al. (1970).
Internal damping was computed from the stress-strain hysteresis loops such as that shown in Fig. 11, a loop for the sample with no reinforcements. Either the complex modulus formulation or the area method of Seed et al. (1970) was used, depending on the strain level and the shape of the loop as previously discussed. Internal damping is plotted against shear strain in Fig. 12 for all of the specimens. From the data, it can be inferred that an increase in the magnitude of the shear strain results in an increase in the internal damping. At strains less than 5%, there is no perceivable difference between the behavior of the specimens. It was also noted that the internal damping was not affected by frequency at any strain level.
Response at High Strain Levels At higher strain levels, the behavior of the samples will be dominated by
the formation of shear bands. Plastic shear deformation will occur in these bands along planes that do not stretch during deformation. These planes are termed the zero extension planes (Roscoe 1971). The orientation of these bands is very important with respect to the deformation of the material. The angle between the zero extension planes is 90° ± v where v = angle of dilation. This angle is similar to the friction angle of the material, but it has a broader definition that describes the frictional and dilational behavior of the soil. The larger this angle, the greater are the frictional and dilational characteristics of the material. Formation of shear bands will result in the accumulation of shear strains of far greater magnitude than the average overall shear strains because the dense material in the shear band has been strained-softened as it dilated. These shear bands undermine the integrity of the material. Inhibition of shear strains along these shear bands should result in increased ductility of the material (Shewbridge 1987), yet, at the lower strain levels thus far discussed, this behavior has not been observed. For the final test, a large strain level, approximately 7.5%, was used, a level at which shear bands could form in a typical unreinforced soil sample.
As was expected, the unreinforced sample (see Fig. 13) produced distinct shear bands. These bands were located in a helical spiral around the sample in opposite directions. Both backward and forward slipping motion was observed in the shear bands. The sample tended to accumulate strains that shortened it, following an initial lengthening due to dilation as shown in Fig. 14, which is a plot of axial strain versus time (positive strain is shown for compression). Softening in the shear bands from the dilation resulted in degradation of the sample's quality, which required that the test be stopped after the first 15 cycles.
The behavior of the uniaxially reinforced sample at high strains was quite different. The sample tended only to dilate and did not collapse as did the unreinforced sample. In fact, the sample did not appear to be damaged after
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s ^ ^
CO
a) $-
t o
CO
100.0
50.0 -
0.0
-50.0
-100.0 -0.06 -0.04 -0.02 . 0.00 0.02 0.04 0.06
Shear Strain FIG. 11. Typical Hysteresis Loop for Unreinforced Specimen
0.01 0.1 1 0.40
^ 0 . 3 0
C5
0) so
• 0.20
0.10
0.00
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
0.001 0.01
Shear 0.1
Strain FIG. 12. Internal Damping versus Shear Strain
the test. As such, more tests were conducted on this sample; it was tested a total of five times, with 30 cycles for each test, and there was no significant collapse of the specimen. The sample could not be tested to destruction due to the limited displacements (±15°) allowed by the testing apparatus. The sample initially dilated and then stabilized for the remainder of the test
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la)\
i f 111
IS
• \
'ihl
\, '
V s - .
.m:*\ «m , - . . - ." j t-r.*:
FIG. 13. Unreinforced Specimen after Testing with Helical Shear Bands
(Fig. 14). Helical shear bands were not noted. However, the formation of vertical shear bands was observable during the test (Fig. 15). These shear bands were spaced equidistant from and parallel to the reinforcements. These shear bands were oriented in directions complimentary to the orientations of the helical bands of the unreinforced sample (90° ± v).
The biaxially reinforced sample's behavior was similar to that of the uni-
0.015
0.010
-0.015 i
-0.020
Time
FIG. 14. Axial Strain versus Time
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Reinforcements
(*) W
FIG. 15. Specimens after Testing: (a) Uniaxial Reinforcements; (b) Biaxial Reinforcements
axial sample in that it could not be tested to destruction. The sample initially dilated and then stabilized (Fig. 14). However, no shear bands were observed in the specimen (Fig. 15). The ductility of the sample was so enhanced that destruction of the specimen by the loading system was not possible. Furthermore, no signs of deterioration of the specimen were observed after 240 cycles.
