The Effect of Soil Depth and Box Culvert Geometry on the Static Soil-Culvert Interaction

Full text

The Effect of Soil Depth and Box Culvert Geometry
on the Static Soil-Culvert Interaction
O. Abuhajar, Ph.D.1; T. Newson, Ph.D.2; and
H. El Naggar, Ph.D.3
1Geotechnical Research Center, Dept. of Civil and Environmental Engineering, Western Univ.,
1151 Richmond St., London, ON, Canada N6A 5B9. E-mail: oabuhajar@gmail.com
2Geotechnical Research Center, Dept. of Civil and Environmental Engineering, Western Univ.,
1151 Richmond St., London, ON, Canada N6A 5B9. E-mail: tnewson@eng.uwo.ca
3Geotechnical Research Center, Dept. of Civil and Environmental Engineering, Western Univ.,
1151 Richmond St., London, ON, Canada N6A 5B9. E-mail: helnaggar@eng.uwo.ca
Abstract
Soil arching is major factor that often controls the response of box culverts to static loads. The
effect of relative stiffness between the culvert and the surrounding soil is critical in considering
the complex soil-culvert interaction (SCI). In practice, box culverts can have different
characteristic length scales in relation to vertical and horizontal soil stress changes. Therefore,
the effect of box culvert size was investigated experimentally and numerically. In the present
study, centrifuge tests were conducted to investigate the SCI considering the height of soil
column and culvert geometry. The results were used to calibrate a numerical model to further
investigate the response of the box culverts to static loads. The results have been evaluated for
bending moment, soil pressure and soil culvert interaction factors. These results were then used
to establish charts that may be employed to assess design values for box culverts.
INTRODUCTION
The study of soil-culvert interaction problems are considered to be one of the more complex
challenges in the field of soil-structure interaction. There are several factors affecting the
response of box culverts to the surrounding soils. These include the effect of soil arching
phenomena, which can lead to an increase or decrease in the soil pressures around the box
culverts due to the stress redistribution. Box culverts tend to have stress concentrations at the
corners, in contrast to other typical culvert shapes. Other factors that may affect the soil culvert
interaction are the geometrical configuration and thicknesses of the box culvert, as well as the
properties and the depth of soil around them. The focus of this paper is on the geometric effects
of box culverts constructed with the embankment installation method.
Soil culvert interaction (SCI) should be considered when defining the static loads as a
culvert, to ensure safe and economic culvert designs. The soil culvert interaction factor Fe is
defined as the ratio of the actual (measured) soil pressure to the theoretical soil pressure. Once
these factors are defined, the actual soil pressure attracted to the box culvert can be calculated.
The theoretical vertical soil pressure v
σ
is usually obtained by multiplying the unit weight of the
soil column
γ
s above the box culvert times its height H. To account for the effect of soil arching,
the actual vertical soil pressure on the culvert top slab may be given by the theoretical vertical
pressure multiplied by the soil culvert interaction factor Fe. The actual vertical soil pressure can
then be given by:
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HF Sev
γ
σ
=
Several researchers (Marston and Anderson (1913), Clarke (1967), Bennett et al. (2005),
Spangler (1947), Katona et al. (1981, 1982), Tadros et al. (1989), Kim and Yoo (2005), Kang et
al. (2008) and others) and standards (AASHTO (2002) and CHBDC (2006)) have investigated
soil culvert interaction considering the effect of soil height above the box culvert. All of these
studies were presented in terms of a dimensionless number (H/Bc). Figure 1 defines the symbols
H and Bc.
Figure 1. Schematic diagram for centrifuge tests (All units are in mm).
