Stress-strain analysis in the soil sample during laboratory testing.
Verveckaite, Neringa ; Amsiejus, Jonas ; Stragys, Vincentas 等
Abstract. During the determination of soil strength and
compressibility in a laboratory by different apparatus soil is loaded in
a different way. It has an influence on stress-strain distribution in a
sample. Some factors are not evaluated during the results
interpretation, for example, friction between soil and device metal
parts. The finite-element method analysis also shows that during
triaxial, oedometer, shear box tests distribution of stress and strain
in the sample is non-uniform. A special apparatus was designed and used
for determining horizontal component of stress in the cross-section of
the sample. It was determined for sands that horizontal component of
stress in the cross-section centre is significantly smaller than at the
edges. Increasing load plastic deformations are developing not in the
whole sample but in particular places. If we know a real distribution of
stress and strain in the sample, it is possible to determine the soil
strength and deformation parameters in a more precise way or to rate the
influence of different factors on soil properties.
Keywords: stress-strain distribution, triaxial test, oedometer,
shear box, finite-element method, soil strength.
1. Introduction
Mechanical properties of soils were determined by different types
of apparatus. Different stress-strain distribution was obtained using
various apparatus. Boundary conditions are not distinct when analysing
soil samples by oedometer and shear box. It is not obvious a stress
distribution within oedometer soil specimen when applying vertical load,
a part of vertical load descends to shear ring pane. Not all vertical
load applied to top of shear box specimen is transmitted to soil. It is
not evident the regularity of change of normal stresses on a shear
plane. If an oedometer test is analysed, stress distribution in soil
sample depends on the following: manner of load transmission, manner of
ring embedding in apparatus (with fixed or floating ring). If a shear
box test is analysed, the stress distribution depends on the following:
manner of vertical load transmission, position of the mobile part of
shear ring, horizontal displacement of the mobile part of the ring.
It is more precise to model stress and strain state in the soil
using a triaxial apparatus. Triaxial test is the most widely used test
method for determining the strength and stress-strain properties of
soil. It is assumed that soil specimen deforms uniformly during the
test. The uniformity of stress and strain in the specimen is the main
idea of triaxial test. However, it is not often the case that a specimen
in triaxial apparatus deforms uniformly during the test. Non-uniformity
can be caused by the end restraint, insufficient drainage, membrane
effects, soil compression, preparation of soil specimen, self-weight
[1-5]. What is stress and strain distribution in soil sample, when a
load is transmitted in a provided way? What influence does a
non-uniformity have on the strength and stress-strain parameters of
soil?
2. Review of references of laboratory testing
K. H. Head suggests that due to friction near the platens a
so-called ,dead zone" is formed between triaxial sample ends and
the platens [6]. That "dead zone" causes a non-uniform
distribution of stress and strain and non-uniform distribution of pore
pressure in case of an undrained test. The use of lubricated ends
eliminates "the dead zones" and protects from a wrong increase
in measured strength due to the end restraint. The length of sample
should be decreased from the standard ratio of height and diameter 2:1
to 1:1. This decrease is necessary to ensure an effective lubrication.
It influences a more uniform stress and strain distribution [6-8], the
sample may retain its cylindrical shape even at large strains. Measured
strains and volume changes at failure are larger and probably
demonstrate the behaviour of soil mass than those from conventional
tests. Another advantage of eliminating friction is that it decreases
lateral forces on triaxial cell piston [6].
Triaxial sample end restraint protects the sample from moving
sideways freely and causes shear stress at the ends of the sample. Thus,
the stress and strain states are not uniform within the sample.
Therefore there are difficulties in interpreting test results.
Experiments findings showed an influence of end restraint on shear
strength of soils. The difference between the shear strength of soil
sample, when the friction is eliminated by lubricating platen ends, is
not considerable if the height and diameter ratio (H/D) of the sample is
larger than 2. Norris, Lee, Lade, Rowe ir Barden found that in the
samples with a H/D ratio of 2 to 1 without friction the volume change at
failure is higher by about 1-1,5 % than in those without lubricated ends
[6]. Bishop ir Green voted that the dilations of soils are practically
the same both in lubricated ends and in those with the friction [9].
These discrepancies may be caused by the difference of axial strains at
failure. Discrepancies of this kind were also found in previous studies.