The behavior of the material can be compared by reviewing the torsional stress-strain hysteresis loops of the unreinforced, uniaxially reinforced, and the biaxial reinforced samples, shown in Figs. 16, 17, and 18, respectively. The unreinforced sample (Fig. 16) had a higher initial modulus, which, however, began to decrease in stiffness until the test had to be stopped. The term "modulus" is used very loosely because the material's behavior was primarily a function of the deformation on the shear bands and was not reflective of a true modulus for the entire specimen. In contrast, the reinforced samples (Figs. 17 and 18) exhibited hysteresis loops that have different shapes than the unreinforced sample and had stabilized after several cycles. The shape of these hysteresis loops indicates that the material behavior had evolved from visco-elastic to elasto-plastic.
To compare the behavior of the three different specimens, a plot of cycle-normalized shear modulus versus number of cycles is shown in Fig. 19. The cycle normalized shear modulus was calculated by dividing the value of the modulus at any cycle by the modulus for the eighth cycle. The eighth cycle was chosen because automatic gain control of the control software could stabilize the strain amplitude after eight cycles, and then hold it constant throughout the remainder of the test. The eighth cycle was an arbitrary number of cycles chosen only due to the nature of the test device and software. The modulus of the unreinforced sample degraded as the test proceeded until the test had to be stopped. The reinforced sample's moduli exhibited little-to-no degradation and, in the case of the biaxially reinforced specimen, the modulus was slightly enhanced as the test proceeded.
In Fig. 20, the damping, calculated by the area method proposed by Seed
Vertical Shear Bands
Reinforcement
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Shear Strain 0.06
FIG. 16. Hysteresis Loops for Unreinforced Specimen
-0.06 -0 .04 -0 .02 0.00 0.02 0.04
Shear Strain 0.06
FIG. 17. Hysteresis Loops for Unlaxially Reinforced Specimen
et al. (1971), is plotted versus the number of cycles. The damping for each specimen is of a different magnitude and diminishes slightly during the test. The reinforced specimens exhibit lower damping ratios, indicating less plastic behavior and increased ductility.
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125.0 - j
100.0 -
r? 75.0 -_
§ : ^_- 50.0 -
$ 25.0 -a> :
-fe o.o --CO :
-25.0 :
ft> - 5 0 . 0 -
CO -75.0 -_
-100.0 :
- 1 2 5 . 0 4 ' ' ' I ' I -0 .06 -0.04 -0.02 0.00 0.02 0.04 0.06
Shear Strain
FIG. 18. Hysteresis Loops for Blaxlally Reinforced Specimen
Interpretation In an elastic layered material, the average modulus of a sample could be
predicted by considering the contribution from each constituent. The contribution would be proportional to the amount of the material, typically in a volumetric ratio. For the reinforced specimens constructed for these tests, the concentration of reinforcements is less than 1%. As such, it would be expected that the dynamic shear modulus would show little, if any change, due to the reinforcements.
However, at large strain levels, the ultimate strength and ductility of the composite will be altered more significantly by the more resistant constituent. Thus, reinforcements have an influence on the ductility of the samples in proportions greater than their volumetric proportions. Formation of shear bands in the unreinforced sand limits the strength of the material. Reinforcements inhibit the formation of these shear bands. In the unreinforced specimen, two sets of complimentary shear bands can form, which limit the ductility of the material. For the uniaxially reinforced specimen, the formation of one set of shear bands is inhibited, but deformation can still occur on the complimentary shear bands. Ductility is enhanced, but large plastic shear strains can still accumulate. In the biaxially reinforced specimen, long continuous shear bands are completely inhibited. The ductility of the material is thus significantly improved, and the sample maintains structural integrity.