In this paper, the response of the box culvert was investigated in a series of experimental
centrifuge tests and used to validate a numerical model using FLAC 2D. A brief description of
the centrifuge tests and development of the numerical model and its calibration/verification is
introduced. A detailed parametric study was then performed to investigate the static SCI and the
results are presented in the form of static bending moment, static soil pressure and soil culvert
interaction factors. To consider the effect of culvert size, the soil fill height was fixed and the
culvert width was changed and the ratio of the culvert width Bc to soil height H (Bc/H) was
investigated. Data for two different culvert wall thicknesses is presented in terms of the change
in culvert thickness to width ratio (t/Bc); referred to in the paper as thin (t/Bc = 0.06) or thick
(t/Bc = 0.12) culverts respectively.
CENTRIFUGE MODELING
A series of static centrifuge tests was conducted at the Rensselaer Polytechnic Institute (RPI)
geotechnical centrifuge facility to investigate the effect of soil arching on the response of box
culverts. Four tests were performed involving two box culvert thicknesses buried within dry
Nevada Sand at two relative densities. The static centrifuge model tests were focused on the
response of the box culverts to the self-weight of the sand. The data collected from the sensors
during the centrifuge tests were interpreted to investigate the main features of the soil-culvert
interaction.
The 120-Nevada Sand used as a test bed was fine, uniform, and clean. It was a poorly
graded sand (SP) with particle sizes in the range of 0.075 to 0.550 mm. The maximum and
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minimum unit weights
γ
max and
γ
min were 16.77 kN/m3 and 14.85 kN/m3. The critical state and
peak friction angles are 32o and 40o.
Several factors were considered in the choice of the culvert model, including: the culvert
sizes and wall thickness, height of the centrifuge box, and g-level that the centrifuge test would
run at. Based on these factors, a hollow square box aluminum tubes were used to construct the
model box culverts. They had external dimension of 76.2 mm, one was 6.35 mm thick (termed
‘thick’ culvert) and the other was 3.18 mm thick (termed ‘thin’ culvert). At 60g in the centrifuge,
these dimensions will be 4.572 m external dimension and 0.533 m and 0.267 m thick.
Strain gauges and tactile pressure sensors were used to investigate the static bending
moment and soil pressure around the box culverts. The locations of these instrumentations were
kept constant in all of the tests. Strain gauges were used to measure strains on the outside and
inside faces of the box culvert top slab and side wall. These strain measurements were then
converted into bending moments using calibration factors. Two tactile pressure sensors were
used for each culvert model to measure the vertical soil pressure on the top slab and horizontal
soil pressure on side wall of the culvert. It should be noted that this sensor has the ability to
measure the normal stress only (i.e. it does not account for shear stresses).
The centrifuge rigid box used in all tests was 355.6 mm in height. The sand was placed in
25.4 mm layers using a raining technique (air pluviation) alone to achieve the 50% relative
density. To achieve the 90% relative density, each sand layer was additionally tamped after air
pluviation. The models were built after placing the empty centrifuge box on the centrifuge
platform. As the required level of sand under the culvert model was reached, the tactile pressure
sensors were connected to the culvert model and placed in the middle of the centrifuge box. All
strain gauges and tactile pressure sensors were then checked and connected to the data
acquisition system. Figure 1 shows the general configurations of the test models. Tests 1 and 2
were for the thick culvert with sand density of 90% and 50 %, respectively. Tests 3 and 4 were
for the thin culvert with sand density of 50% and 90%, respectively.
The centrifuge was accelerated incrementally and held at the following acceleration
levels, 10g, 20g, 30g, 40g, 50g and 60g to check the stability of the sensor readings. Data from
all of the sensors were recorded continuously during the tests. Further details regarding the
centrifuge tests are presented by Abuhajar (2013) and Abuhajar et al. (2015a).
NUMERICAL MODELING
Numerical modeling was conducted to investigate the effects of several parameters on soil
culvert interaction, and one of them being the effect of the box culvert size. The numerical
models in FLAC 2D consisted of two main materials: the box culvert and the soil. The box
culvert was modeled using liner elements, which were assumed to behave linearly. Hence the
linear elastic model was used to simulate their response. The soil response was expected to cover
a range of elastic and plastic deformations; hence an elastic-perfectly plastic model with Mohr-
Coulomb failure criterion was used to simulate its nonlinear behaviour. The model was based on
plane strain conditions and was formulated in terms of effective stresses. The material behaviour
was modeled by using elements that obey assigned linear or nonlinear stress-strain behaviour in
response to the forces and boundary conditions (Abuhajar (2013) and Abuhajar et al. (2015a)).
