Stress-strain analysis of sand subjected to triaxial loading/Smelio bandinio itempiu ir deformaciju analize tiriant triasio slegio aparate/Triaksialajai slogosanai paklautas smilts sprieguma--deformacijas analize/Kolmeteljelisele survele allutatud liiva pingedeformatsioonide analuus.
Dirgeliene, Neringa ; Norkus, Arnoldas ; Amsiejus, Jonas 等
1. Introduction
Necessary information on soil properties, needful for designers and
constructors, is obtained in each construction site by examining the
physical and mechanical properties of soils (Chang, Meidani 2012;
Sulewska 2012). A triaxial test is recognized to be the most widely used
method for the determination of the strength properties and the
stress-strain state of soil. However, applying this test one should
evaluate the peculiarities of the testing equipment in terms of the
actual boundary conditions. One of these peculiarities to be taken into
account in triaxial test is the effect of end restraint prescribing
impact to the stress and strain distribution in sample. This
subsequently results in an accuracy of the determined mechanical
properties of soil. Thus, the contact between the testing equipment and
the ends of soil specimen should be properly evaluated. The developed
friction at the ends of specimen limits the deformation of the specimen
ends. It obviously has the influence to a non uniform distribution of
the stress and strain inside the specimen and finally affects the test
results. The finite-element method simulation as well as the analysis of
experimental tests yields that stress and strain distribution within the
sample is non-uniform during triaxial testing (Airey 1991; Jeremic et
al. 2004; Liyanapathirana et al. 2005; Peric, Su 2005; Sheng et al.
1997; Verveckaite et al. 2007; Widulinski et al. 2009). Thus, one must
identify the actual stress and strain distribution in the soil sample
when a load is transmitted in a provided way. Summarizing, one should
determine the influence of the non-uniformity of stress and strains for
the shear strength parameters.
One can list many investigations in this field with the different
results, proposals, recommendations. Yang and Ge (2012) concluded that
the influence of the end effect decreases with an increasing distance
away from the specimen ends. It is known from simulation results that
the stress and strain distributions are uniform within the 1/3 zone in
the middle of the specimen, thus, one can employ this finding to reduce
the influence of the end effect. One cannot directly use the force
applied to the specimen before it is deducted when measuring and
calculating the stress of the zone. By calculating and comparing, it is
suggested that the vertical stress applied to the specimen should be
deducted 10% for calculating the stress within the 1/3 zone in the
middle of the specimen, the horizontal stress remaining the same (Yang,
Ge 2012).
Liyanapathirana and his co-workers studied the effect of end
restraint on the non-homogeneous behavior of the triaxial specimen. They
obtained results for the ideal case, i. e. when there is no friction
between the triaxial specimen ends and the platens. The obtained results
have been compared with the one of the case where no displacement is
allowed between the specimen and the ends platens. It could be seen that
a destruction phase of the structured soil has not been influenced
significantly by the ends restraints but after the destruction, during
the hardening, the stress-strain behavior of the soil was significantly
influenced by the end restraint effect (Liyanapathirana et al. 2005).
Su and his colleagues concluded that the stress in the specimen
with free ends was uniform while the stress in the specimen with the
fixed ends was not uniform for non-dilative soil. The investigations
showed that the end restraints influence only the stress distribution in
the specimen and do not influence the overall behaviour in case of the
drained triaxial tests (Su et al. 2011).
Bishop and Green proved that the same strength properties are
obtained even for dense sand sample when the sample dimensions ratio is
1 and 2 in the case when the friction is eliminated at the top and the
bottom of the sample (Bishop, Green 1965). Other researches state that
the eliminating of the friction by applying the silicone between two
rubber membranes is a sufficiently reliable method to eliminate the
friction, that develop between the sample and the platens during the
triaxial test (Tatsuoka et al. 1984). Rowe and Barden (1964) found that
the usage of lubricated end platens led to a much greater uniformity of
stress and deformation during the test.
