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Simulating tunnel behaviour in interbedded shale and limestone using FDEM

Authors:
  • Geomechanica Inc.

Abstract and Figures

Standard laboratory testing of Georgian Bay shale and limestone was performed for an ongoing tunnel project in the Greater Toronto Area (GTA). The laboratory results were used as calibration targets to determine input parameters for numerical simulations using the Irazu simulation package based on the hybrid finite-discrete element method (FDEM). Using the calibrated input parameters, the deepest alignment of the proposed tunnel was simulated using Irazu and PLAXIS. The Irazu results predicted the liner displacement to be approximately two to three times greater than the PLAXIS simulation results. In the PLAXIS models, the presence of strong and stiff limestone layers restricted the liner deformation because they bear some of the load, but this effect did not appear in the Irazu models. Our results demonstrated that the high stiffness contrast between shale and limestone drastically altered the prescribed stress field as the limestone tends to bear more of the geostatic load.
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For more information on Irazu: http://irazu.geomechanica.com/
Simulating tunnel behaviour in interbedded shale and limestone using
FDEM
Johnson Ha, Giovanni Grasselli & Karl Peterson
Department of Civil Engineering University of Toronto, Toronto, ON, Canada
Bryan S. A. Tatone, Andrea Lisjak & Omid K. Mahabadi
Geomechanica Inc., Toronto, ON, Canada
Giuseppe M. Gaspari
Geodata Engineering SpA, Toronto, ON, Canada
ABSTRACT
Standard laboratory testing of Georgian Bay shale and limestone was performed for an ongoing tunnel project in the
Greater Toronto Area (GTA). The laboratory results were used as calibration targets to determine input parameters for
numerical simulations using the Irazu simulation package based on the hybrid finite-discrete element method (FDEM).
Using the calibrated input parameters, the deepest alignment of the proposed tunnel was simulated using Irazu and
PLAXIS. The Irazu results predicted the liner displacement to be approximately two to three times greater than the PLAXIS
simulation results. In the PLAXIS models, the presence of strong and stiff limestone layers restricted the liner deformation
because they bear some of the load, but this effect did not appear in the Irazu models. Our results demonstrated that the
high stiffness contrast between shale and limestone drastically altered the prescribed stress field as the limestone tends
to bear more of the geostatic load.
1 INTRODUCTION
Tunnel design in interbedded Georgian Bay shale and
limestone must account for numerous mechanisms
including, time-dependent rock deformation, high
horizontal in-situ stresses and shear-induced slipping
along bedding planes. The shale is typically weak (<30
MPa) and behaves as a transversely isotropic medium; in
contrast, the interbedded limestone is relatively strong
(125175 MPa) and behaves isotropically.
To aid in understanding these complex mechanisms,
numerical modelling is required. The hybrid finite-discrete
element method (FDEM) was employed in this study
because of its ability to model realistic deformation and
fracturing of geomaterials (Lisjak, Tatone, et al. 2014,
Lisjak, Figi and Grasselli 2014, Mahabadi, Lisjak and
Munjiza, et al. 2012). FDEM has been applied to model
tunnel behaviour in a clay shale in Switzerland (Lisjak, Figi
and Grasselli 2014, Lisjak, Garitte, et al. 2015, Lisjak,
Grasselli and Vietor 2014). The commercial software,
Irazu, based on the FDEM by Geomechanica Inc. was
used in this work (Mahabadi, Lisjak and He, et al. 2016).
The software contains features permitting realistic
simulation of shale tunnel behaviour including: (i) in-situ
stress, (ii) excavation, (iii) rock support and (iv)
transversely isotropic materials.
As with any modelling software, the quality of the
simulation results rely on the accuracy of the material
parameters used in the model. To calibrate the
geomaterials, standardized rock laboratory tests were
performed which include: (i) Brazilian disc tests, (ii)
uniaxial, and (iii) triaxial compression tests to quantify the
Young’s modulus, Poisson’s ratio, strength envelope and
tensile strength of the material. Once the rock was
characterized, FDEM was calibrated via a trial-and-error
process, thus reproducing the laboratory tests in terms of
strength, deformation and fracture pattern.