The numerical models simulated the centrifuge tests at prototype scale. The soil was
modeled using continuum zones and each zone divided into small grids. The finite difference
grids used around the box culvert were square in shape; they were rectangular elsewhere as
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shown in Figure 2. The density of the grid was increased around the box culvert to improve the
accuracy of solution. Several trials were performed to refine the grids until there was no
noticeable change in the results. The boundary conditions for the numerical model simulated the
same conditions as the centrifuge tests. The base of the model was fixed in the x and y directions,
while the side boundaries were fixed in the x direction only.
Figure 2. Numerical grid and model components.
Model Parameters. The experimental data were used to calibrate/verify the numerical models.
The box culvert was modeled using linear elastic elements, with mass density of 2700 kg/m3,
elastic modulus of 69 GPa and Poisson’s ratio of 0.33. Two box culvert thicknesses were used:
0.533 m and 0.267 m for the thick and thin culverts. A nonlinear elastic-plastic material using the
Mohr-Coulomb failure criterion with a non-associated flow rule was used to model the dry
Nevada Sand. The initial mass densities obtained during centrifuge tests for the 50% and 90%
relative density were used in the analysis. The sand elastic modulus used was 10 and 30 MPa for
the 50% and 90% relative density cases. The soil Poisson’s ratio, critical state friction angle and
dilation angle were 0.28, 32o and 8o, respectively. To avoid numerical instability, a small value
of cohesion (1 kPa) was used as recommended by Kanungo (2008).
Numerical Model Calibration. To calibrate the numerical model, the sand and culvert
properties stated above were assigned to the respective elements, and the analysis was repeated
numerous times to examine the effect of different soil and interface parameters on the results in
comparison with the centrifuge results of Test 1. The model that achieved best fit with the
experimental results was verified by applying it to all of the other centrifuge tests. The
calibrated/verified model was then employed to conduct parametric studies of the static
performance of box culverts in sand. Two main parameters were considered in the
calibration/verification process. These parameters are the bending moment and soil pressure. As
the results from all of the tests were generally similar, only the results from Test 1 are shown
here.
The calculated bending moments from the numerical model were compared with the
corresponding values evaluated from the strain gauge measurements on the top slab and side wall
during the centrifuge tests. Figure 3(a) compares the calculated bending moments obtained using
the numerical model that provided in the best fit with the measured data, with those obtained
from the centrifuge results for the top slab of Test 1. The agreement between the calculated and
measured bending moments on the top slab was sufficiently accurate.
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Fi
g
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T
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etween
The cal
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r
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ending
calculat
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less than
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The vali
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(2015b)
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ings, but st
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a
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in box cul
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rmine the
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asured dir
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soil press
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r
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amine thei
r
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r (2013) an
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)
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o
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ontact pres
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r
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. The para
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d
Abuhajar
soil above i
t
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e size of th
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i
de range o
f
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ly. The hei
g
0.5 and 0.
6
m (t/Bc =
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the soil c
u
the next se
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ending mo
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ents
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ures.
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Bending Moment. Figure 4 presents the effect of Bc/H ratio on the bending moment diagrams
of the top slab for the thick (t/Bc = 0.12), and thin (t/Bc = 0.06) culverts; Figure 5 shows their
effects on side wall. Even though it is hard to compare bending moments that do not have the
same culvert width, the bending moment diagrams on the top slab and for both culvert
thicknesses show that as the Bc/H ratio increase, the bending moment values increase, which is
consistent at the edge and center. Similar observations were noted in terms of the effect of
thickness on the values of bending moment. As the thickness decrease, the bending moment
values decrease and vice versa.