Current investigation analyses the influence of constraining
horizontal displacements at the sample top and bottom on the soil shear
strength parameters. The method to reduce this effect is proposed. The
influence of movable support on the soil shear strength parameters is
analyzed experimentally. This influence and the distribution of stress
and strain within the soil sample were also simulated using the
commercial program COSMOS/M (Finite Element Analysis System, Version
1.75).
2. Experimental analysis
2.1. Identification of tested soil
The type of soil analyzed experimentally and via numerical
simulation is sand. According to Unified Soil Classification System it
is recognized as poorly-graded sand with fine SP-SM. Sand is described
by the following properties: the uniformity coefficient is 3.03, the
curvature coefficient is 1.47, the specific gravity of soil particles is
2.671 g/[cm.sup.3], the maximum void ratio is 0.745, the minimum void
ratio is 0.502.
2.2. Triaxial testing
The testing process of the dense sands by triaxial apparatus leads
to the formation of a shear plane. The specimen parts, located below and
above this plane, displace in respect of each other not only in the
vertical but also in the horizontal directions. The horizontal
displacements are resisted by the friction forces, developed between the
specimen ends and the platens on the top and the bottom of the specimen.
Thus, the normal and the shear stresses are induced at the specimen
ends. This should be taken into account when determining shear strength
parameters of the soil. When the horizontal displacements at the ends of
the specimen are constrained, the larger normal stress magnitudes are
necessary for the sheared specimen parts to displace in respect of each
other. The ability of free displacements at the specimen base cancels
here the shear stress.
The triaxial tests were performed at Geotechnical Research
Laboratory of Vilnius Gediminas Technical University on the specimens of
height/diameter ratio of H/D = 2. The tested samples of low water
contents (W = 6%) have been performed by compacting. The triaxial tests
have been performed for the samples of two densities, namely: for the
dense sands of density p = 1.871 g/[cm.sup.3] and the void ratio e =
0.51; and that of the loose sands of the p = 1.610 g/[cm.sup.3] and the
e = 0.74. Each type of the prepared samples has been cut leastwise three
times. The boundary conditions being employed for the tests are: in the
first case, when the sample top cap can turn and the friction between
specimen ends and the platens is not eliminated (Fig. 1a); and in the
second case, when the above described friction is eliminated. The
friction at the ends of specimen is eliminated by introducing the
movable support which allows the lateral displacement of the specimen
base in any direction at the horizontal plane (Fig. 1b).
[FIGURE 1 OMITTED]
In the first case of the sample with the fixed ends, when the
failure plane starts to develop the vertical component of the stress
decreases evenly as the axial strain increases. In the second case of
the free ends when the axial strain of 4-5% is achieved a significant
vertical component of stress decrease is observed, after that the curve
declines insignificantly and remains stable (Fig. 2). The deviatoric
stress achieves the minimal value for dense samples with free ends
faster than for the sample with the fixed ends. In this case the
vertical component of the stress is by 16% smaller when compared with
the vertical stress for the sample with the fixed ends (here the
relative axial deformation [[epsilon].sub.1] is equal to 15%)
(Dirgeliene et al. 2007a; 2007b).
When testing the loose sand samples of ratio H/D = 2 for the fixed
and the free ends cases, the shape of the graphs of strain versus stress
[[epsilon].sub.1] = f([[sigma].sub.1] - [[sigma].sub.3]) is similar in
both cases under consideration (Fig. 3). The vertical component of the
stress varies similarly for the fixed and the free ends. The deviatoric
stress increases until the axial relative strain magnitude reaches 12%.
2.3. Determining the shear strength parameters
The mean value of the stress vertical component for the dense
samples in the case of free ends is approximately 16% less compared to
the one obtained by the standard triaxial apparatus (fixed ends) for
relative 15% axial deformation to be reached (Table 1). The obtained
residual mean values of [[sigma].sub.1] for the loose samples are
similar in the case of the standard and the improved triaxial
apparatuses to be employed.