Raju and his colleagues presented more reliable explanations for the
non-uniform results [10]. For the samples without end lubrication, axial
strains and volume change occur in the middle of the sample, while their
calculations are made according to the values of the whole sample, even
though the latter being lower than those in the middle part.
Lubrication method using silicone between two rubber membranes is
sufficiently reliable to eliminate ends restraints in triaxial tests
[11, 12]. Scientists have carried out triaxial tests using Fulung,
Ottawa and Tamsui River types of sands. Higher peak strengths were
generally found in the samples without lubrication. The volume change of
the triaxial sample at failure of the sands, either dilation or
compression, were higher in the samples with lubricated ends. The volume
changes of the samples with and without end lubrication were rather
uniform within the sample before the peak strength was reached. When the
peak strength was reached and past, the volume change of the samples
without end lubrication at the middle part of the sample was
considerably higher. The particle breakage within the samples with
lubricated end platens was quite uniform throughout the test. However,
the particle breakage within the samples without end lubrication was
significantly larger at the middle portion of the sample and after the
peak [13].
D. W. Airey suggests that the non-uniformity occurs in triaxial
samples during isotropic consolidation. He says that uniformity of
deformation and stress can only be guaranteed if frictionless ends are
used [14]. B. Jeremic, Z. Yang, S. Sture investigated the behaviour of
elastic-plastic specimens during testing in a triaxial apparatus. They
determined that inclination of end platens can cause significant
non-uniformities in stress distribution inside specimen. The specimen is
distorted sideways and the plastic zone remains signficantly larger than
in the level end platen case. Development of plastic zone for a specimen
with fixed level end platens begins in the centre of specimen and
remains sizable to the specimen ends. For the specimens with friction
ends the plastic zone begins at the ends and extends to middle of
specimen [15].
H. Sun, J. F. Chen, X. R. Ge used computerised tomography or
commonly known by its medical name an X-ray scanner. This method
demonstrates the attenuation of an X-ray through soil specimen and
density of specimen. The lower density bands appeared in the centre of
specimen at a lower confining pressure and spread to the edge due to the
weak internal coherence. At higher confining pressure the lower density
bands started at the edge and spread to the centre of triaxial due to
strong internal coherence of sample [16]. J. Otani, T. Mukunoki, M.
Yoshimura also used a new triaxial compression apparatus with the X-ray
scanner. After consolidating, the initial condition of the soil sample
was scanned before applying the compression. According to the initial
condition, the sample cannot be considered homogeneous due to
insufficient saturation. The density around the centre of the sample is
lower. Low density zones first form in the middle of the sample, later
other low density zones start and that is strain localisation. It is
simple to determine that the local shear areas start from the bottom of
the sample and expand towards the sample edges. With the strains
increasing, the shear areas of the sample go upwards, from the bottom of
the sample towards the top [17].
C. K. Januskevicius and E. Vey for measuring stresses and strains
in triaxial sample used embedded gauges. Samples were 12,7 cm in
diameter and 27,94 cm in height. They were vertically loaded at constant
rate of strain and confined by horizontal pressure. The results of the
strain-gauge show a non-uniformity of strain along the sample length. In
the middle of the sample (at its axis) the strains were found to be
higher than the average ones, whereas they were less than average at the
distance of 3,81 cm from the sample ends (at its axis). The strains were
lower at the distance of 3,81 cm from the sample bottom than at the same
distance from the sample top (at its axis). The strains were gauged at
mid-height of the sample and at 3,81 cm radius positions from the sample
centre. The results demonstrate that strains at the centre of the sample
are only insignificantly larger than those at the edge [18].
A. Drescher and I. Vardoulakis maintain that bulging and shear band
formation depend on the sample density. A dense sample is more sensitive
to these forms of strain inhomogeneities [19]. Researchers presume that
in medium-dense and loose sand samples shear bands do not appear and the
sample may be deformed uniformly even at large strains [20].
Strength and deformation parameters depend on stress-strain
distribution in the case of the symmetrical stress distribution about
axis. Using stresses [[sigma].sup.2] = max and in other case
[[sigma].sup.2] = min, different soil mechanical properties are obtained
[21].