The calibrated input parameters of the rock were used
to model a large-scale tunnel in the Greater Toronto Area
(GTA). The influence of different geological conditions on
the failure mechanisms of the tunnel was investigated. In
addition, modelling using the finite-element method (FEM)
was also undertaken using PLAXIS and compared to the
Irazu results.
2 LABORATORY TESTS AND RESULTS
The uniaxial and triaxial compression tests were conducted
at the University of Toronto and Geomechanica/Geodata’s
West Vaughan Project Warehouse, while the Brazilian disc
tests were exclusively performed at the University of
Toronto. All tests performed on Georgian Bay shale were
prepared from core with the bedding plane oriented
perpendicular to the core axis.
2.1 Brazilian Disc Testing
These tests were performed using a 50kN ELE Tritest
electro-mechanical loading frame equipped with a 20 kN
load cell (Figure 1a). In cases where the load cell capacity
was exceeded, a stiff loading frame equipped with a 450
kN load cell was used (Figure 1b).
The indirect tensile strength, σt, was calculated using:
t2P
σ
πDt
, [1]
where P is the peak load, D is the diameter of the sample
and t is the thickness of the sample (Bieniawski and
Hawkes 1978).
Figure 1. Apparatus used to perform Brazilian disc tests:
(a) 50kN ELE Tritest electro-mechanical loading frame and
(b) stiff loading frame.
The distributions of valid Brazilian disc tests for shale
and limestone are shown in Figure 2. The average and
standard deviation for shale and limestone are 2.14 ± 0.76
MPa and 10.07 ± 4.10 MPa, respectively.
Figure 2. Distribution of measured indirect tensile strength
of shale and limestone.
2.2 Uniaxial and Triaxial Compression Tests
The uniaxial and triaxial compression tests were performed
on a stiff loading frame (Figure 3a) and the MTS loading
frame (Figure 3b) at the University of Toronto. Alternatively,
the majority of the tests were performed on
Geomechanica’s loading frame (Figure 3c) at the West
Vaughan Project Warehouse. For triaxial compression
tests, a HQ-sized (61 mm diameter) Hoek cell was used
and the confining pressure (σ3) was applied and
maintained using a manual hydraulic pump.
A summary of the measured shale and limestone
properties are shown in Table 1. The strength envelopes of
tested shale and limestone are shown in Figure 4a and
Figure 4b, respectively. RocLab (v1.031) by Rocscience
was used to fit the equivalent Mohr-Coulomb (MC) failure
criterion using the Marquardt-Levenberg fitting method with
the shale test results (Rocscience n.d.). The equivalent MC
criterion was used because Irazu uses a similar strength
criterion. Since the majority of tests were performed under
6 MPa confinement, the test at 10 MPa confinement was
ignored from the fitting.
A similar attempt to fit the limestone data using RocLab
was undertaken, but was unsuccessful. RocLab has a cap
on the Hoek-Brown (HB) criterion fitting parameters and
thus was unable to properly fit the data. In this case, the
cap was met because the tested range of confining
pressures was too narrow. To properly fit the data using the
HB criterion and subsequently derive the equivalent MC
parameters, the range of confining pressures should be
from zero to half the intact uniaxial compressive strength
(Hoek and Brown 1997). Due to the confining pressures
being on the low end of the recommended range, the HB
fit would be virtually linear and, as a result, the data was
fitted using linear least squares.
Table 1. Measured mean density and elastic properties ±1
standard deviation.
Property
Shale
Limestone
Density (g/cm3)
2.60 ± 0.02
2.63 ± 0.04
Young’s modulus (GPa)
3.80 ± 1.73
29.06 ± 7.66
Poisson’s ratio
0.30 ± 0.07
0.18 ± 0.04
Figure 3. Photos of (a) stiff loading frame, (b) MTS loading
frame and (c) Geomechanica’s loading frame.
Figure 4. Laboratory results, simulation results and MC
strength envelope for (a) Georgian Bay shale and (b)
limestone.