Further, the modulus and damping ratio of the material is defined on a strain basis, yet in most empirical methodologies, measurement of strain is generally determined by measuring overall sample displacement. This common methodology does not consider the effect of localized deformations and thus gives values of moduli and damping that are not rigorous. However, in the tests on reinforced samples, reinforcements have inhibited the for-
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i i-oo s
?> 0 .95
fe;0.90
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i i i i i i I i i i i i i i i i i i i i i ) i i i i ' i • i * i i i i i i i
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
I I I I I I I I | I I I I I I I I I | I I / I I I ! I I J I I
5 10 15 20 ,..T-j.„f , r n T | ) ) | | ) ) ; i J ) ) |
25 30 35
FIG.
Number of cycles
19. Cycle Normalized Dynamic Modulus versus Number of Cycles
0.36
0.34
a;
C5
0.32
• 0.30
0.28
0.26 < I I 1 1 I 1 I | I I I
5 9 13
BIAXIAL reinforcements UNIAXIAL reinforcements NO reinforcements
i i i i i i 17 21 25
Number of cycles
FIG. 20. Damping versus Number of Cycles for Three Types of Specimens
mation of shear bands. As such, values for moduli and damping are actually more consistent, as the measurements and calculation of shear strains are accurate. Increase in modulus and decrease in damping ratio for the reinforced sample is more a reflection of the behavior of the sand with the dominating effect of deformation localization removed, than an effect of the reinforcements on the measured values of these parameters. Reinforcements do not alter the behavior of the soil, they restrict localization effects and,
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thus, increase the ductility of the material. Based on these experimental results, a simplified analysis procedure for
reinforced soil structures subjected to dynamic loadings can be considered. For reinforced soil material made with a low concentration of oriented reinforcements, the soil's behavior will dominate the behavior of the composite at strains less than those associated with the soil's peak strength. For a dense compacted sand, peak strength might occur somewhere between 2— 5%. Thus, to analyze the dynamic response of a structure composed of a material similar to these test specimens, such as a reinforced soil wall, the dynamic model could be developed and analyzed for dynamic response up to limiting strain values using the dynamic properties of the soil alone. Often, strains greater than 2-5% are not acceptable for the performance of the structure, and thus the analysis would yield results in the realm of required information.
If the strains calculated from the dynamic loading analysis are in excess of the strains associated with the nonreinforced peak strength of the soil, then a pseudo-static stability analysis of the structure such as a Newmark (1965) type of analysis could be performed. This method considers three forces acting upon a potentially deforming mass: (1) Driving force due to the weight of the soil; (2) a driving force due to the dynamic loading; and (3) resisting forces due to the soil and the reinforcements. Using limit equilibrium methods, such as found in Duncan et al. (1985), a dynamic yield acceleration can be determined. The dynamic yield acceleration is that acceleration which, when multiplied by the total weight of the sliding mass, gives a large enough driving force due to inertia to make the factor of safety equal to one. That the total driving force becomes equal to or greater than the total resisting force does not, by itself, necessarily imply "failure." Rather, because of the transient nature of the inertial driving force, finite displacements accrue on the failure surface. The magnitude of the displacements are dependent on the level and duration of the dynamic loading.
There are three main parameters affecting the dynamically induced displacement of a given block: (1) The dynamic yield acceleration; (2) the maximum acceleration induced by the dynamic loading; and (3) the duration of the dynamic loading.
If values of these three parameters are known, it is possible to estimate a range of dynamically induced displacements for the sliding mass. For seismic loadings, the charts of Newmark (1965), Franklin and Chen (1969), and Makdisi and Seed (1979) can be used to estimate the seismically induced displacements. For other loadings, double integration methods can be used (Goodman and Seed 1966).