The variation coefficients for the stress vertical components for
two densities of sample obtained by both types of apparatuses are
presented in Table 2. The analysis of these data confirms that the
variation coefficients are similar in all cases under investigation.
This is valid for the loose and the dense samples, also and for the free
and the fixed ends. Thus, the movable support has no influence on the
variation of [[sigma].sub.1].
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
The mean values of the angle of internal friction for the dense
sand, obtained from max values of the vertical stress [[sigma].sub.1] in
both considered cases of testing, differ insignificantly. The max
difference is 5%. The mean values of cohesion are approx 31% less for
the samples with the free ends. The values of the angle of internal
friction obtained from the values of the vertical component of stress
when [[epsilon].sub.1] = 15% for the sample with the free ends are up to
15% smaller compared to those obtained from the sample with the fixed
ends (Table 3). The values of cohesion are less by approx 42%
(Dirgeliene et al. 2007a; 2007b).
The calculated mean residual values of tan[phi] for the loose sands
are similar both of the standard and the improved apparatuses test
results. The values are approx 6% larger for the samples tested with the
movable support, and the mean residual values of c are approx 35% larger
for the samples tested with the fixed support. Thus, the movable support
has no essential influence on identifying the shear strength parameters
of the loose sands by processing the data tests.
[FIGURE 4 OMITTED]
3. Theoretical analysis of stress distribution in soil sample via
numerical simulation
The stress and strain state of soil sample has been simulated
applying the computer software COSMOS/M (Finite Element Analysis System,
Version 1.75). The nonlinear behavior of the sample is described by the
Drucker-Prager physical model of an elastic-perfectly plastic material.
The material of the discrete model of the sample responses in an elastic
way prior the yielding limit is reached, after that it responses in the
perfectly plastic way. The yielding criterion is described by:
F = 3[alpha][[sigma].sub.m] + [bar.[sigma]] - k = 0, (1)
where [alpha] and k - material constants; [[sigma].sub.m]-the
stress mean value; [bar.[sigma]]--an effective stress value. The
constants [alpha] and k obtained experimentally.
During the triaxial test the specimen of the dense soil is sheared,
the sand particles at the shear plane slide in respect of each other,
i.e. the dilatancy effect is recognized. The dilatancy causes the change
of the sample volume during shear. The dense soil is loosened at the
shear plane (volume), i.e. its properties change and therefore the
material properties here are different when compared to that of the
remaining volume of the tested specimen. Thus, the different material
properties have been chosen for the specimen and the shear plane
(volume), respectively (Table 4) to represent the actual situation.
The description of the discrete model of the soil specimen and the
loading for simulation of the triaxial test is given below. The geometry
of the specimen: the diameter D = 5 cm, the height H = 10 cm (Fig. 4).
The specimen is divided to solid tetrahedral finite elements with four
nodal points of three degrees of freedom. Two different design schemes
have been simulated. At the first case the nodes at the top plane of the
specimen are subjected by the equal vertical displacement [u.sub.y] =
const, while that of the nodes at the bottom plane are fixed. The
horizontal (lateral) displacements of the top and the bottom plane of
the specimen are also fixed. The pressure is applied isotropically to
the soil sample. In the second case the design scheme corresponds to the
first case described above with an exception that the horizontal
displacements at the bottom plane are free (the movable support links
are introduced).
The comments on the obtained numerical simulation results are given
below. The shear stress components develop in the contact planes of the
platens and the specimen ends. The latter does not allow (or constrain)
the horizontal displacements in the case when the changed properties at
the shear plane (volume) are introduced to the design scheme (Figs 5, 7,
9, 11). The horizontal displacements develop in the case when the design
scheme of the free bottom plane is employed (Figs 6, 8, 10, 12).
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
The case of the free ends yields the stress vertical component
reduction of approx 10% when compared to the case of the fixed ends
(Fig. 13). The similar data have been obtained experimentally.