(a) (b)
Figure 4. Effect of Bc/H ratio on the bending moment on the top slab: (a) (t/Bc = 0.12),
(b) (t/Bc = 0.06).
(a) (b)
Figure 5. Effect of Bc/H ratio on the bending moment on the side wall: (a) (t/Bc = 0.12),
(b) (t/Bc = 0.06).
On the side wall, the effect of thickness is more pronounced. For the thick culvert, all of
the bending moment values are positive, and at the edges most of the bending moment values are
very close. At the center, two groups of bending moment values were observed to give very close
values and these groups are those resulted from Bc/H ratios of 0.1, 0.2, and 0.6 and the Bc/H
ratios of 0.3, 0.4, and 0.5. Different behaviour was observed for the thin culvert, with all of the
bending moment values resulting from the Bc/H ratio of 0.1, 0.2 and 0.3 being positive and
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showing an increase in the bending moment as the Bc/H ratio increases, and being more
pronounced in the center. For the Bc/H ratios of 0.4, 0.5, and 0.6 the bending moment at the top
and bottom corners are all positive, while at the center are all negative. This indicates that for the
Bc/H ratio of 0.1, 0.2 and 0.3 the behaviour of the thick and thin culvert was similar and the
difference appears more as the ratio increases. Despite the similarities and differences between
the thick and thin culvert bending moments in terms of the shape, the bending moment values
observed from thick culvert are still higher than the thin culvert similar to the top slab.
Soil Pressure. Figure 6 presents the effect of Bc/H ratio on the soil pressure diagrams on the top
slab for the thick (t/Bc = 0.12), and thin (t/Bc = 0.06) culverts, while Figure 7 shows their effects
on the side wall. The vertical soil pressure on the top slab for the thick culvert has a parabolic
shape with increases at the edge and decreases at the center; this is more pronounced for the
Bc/H ratio of 0.3 and above. The difference between the vertical soil pressure for the Bc/H ratios
of 0.1, 0.2, and 0.3 are very small and start to appear more for increasing Bc/H ratios. For the
thin culvert, the vertical soil pressure for the Bc/H ratio 0.1 is uniform and that is because of the
small size of the culvert. For the Bc/H ratios greater than 0.1, all of the vertical soil pressure
diagrams have a parabolic shape and it is clear that as the Bc/H increase, the vertical soil pressure
increases at the edge and decreases at the center. Also the difference between the vertical soil
pressures between the Bc/H ratios is very similar. Generally, the parabolic shape of the vertical
soil pressure is a function of the thickness of the culvert, and therefore as the culvert thickness
increases the difference between the vertical soil pressure values at the edges and at the center
decrease and vice versa.
(a) (b)
Figure 6: Effect of Bc/H ratio on the soil pressure on the top slab: (a)(t/Bc = 0.12),
(b) (t/Bc = 0.06)
On the side wall, the horizontal soil pressure from all Bc/H ratios shows increases with
depth, with some peak values close to the top and bottom corners. For the thick culvert, the
horizontal soil pressure increases as the Bc/H ratio increases, while for the thin culvert, this
applies only at the top and bottom corners. At the center of the side wall, the horizontal soil
pressure increases as the Bc/H ratios of 0.1, 0.2, and 0.3 increases, while for higher Bc/H ratios
an undulating shape appears at the center. This causes the horizontal soil pressures obtained from
Bc/H ratios of 0.1, 0.2, and 0.3 to be very close to 0.6, 0.5, and 0.4 respectively. This behaviour
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was observed in the thin culvert, while the thick culvert shows a uniform increase in the
horizontal soil pressure with depth.
(a) (b)
Figure 7. Effect of Bc/H ratio on the soil pressure on the side wall (a) (t/Bc = 0.12),
(b) (t/Bc = 0.06).