D. Sheng and his colleagues analysed soil stress and strain
inhomogeneities in a triaxial sample caused by end restraint in drained
and undrained tests, also at insufficient drainage in drained test. The
stresses were measured in three aspects. First, when the soil sample
end, contact with platen is perfectly smooth, without friction (SC) (Fig
1 a, d, g); secondly, when there is friction (FC) between sample ends
and platens (Fig 1 b, e, h); and third, when the contact is rough (RC)
(Fig 1 c, f, i). For that purpose a slightly overconsolidated Swedish
clay was used. The analysis was carried out using numerical method, by
applying the commercial program ABAQUS [1].
[FIGURE 1 OMITTED]
The distribution of effective axial stress [[sigma]'.sub.a]
(kPa) in specimens for drained tests can be seen in Fig 1 (qualitative
view). The maximum values of effective axial stresses take place at the
end edges and the minimum near-1/2 height of the lateral surface of the
specimen [1].
3. Theoretical analysis of stress distribution in soil specimen
using a numerical method
In order to analyse stress and strain distribution in soil sample
during the oedometer and shear box tests a numerical modelling was
carried out by COSMOS/M programme. A finite element models are presented
to simulate the testing procedures. The formulation of a linear static
problem for solution by the displacement method is fully described by
the matrix equation:
[K]{U} = {K}, (1)
where[K] K - the structural (assembled) stiffness matrix; {U}--the
vector of unknown nodal displacements; {F}--the load vector.
The nodal displacements are calculated according to formula:
{U} = [[K].sup.-1] {F}. (2)
Then the vector of elements nodal displacements {[u.sub.k]} is
found. By using the matrix of elements geometrical properties
[[B.sub.K]], the elements deformations are obtained:
{[[epsilon].sub.K]} = [[B.sub.k]]{[u.sub.K]}. (3)
The stresses of finite elements are obtained, when elements
deformations and stiffness matrix of elements material [[K.sub.elem]],
in which the information about characteristics of material elasticity E
, [upsilon], is known:
{[[sigma].sub.K]} = [[K.sub.elem]] {[[epsilon].sub.K]}. (4)
The following parameters were used for the sand specimen:
elasticity modulus--20 MPa, mass density--1,85 Mg/[m.sup.3],
Poisson's ratio--0,3. The oedometer sample, 0,0875 m in diameter
and 0,035 m in height, was discretised into 4-node two dimensional
elements (Fig 2) with symmetric loading. Only two translational degrees
of freedom per node were considered for structural analysis. Two
principal diagrams were designed: a) oedometer with fixed ring and stiff
stamp, displacement const [u.sub.y] = uniformly vertically acts on the
top surface of the oedometer; b) oedometer with fixed ring and flexible
stamp, vertical component of stress [[sigma].sub.y] = cons uniformly
acts on the top surface of the oedometer.
[FIGURE 2 OMITTED]
The soil sample of the shear box with a stiff stamp, 0,071 in
diameter and 0,035 in height, was discretised into 4-node tetrahedral elements (Fig 3). Three translational degrees of freedom per node were
considered for structural analysis. Displacement const [u.sub.y] =
uniformly vertically acted on the top surface of the model.
[FIGURE 3 OMITTED]
While analysing the oedometer sample by numerical method, it was
found that shear stress distribution in the sample was not uniform, and
thus the vertical component of stress distributed also non-uniformly
(Fig 4). A part of vertical load descends to shear ring pane. When the
oedometer was analysed with the fixed ring and stiff stamp, it was
obvious that whole sample is affected by the stress of compression and
the upper corner by the stress of tension (Fig 5).
[FIGURES 4-5 OMITTED]
The stress and strain distribution in soil sample during the shear
box test is non-uniform [22-25]. Applying a numerical method, horizontal
soil displacements [u.sub.x] in soil sample during shear box tests were
analysed using stiff and flexible stamps. In Fig 6 we can see that the
horizontal soil displacements [u.sub.x] in shear box distribute
non-uniformly. Fig 7 shows that vertical soil displacements reflect
stamp turn: one side of stamp rises, the other--falls. Experiments
demonstrate the same results.