4. Conclusions
1. The movable support between the sample base and the platen
aiming to reduce the friction between the specimen and the platen is
recommended to be introduced for obtaining the more accurate soil shear
strength parameters from triaxial test results.
2. The experimental investigations showed that the mean values of
the angle of internal friction obtained by processing the values of
[[sigma].sub.1] when [[epsilon].sub.1] = 15% are approx 15% smaller in
the case of the free specimen ends compared to the one of the fixed
specimen ends. The mean values of the residual cohesion are smaller by
42%.
3. The FEM simulation showed that the shear stress components
develop in the contact planes of the platens and the specimen. They do
not allow the horizontal displacements. This is not evaluated when
processing the test data to determine the shear strength properties of
the tested soil.
4. The FEM simulation proved that the case of the free specimen
ends yields the stress vertical component reduction of approx 10%
compared to the case of the fixed ends. The similar results have been
obtained experimentally.
The equipment and infrastructure of Civil Engineering Scientific
Research Center of Vilnius Gediminas Technical University were employed
for investigation.
doi:10.3846/bjrbe.2013.04
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699-707. http: <//www.ejge.com/2012/Ppr12.063alr.pdf>. Received 16
April 2011; accepted 8 February 2013
Neringa Dirgeliene (1) ([mail]) Arnoldas Norkus (2), Jonas Amsiejus
(3), Sarunas Skuodis (4), Daiva Zilioniene (5)
(1 2, 3, 4) Dept of Geotechnical Engineering, Vilnius Gediminas
Technical University, Sauletekio al.11, 10223 Vilnius, Lithuania (5)
Dept of Roads, Vilnius Gediminas Technical University, Sauletekio al.11,
10223 Vilnius, Lithuania E-mails: 1neringa.dirgeliene@vgtu.lt; (2)
arnoldas.norkus@vgtu.lt; (3) ajonas@vgtu.lt; (4)
sarunas.skuodis@vgtu.lt; (5) daizil@vgtu.lt
Table 1. Mean values of stress vertical component
[[sigma].sub.1]
Void Stress Values of Values of variation
ratio, e vertical [[sigma]. coefficients V
component sub.3], of stress vertical
[[sigma]. kPa components
sub.1] [[sigma].sub.1], kPa
Standard Improved
apparatus apparatus
0.51 Peak 50 304.71 308.79
100 541.92 489.94
200 907.61 930.93
Residual 50 234.30 197.71
100 423.32 339.75
200 760.05 661.73
0.74 Residual 50 183.15 184.10
100 331.93 309.51
200 583.35 601.15
Table 2. Variation coefficients of stress vertical component
[[sigma].sub.1]
Void Stress Values of Values of variation
ratio, e vertical [[sigma]. coefficients V of
component sub.3] stress vertical
kPa component
[[sigma].sub.1], kPa
Standard Improved
apparatus apparatus
0.51 Peak 50 0.090 0.103
100 0.088 0.114
200 0.053 0.037
Residual 50 0.072 0.061
100 0.047 0.057
200 0.020 0.036
0.74 Residual 50 0.064 0.065
100 0.058 0.056
200 0.024 0.048
Table 3. Mean values of the soil shear strength parameters [phi]
and c
Void Shear strength Mean values of soil
ratio, e parameters shear strength
parameters
Standard Improved
apparatus apparatus
0.51 Peak [phi], [degrees] 36.6 38.0
c, kPa 30.9 21.4
Residual [phi], [degrees] 33.6 31.0
c, kPa 17.8 10.3
0.74 Residual [phi], [degrees] 26.8 28.3
c, kPa 17.7 11.4
Table 4. Soil strength parameters
Soil properties For For
whole failure
specimen plane
Elasticity modulus E, MPa 50.0 30.0
Poisson's ratio [upsilon] 0.30 0.45
Angle of internal [phi], 37.9 30.0
friction [degrees]
Cohesion c, kPa 26.0 17.0