Soil Culvert Interaction Factors. Figure 8(a) presents the effect of Bc/H ratio on the soil
culvert interaction factors on the top slab for the thick (t/Bc = 0.12) and thin (t/Bc = 0.06) box
culverts. The results show that Fev values from both thicknesses for the edge at low Bc/H values
(0.1 and 0.2) are less than 1.0 and the Fev values increase after that above 1.0 as the Bc/H
increase. This means that at low Bc/H ratio values, the vertical soil pressure is less than the
theoretical vertical soil pressure, and as the Bc/H ratio increase, the vertical soil pressure is
greater than the theoretical vertical soil pressure. At the center, the results are opposite and all the
Fev values are less than 1.0 and continue to decrease as the Bc/H ratio increase. This indicates
that the vertical soil pressure at the center is less than the theoretical vertical soil pressure and
this reduction increases as the Bc/H ratio increases. The Fev values at the edge from thin culvert
are larger than those from the thick culvert, while at the center, the Fev values for the thin culvert
are less than the thick culvert.
Figure 8(b) illustrates the effect of Bc/H ratio on the soil culvert interaction factors on the
side wall for the thick (t/Bc = 0.12) and thin (t/Bc = 0.06) box culverts under the conditions of at
rest soil pressures. Generally, all of the Feh values for the top and bottom corners are greater than
1.0, which means that all the horizontal soil pressures are larger than the theoretical horizontal
soil pressure “at rest” condition. The Feh values for the thick and thin culverts show an increase
in their values as the Bc/H ratio values increase except in the case of the bottom corner of the
thick culvert where it shows some reduction. The difference between Feh values for the top and
bottom appears as the Bc/H ratio values increase. At the bottom, the Feh values for the thick and
thin culverts show a decrease in the Feh values up to Bc/H ratio of 0.2 and after that the thick
culvert continue to decrease as the Bc/H ratio values increase, while the thin culvert increase as
the Bc/H ratio values increase. This observations show that the thick culvert Feh values less than
Bc/H = 0.3 are higher than the thin culvert and vice versa after that.
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(a) (b)
Figure 8: Effect of the thickness and the ratio Bc/H on the soil culvert interaction factors Fe on
(a) the top slab, (b) the side wall at rest pressure Ko
CONCLUSION
Four centrifuge tests were conducted to examine soil culvert interaction under static loadings.
The results from these tests were used to calibrate and verify numerical model established using
the computer program FLAC 2D. The verified model was used to perform a static parametric
study to investigate the effect of the change in box culvert size on the soil culvert interaction
factors.
Several factors were investigated during the static parametric study. These parameters
include: bending moment, soil pressure, and soil culvert interaction factors. The effect of the box
culvert size is presented in the form of dimensionless numbers Bc/H. The effect of the change in
Bc/H ratios on the soil culvert interaction factors was investigated. The Bc/H ratio results show
an increase at the edges and decrease at the center of the top slab as the Fe values increase. At the
top of the side wall for the thick and thin culverts and also at the bottom, but for the thin culvert
only there are increase in the Fe values as the Bc/H ratio increase, while the thick culvert at the
bottom decrease as the Bc/H increase. It was observed that the Fe values obtained from the Bc/H
ratio increase at the edges of the top slab as the thickness decrease while at the center decrease as
the thickness decrease and similar observation was made for the side wall.
Considering the geometrical culvert configuration and soil properties used in this static
parametric study, several design charts are proposed. These design charts can be used to define
the Fe values at specific extreme points on the box culvert for different cases. These results can
be used to evaluate the actual soil pressure distribution around box culverts under static loads.
Using these soil pressure values, the static bending moment can be easily calculated.
ACKNOWLEDGEMENT
The authors would like to show their great appreciation and thanks for all the staff members of
the Geotechnical Centrifuge Facility at Rensselaer Polytechnic Institute (RPI), Troy, NY, USA,
for providing all the help and support during the experimental part of this research. The authors
also would like to thank NSERC for their financial support.
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of static soil pressures on box culverts.” Can. Geotech. J., 52(11), 1637–1648.
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of box culvert geometric configuration and soil properties”. Computers and Geotechnics,
69: 219–239. doi:10.1016/j.compgeo.2015.05.005.
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