3 MODEL CALIBRATION
In Irazu, the input parameters pertain to microscale
properties, resulting in the emergence of macroscale
properties such as indirect tensile strength and uniaxial
compressive strength. Although FDEM is able to model
realistic deformations and fracturing, generally, laboratory
results cannot be used as direct inputs, and some input
parameters cannot be measured (Tatone and Grasselli
2015). Thus, a calibration procedure was necessary to
accurately determine the micro-mechanical input
parameters to reproduce the macro-behaviour observed in
the laboratory. Using a trial-and-error process, the input
parameters are adjusted until the macro-properties
observed in the model emulate the laboratory results. In a
successful calibration, the set of input parameters will
reproduce the (i) Brazilian disc tests, and (ii) uniaxial
compression tests in terms of stiffness response, strength
and fracture pattern.
The UCS/biaxial simulations were constructed based
on typical sample dimensions of 61 mm diameter and 122
mm length. The Brazilian disc simulations for Georgian Bay
shale are 61 mm diameter, while limestone simulations are
48 mm diameter. It should be noted that the limestone
calibration was preliminary and not yet complete, but an
estimate of the input parameters was used in subsequent
modelling. A nominal mesh element edge length of 1.5 mm
was adopted for all calibration simulations because a
tradeoff was made between element size and accuracy to
reduce overall computation time. Since the main objective
of this work was to model tunnel behaviour, a nominal
mesh element edge length greater than 1.5 mm would be
adopted, thus reducing the need to have fine meshes for
the calibration models. An effective loading rate of 0.10 m/s
was used, which is much greater than laboratory tests, but
it has been found that simulated compressive strengths
and indirect tensile strengths typically converge at effective
loading rates of approximately less than 0.25 m/s (Tatone
and Grasselli 2015, Lisjak, Tatone, et al. 2014).
The shale behaves as a transversely isotropic medium,
thus it is necessary to characterize the shale at different
bedding orientations. Unfortunately, shale samples with
different bedding orientations were not available for testing
and thus cannot be calibrated with FDEM in this work.
Therefore, input parameters of the shale with bedding
planes oriented parallel to the direction of loading were
derived from peer-reviewed literature and experience with
handling the material. The calibrated input parameters are
summarized in Table 2 and the emergent simulation
properties are summarized in Table 3. Typical fracture
patterns from laboratory uniaxial compression tests and
Brazilian disc tests are compared to the simulation results
in Figure 5.
Figure 5. Comparison of typical fracture from uniaxial
compression and Brazilian disc tests between (a)
simulation results and (b) laboratory photos for shale.
Table 2. Calibrated Irazu input parameters for Georgian
Bay shale and limestone.
Parameter
Elastic triangular elements
Density (g/cm3)
Young’s modulus to bedding
(GPa)
Poisson’s ratio to bedding
Time step size (ms)
Crack elements
Tensile strength to bedding
(MPa)
Tensile strength to bedding
(MPa)
Cohesion to bedding (MPa)
Cohesion to bedding (MPa)
Mode I fracture energy to
bedding (J/m2)
Mode I fracture energy to
bedding (J/m2)
Mode II fracture energy to
bedding (J/m2)
Mode II fracture energy to
bedding (J/m2)
Intact friction angle (°)
Normal contact penalty
(GPa·m)
Tangential contact penalty
(GPa/m)
Fracture penalty (GPa)
aEstimated based on experience with the material.
bStrength ratio from point load tests.
cFracture energy ratio from Lisjak, Figi and Grasselli (2014).
d10x Young’s modulus.
Table 3. Emergent properties of Irazu simulation.
Property
Shale
Limestone
Young’s modulus (GPa)
3.56
29.57
Poisson’s ratio
0.31
0.24
Uniaxial compressive strength
(MPa)
20.33
164.9
Triaxial compressive strength
(MPa) (σ3 = 2.5 MPa)
31.97
214.69
Triaxial compressive strength
(MPa) (σ3 = 5.0 MPa)
43.54
260.67
Indirect tensile strength (MPa)
1.99
8.37
4 TUNNEL MODELLING
The tunnel in the GTA is intended to be excavated using
an earth pressure balance machine (EPBM). The deepest
portion of the alignment, approximately 46 m from surface
to the crown of the tunnel (approximately 1.2 MPa), was
modelled assuming a linearly varying vertical stress.
Situated in the GTA, the horizontal to vertical in-situ stress
ratio, k, is typically greater than one (Lo 1978).