The formation of shear zones will influence which shear strength will be mobilized in the soil and at the soil-reinforcement interface. If shear zones can form in the reinforced soil mass, then the post-peak critical state friction angles should be used for determining both the soil and soil-reinforcement interface shear strengths. The dynamic yield acceleration is thus determined using post-peak friction angles in the reinforced and nonreinforced regions of the slope (Fig. 21). If shear zone formation is restricted by either the reinforcements or by kinematics, then the peak friction angle can be used for determining the soil shear strength. For regions outside of the mass being analyzed, it is recommended that the post-peak friction angles be used for the soil and soil-reinforcement interfaces. The dynamic yield acceleration is
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<")
YPost Peak
Strain
(b)
—.—<&--***=*
Strain
<c) (d)
FIG. 21. Newmark-Type Analysis of Reinforced Soil Structures: (a) Analysis if Shear Bands Form in Sand; (b) Postpeak Shear Strength of Sand; (c) Analysis if Shear Bands Do not Form in Sand; (d) Peak and Postpeak Shear Strength of Sand
thus determined using peak friction angles for the sand in the reinforced mass and post-peak friction angles for the soil-reinforcement interfaces and the soil outside of the mass (Fig. 21). The reinforcements thus inhibit displacement through direct mechanical resistance and by inhibiting shear zone formation. These results suggest that when analyzing failed structures, the use of peak friction angles may be more appropriate in the reinforced mass when attempting back-calculations for measured displacements. It would be more prudent, however, for the design engineer to use post-peak friction angles throughout the reinforced and nonreinforced soil masses for pull-out-type failures.
CONCLUSIONS
The test results indicate that confining pressure and strain level affect the dynamic shear modulus and damping of unreinforced and reinforced sands. Frequency has no effect on the dynamic shear modulus or damping for unreinforced or reinforced sands. The results from these tests indicate that reinforcements do significantly change the ductility of the material. The reinforcements altered the mode of deformation of the sand, inhibiting the formation of shear bands at high strain levels, which resulted in a more ductile material with altered volume change characteristics and more uniform displacements throughout the sample. These results indicate that the effect of the reinforcements on the dynamic properties of the reinforced soil composite can be ignored when analyzing the dynamic response at small strains,
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but should be considered when evaluating the dynamic response at larger strains.
ACKNOWLEDGMENTS
This research was in part funded by the National Science Foundation PYI award to Nicholas Sitar, Grant No. CEE 8352147. The writers also gratefully acknowledge the support and advice of Prof. Carl Monismith of the University of California, Berkeley. Opinions, findings, and conclusions expressed in this paper are those of the writers and do not necessarily reflect the views of the National Science Foundation.
APPENDIX. REFERENCES
Duncan, J. M., Low, B. K., and Schaefer, V. R. (1985). STABGM: A computer program for slope stability analysis of reinforced embankments and slopes. Department of Civil Engineering, Virginia Tech, Blacksburg, Va.
Franklin, A. G., and Chang, F. K. (1977). "Earthquake resistance of earth and rock-fill dams." Misc. Paper S-71-17, Report 5, Soils and Pavements Laboratory, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss.
Goodman, R. E., and Seed, H. B. (1966). "Earthquake-induced displacements in sand embankment." J. Soil Mech. and Found. Engrg. Div., ASCE, 92(2), 125— 146.
Lazan, B. J. (1968). Damping of materials and members in structural mechanics. Pergamon Press, Oxford, England.
Makadisi, F. I., and Seed, H. B. (1978). "Simplified procedure for estimating dam and embankment earthquake induced deformations." J. Geotech. Engrg., ASCE, 104(7), 849-867.
Mitchell, J. K. (1981). "Soil improvement: State of the art." Session 12, Proc. 10th Int. Conf. on Soil Mech. and Found. Engrg., Stockholm, Sweden.
Newmark, N. M. (1965). "Effects of earthquakes on dams and embankments." Geo-technique. 15(2), 139-160.
Roscoe, K. H. (1970). "The influence of strains in soil mechanics." Geotechnique, 20(2), 129-170.
Seed, B., and Idriss, I. M. (1970). "Soil moduli and damping factors for dynamic response analyses." Report No. EERC 70-10, Earthquake Engrg. Res. Ctr., Berkeley, Calif.
Shewbridge, S. E. (1987). "The influence of reinforcement properties on the strength and deformation characteristics of a reinforced sand," thesis presented to the University of California, at Berkeley, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Sousa, J. (1986). "Dynamic properties of pavement materials," thesis presented to the University of California, at Berkeley, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Sousa, J., and Monismith, C. (1987). "Dynamic response of paving materials." Transp. Res. Rec, 1136, 57-68.
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