[FIGURES 6-7 OMITTED]
4. Experimental analysis of stress distribution in soil specimen
during triaxial tests
4.1. Physical and mechanical properties of testing soil
The fine sand was tested in this work. Name of soil according with
Unified Soil Classification System is poorly-graded sand with fine
SP-SM. Particles of sand are of rounded shape. The sand has an
uniformity coefficient of 3,03, curvature coefficient 1,47, a specific
gravity of soil particles of 2,671, maximum void ratio of 0,7446,
minimum void ratio of 0,5706, density index 0,83. The grain size
distribution of the sand is shown in Fig 8.
[FIGURE 8 OMITTED]
The specimens for triaxial tests were prepared of dry sand with 6 %
wet by tamping. The diameter of specimens was nearly equal to 5 cm, the
height-diameter ratio being about 2. All shear tests were performed in a
conventional triaxial cell under undrained (CU tests) conditions. The CU
tests were performed under cell effective pressure [[sigma].sup.3] of 50
kPa, 100 kPa, 200 kPa. The experiments were carried out at a constant
cell pressure and constant axial strain rate. During the test, axial
deformation, volume change, force and pore water pressure were measured
(Figs 9, 10). Fig 9 shows the relationship between a principal stress
difference ([[sigma].sup.1]- [[sigma].sup.3]) and axial strain
[[epsilon].sub.a] of CU test. The deviator stress at failure is found to
increase with cell pressure. This increase becomes progressively smaller
as the water in the voids is compressed, and ceases when the stresses
are large enough.
[FIGURES 9-10 OMITTED]
The loading path is given in Fig 11. The Mohr envelope for a series
of CU tests on sand specimens is shown in Fig 12. c and [phi] according
to experimental findings were calculated.
[FIGURES 11-12 OMITTED]
4.2. Experimental tests
In geotechnical laboratory a device was designed to analyse the
non-uniform deformation of soil specimen caused by such factors as
compaction of soil specimen, soil grading, shape and orientation of soil
particles, magnitude of load etc.
The device consists of a metal cylinder of 15 cm in diameter and 45
cm height--1, inside which there is a rubber membrane--2. Metal cylinder
was filled with a soil--3. Inside the soil a steel strip--4, 20 mm width
and 0,5 mm thick was inserted. Steel strip was placed in 3 positions: in
the centre of soil A and in the sides of soil B and C (Fig 13). After
cylinder was filled with a soil, ends of cylinder were closed with
plates--5. Plates were screwed in order to stop leakage of air pressure.
A sample of soil was subjected to an air pressure in the horizotal
direction. Then the steel strip was loaded step by step by a vertical
load till finally pulled out.
[FIGURE 13 OMITTED]
The same fine sand like triaxial sample and fine granite crushed
stone of 0,5 mm were used for experiments. Soil samples were compacted
and subjected to 25 kPa, 50 kPa, 75 kPa of confining air pressure. The
density of sand 1,7 t/[m.sup.3], void ratio 0,6. The density of fine
granite crushed stone 1,4 t/[m.sup.3], void ratio 0,95. Six experiments
were carried out in one hole applying different confining air pressure.
The experiment findings are presented in Figs 14-18. Experimental
findings show that horizontal component of stress [[sigma].sub.x] inside
soil sample is distributed non-uniformly. Larger horizontal component of
stress was found in the sides of soil specimen (side holes B,C) and
smaller was found in the centre of soil specimen (middle hole-A). Figs
14, 16, 17, 18 show stresses, when steel strip from sand specimen was
pulled out about 1-3 centimeters (continuous line) at first, and finally
(dotted line) was pulled out at a larger force. It is not evident why.
[FIGURES 14-18 OMITTED]
Three types of samples with the density of 1,65 t/[m.sup.3] were
prepared in order to estimate the influence of vertical component of
stress on stress distribution in the soil samples. First sample,
cylinder was full of sand (Figs 14, 16); the second sample, it is
slightly more of sand in the cylinder than its length (Fig 17); the
third sample contains slightly less sand in the cylinder than its length
(Fig 18). Experimental results did not show an obvious influence.
Displacements were not measured.
The pull-out force from fine granite crushed stone was larger,
because its particles had a sharp shape. Thus, friction between a steel
strip and particles of fine granite crushed stone was higher than
between the steel strip and sand.
5. Conclusions
1. Mechanical properties of soils depend on construction of
apparatus. Using different types of apparatus different stress-strain
distribution was obtained.