The model of a tunnel with an inner diameter of 3 m and
a purely elastic 0.25 m thick M50 concrete liner was
constructed. To achieve complete elasticity in FDEM, the
cohesion and the tensile strength of the liner are set to
extremely high values (100 GPa). The liner has a Young’s
modulus of 35 GPa and a Poisson’s ratio of 0.2.
4.1 Irazu Modelling
Using the calibrated input parameters, the influence of
different geological conditions on tunnel stability was
analyzed. On site, the thickest limestone layer encountered
was 0.33 m with the highest frequency of occurrence being
28% of the total core, resulting in about one limestone layer
for every metre depth of tunnel. Three geological
geometries were considered where the first was an
excavation in shale, the second was an excavation in shale
with a 0.33 m thick layer of limestone at the crown (Figure
6), and the third was an excavation in shale with four 0.33
m thick limestone layers intersecting the tunnel (Figure 7).
A k value of two, a shale bedding orientation of 0° from
the horizontal and a stable time step of 1 x 10-8 seconds
were used. The model was bounded by a 50 m x 50 m box
and an anisotropic strength criterion for shale was adopted.
To simulate strength anisotropy, the crack element
orientation, the strength decreases linearly from
perpendicular () to bedding to parallel () to bedding.
Figure 6. Model geometry with one 0.33 m thick layer of
limestone located at the crown of the tunnel with fixed X
and Y boundary conditions.
4.1.1 Irazu Modelling Sequence and Results
For tunnel modelling, several steps are used in Irazu.
Firstly, in-situ stress was applied using the FEM (i.e. no
fracturing) on the unexcavated model until the total kinetic
energy in the model approached zero. Secondly, the new
nodal coordinates were inputted into the FDEM model. The
excavation core was removed from the simulation using the
modulus reduction method, where the Young’s modulus
was linearly decreased over time before removing the
material. The Young’s modulus of the excavation core was
linearly decreased two magnitudes over 75,000 time steps.
The purpose is twofold: (i) to avoid introducing dynamic
effects into the model and (ii) to simulate the time-
dependent three-dimensional face effect as a result of
preconfinement (Lisjak, Grasselli and Vietor 2014).
However, for the planned construction method of the
tunnel, the ground is unsupported for a very short time and
thus the main reason for the core reduction method is for
the first reason. Thirdly, at the same time the material was
excavated, the stress in the liner was reset and the
stiffness was changed to concrete. The FDEM model
results are shown in Figure 8 to Figure 10.
Figure 7. Model geometry with four 0.33 m thick layers of
limestone intersecting the tunnel with fixed X and Y
boundary conditions.
4.2 PLAXIS Modelling
For PLAXIS modelling, similar geometries were adopted
from the FDEM models, except the vertical extent of the
model is 72 m instead of 50 m to simulate the vertical stress
gradient. As a consequence of how the in-situ stress is
applied to the model, the boundary conditions are also
changed. The top boundary is a free surface, the bottom
boundary is fixed in the X- and Y-directions, and the side
boundaries are fixed in the X-direction. The shale and
limestone are treated as isotropic materials and the
properties of shale to bedding were used (Table 2).
4.2.1 PLAXIS Modelling Sequence and Results
Firstly, the model was run until it equilibrates to the desired
in-situ stress and the boundary conditions are applied.
Secondly, the stress reduction method was used. This
method maintains the stress in the cavity to 10% of the in-
situ stress before activating the liner. Thirdly, the liner was
activated and the stress in the cavity was removed. The
PLAXIS results of the three case studies are shown in
Figure 11 to Figure 13.
5 DISCUSSION
The deformation predicted in the Irazu and PLAXIS models
are quite different. In the Irazu simulations, the predicted
liner deformation is approximately two to three times larger
than the PLAXIS prediction. There is a contrasting
prediction when there are limestone layers present. In
Irazu, the limestone layers does not have an effect on the
liner deformation, whereas in the PLAXIS models, liner
deformation slightly decreases with limestone layers. This
is due to the strong and stiff limestone layers carrying a
portion of the geostatic load. In terms of the overall liner
behaviour, the presence of limestone does not have a
pronounced effect on its stability.
In the Irazu models, fracturing was not observed in the
shale model. However, in the models with limestone layers,
shear fractures were observed at the crown of the
excavation. Due to the high stiffness contrast between the
limestone and shale, the stresses tend to concentrate in
the limestone layers, which was observed in both Irazu and
PLAXIS models. The stress in the limestone layer at the
crown exceeded its strength, resulting in the presence of
fractures, thus forming areas of stress concentration. It
should be noted that the presence of limestone layers in
the Irazu models heavily altered the prescribed in-situ
stress state of the model. The observation could have
significant real-world consequences as it brings to question
the reliability of in-situ stress measurements in interbedded
shale and limestone. It would be difficult to accurately
determine the locked-in stress as the high stiffness contrast
in the rock mass would drastically alter the stress field.
The k is greater than one, thus the liner is being
squeezed horizontally, causing tensile stresses to develop
at the sides and compressive stresses at the crown and
floor of the liner. In terms of liner design, it would be
beneficial to have additional rebar support at the sides of
the tunnel to strengthen the liner in tension. However, since
the principal stress field may not always be oriented in the
horizontal and vertical directions, further design analysis
will be required.
6 CONCLUSIONS
Extensive laboratory testing of Georgian Bay shale and
limestone was undertaken for an ongoing tunnel project in
the GTA. The results were used as calibration targets to
determine the appropriate FDEM input parameters to
accurately reproduce the rock behaviour numerically. Upon
successful calibration, the parameters were used to model
large-scale tunnel behaviour. In comparison, Irazu and
PLAXIS generated contrasting results in terms of liner
displacement and stresses in the rock mass. The liner
displacement in the Irazu simulation, but was slightly lower
in the PLAXIS simulations with increasing number of
limestone layers. The presence of limestone did not have
a pronounced effect on the overall behaviour of the liner. It
was found that the presence of a high stiffness contrast in
the rock mass heavily alters the prescribed in-situ stress
field.
The current work can be extended by conducting
additional laboratory tests and simulating variations of the
same model. To improve the characterization of the
limestone, additional Brazilian disc, uniaxial, and triaxial
compression tests are required, subsequently improving
the limestone FDEM calibration. In a more realistic
simulation, the elastic properties of shale will use a
transversely isotropic model. The properties of the liner can
be changed to reflect its actual strength to predict fracturing
and determine if the design is adequate for the proposed
tunnel. In addition, the joints in the segmental liner and rock
mass can be explicitly included to determine their effect on
liner behaviour and overall tunnel stability. Different in-situ
stress magnitudes, orientations, and geological geometries
can be modelled to determine their effects on the tunnel.
7 ACKNOWLEDGEMENTS
We would like to thank Vincenza Floria, head of the urban
tunnelling sub-sector of Geodata Engineering for providing
the PLAXIS model results. This work has been supported
through NSERC Discovery Grant 341275 held by G.
Grasselli, the Mitacs-Accelerate Internship and QEII
scholarship held by J. Ha.
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Figure 8. Irazu simulation of a tunnel in shale. The (a) liner displacement, (b) stress in the X-direction, and (c) stress in
the Y direction are shown, where negative stresses are compressive.
Figure 9. Irazu simulation of a tunnel in shale and one layer of limestone at the crown. The (a) liner displacement, (b)
stress in the X-direction, and (c) stress in the Y direction are shown, where negative stresses are compressive.
Figure 10. Irazu simulation of a tunnel in shale and multiple intersecting limestone layers. The (a) liner displacement, (b)
stress in the X-direction, and (c) stress in the Y direction are shown, where negative stresses are compressive.
Lisjak, A., B. Garitte, G. Grasselli, H. R. Müller, and T.
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Figure 11. PLAXIS simulation of a tunnel in shale. The (a) liner displacement, (b) stress in the X-direction, and (c) stress
in the Y direction are shown, where negative stresses are compressive.
Figure 12. PLAXIS simulation of a tunnel in shale and one layer of limestone at the crown. The (a) liner displacement,
(b) stress in the X-direction, and (c) stress in the Y direction are shown, where negative stresses are compressive.
Figure 13. PLAXIS simulation of a tunnel in shale and multiple intersecting limestone layers. The (a) liner displacement,
(b) stress in the X-direction, and (c) stress in the Y direction are shown, where negative stresses are compressive.