2. The analysis of vertical component of stress [[sigma].sub.y]
using the numerical method demonstrates a non-uniform distribution in
soil specimens during oedometer, shear box and triaxial tests.
3. From the results of tests carried out in the laboratory it is
obvious that horizontal component of stress [[sigma].sub.x] inside
triaxial soil sample is distributed non-uniformly. Larger horizontal
component of stress was found at the sides of soil specimen and smaller
in the centre of soil specimen.
4. Analysis of references show that stress distribution in the
triaxial soil samples are influenced by the friction between platens and
sample ends.
5. Further experiments should be carried out in order to analyse
stress distribution in the sample depending on soil density, soil
grading, shape and orientation of soil particles, magnitude of load,
ways of sample preparation, sample height and diameter ratio, friction
between platens and sample ends etc.
Received 23 Nov 2005; accepted 11 Sept 2006
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ITEMPIMU BUVIO GRUNTO BANDINIUOSE, TIRIANT JUOS LABORATORINEMIS
SALYGOMIS, ANALIZE
N. Verveckaite, J. Amsiejus, V. Stragys
Santrauka
Tiriant gruntu mechanines savybes laboratorijoje kiekviename
aparate bandinys apkraunamas skirtingai. Tai turi itakos itempimu ir
deformaciju pasiskirstymui bandinyje. Interpretuojant bandymo duomenis
ignoruojami kai kurie veiksniai, turintys itaka itempimams
pasiskirstyti, pvz., bandinio galiniu plokstumu ir stampo trinties jegos
tiriant grunta triasio slegio aparate, bandinio ir ziedo trinties jegos
tiriant grunta odometre ar tiesioginio kirpimo aparate. Itempimu
pasiskirstymo skaitiniu modeliavimu nustatyta, kad ne tik odometre bei
tiesioginio kirpimo aparate, bet ir triasio slegio aparate bandinio
itempimo buvis nera vienodas. Sukonstruotas prietaisas, kuriuo galima
ivertinti itempimu horizontaliosios komponentes kitima bandinio
skerspjuvyje. Tiriant smelini grunta gauta, kad itempimu
horizontaliosios komponentes reiksme bandinio centre yra gerokai mazesne
nei taskuose arti soninio pavirsiaus. Plastiniu deformaciju didinant
apkrova atsiras ne visame bandinyje, o tik tam tikruose jo taskuose.
Zinant tikraji itempimu ir deformaciju pasiskirstyma bandinyje, bus
galima tiksliau nustatyti grunto mechaniniu savybiu rodiklius arba
ivertinti tam tikru veiksniu itaka grunto savybiu rodikliams.
Reiksminiai zodziai: itempimu ir deformaciju pasiskirstymas,
triasio slegio aparatas, odometras, tiesioginio kirpimo aparatas,
skaitinis modeliavimas, grunto mechaniniu savybiu rodikliai.
Neringa Verveckaite (1), Jonas Amsiejus (2), Vincentas Stragys (3)
Dept of Geotechnical Engineering, Vilnius Gediminas Technical
University, Sauletekio al.11, LT-10223 Vilnius, Lithuania. E-mail: (1)
nidavera@yahoo.com; (2) jonas.amsiejus@st.vtu.lt;
(3)vincentas.stragys@st.vtu.lt
Neringa VERVECKAITE. PhD student at the Dept of Geotechnical
Engineering at Vilnius Gediminas Technical University, Lithuania.
Research interests: stress and strain distribution in the soil sample
during laboratory testing, triaxial testing.
Jonas AMSIEJUS. Dr Assoc Professor at the Dept of Geotechnical
Engineering at Vilnius Gediminas Technical University, Lithuania.
Research interests: various topics of estimation of soils mechanical
properties, stress and strain state, practical aspects of using
probabilistic and statistical analysis and design methods for structures
and foundations.
Vincentas STRAGYS. Dr Assoc Professor. Head of Geotechnical Dept of
Vilnius Gediminas Technical University. Partner of Framework 5 research
project "GeoTechNet". Vice-Chairman of Lithuanian Association
of Civil Engineers and Chairman of Technical Committee
"Geotechnics" at Lithuanian Standards Board. Research
interests: geotechnical investigation and testing soils, triaxial
testing, research in to standardisation in geotechnics.