Rocscience. n.d. "RocLab 1.0."
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... Lisjak et al. [17], Tatone et al. [25] and Mahabadi [25], μ (kg/(m·s)) 15 [19] d ≤ 54 [20] N/A -10 −6 Lisjak [17] φ17.5 × 35 φ30 0.3 0.1 0.1 Lisjak [16] φ17.4 × 35.8 φ29.7 0.3 0.1 0.1 Lisjak [40] φ36 × 72 φ50.8 × 25.4 1.5 0.1 0.1 Lisjak [11] and Zhao [41] φ50 × 100 -0.7 0.1 0.1 Lisjak [42] and Zhao [43] φ54 × 108 -0.8 -0. 25 Farsi [22] -φ18.56, φ17.69 0.15, 0.1 -0.01 Euser [44] φ60 × 120 φ25. 4 1.0 -0.1 Tatone [25] φ36 × 72 φ50.8 0.75 0.1 0.1 Tatone [7] φ54 × 108 -2 -0.1 Tatone [39] φ50 × 100 φ50 2 0.001 0.004 Ha [45] φ61 × 122 φ61 1.5 -0.1 Knight [46] φ60 × 120 ----Guo [47] φ50 × 100 ---0. et al. [13,24,27,28] only studied the loading rate sensitivity of individual specimen size. ...
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The mechanical response of a test tunnel in Opalinus Clay was characterized.•A strong directionality in the deformational behaviour was measured.•Numerical simulations were used to analyze the evolution of rock mass damage.•The importance of bedding plane slippage and extensional fracturing was shown.
Article
The Opalinus Clay (OPA) is an argillaceous rock formation selected to host a deep geologic repository for high-level nuclear waste in Switzerland. It has been shown that the excavation damaged zone (EDZ) in this formation is heavily affected by the anisotropic mechanical response of the material related to the presence of bedding planes. In this context, the purpose of this study is twofold: (i) to illustrate the new developments that have been introduced into the combined finite-discrete element method (FEM/DEM) to model layered materials and (ii) to demonstrate the effectiveness of this new modelling approach in simulating the short-term mechanical response of OPA at the laboratory-scale. A transversely isotropic elastic constitutive law is implemented to account for the anisotropic elastic modulus, while a procedure to incorporate a distribution of preferentially oriented defects is devised to capture the anisotropic strength. Laboratory results of indirect tensile tests and uniaxial compression tests are used to calibrate the numerical model. Emergent strength and deformation properties, together with the simulated damage mechanisms, are shown to be in strong agreement with experimental observations. Subsequently, the calibrated model is validated by investigating the effect of confinement and the influence of the loading angle with respect to the specimen anisotropy. Simulated fracture patterns are discussed in the context of the theory of brittle rock failure and analyzed with reference to the EDZ formation mechanisms observed at the Mont Terri Underground Research Laboratory.
Article
Some structures built in different rock formations in Southern Ontario have been subjected to various degrees of distress. These case histories include heaves of quarry bottoms, buckling of concrete lining of canal floors, cracking of concrete lining of tunnels at the springline, and long term movement of the walls of unsupported excavations.Inference from these case histories, together with direct measurements of in situ stresses, indicate that high horizontal stresses exist in the Silurian and Ordovician rocks. The magnitude of the maximum stress in the horizontal plane varies from 6–14 MPa depending on the depth and rock type.Excavations in rock relieve the in situ stresses. The stress relief serves as an initiating mechanism for time-dependent deformation to occur leading to the process loosely termed as 'rock squeeze'. It appears, therefore, that due consideration must be given to this prevalent phenomenon for the design of underground structures in rock in this region.
Article
The Hoek-Brown failure criterion was originally developed for estimating the strengths of hard rock masses. Because of the lack of suitable alternatives, the criterion has been applied to a variety of rock masses including very poor quality rocks, which could almost be classed as engineering soils. These applications have necessitated changes to the original criterion. One of the principal problems has been the determination of equivalent cohesive strengths and friction angles to meet the demands of software written in terms of the Mohr-Coulomb failure criterion. This paper summarises the interpretation of the Hoek-Brown failure criterion which has been found to work best in dealing with practical engineering problems.