Natural Hazards
https://doi.org/10.1007/s11069-020-04090-w
ORIGINAL PAPER
Seismic settlement of a strip foundation resting on a dry
sand
Saif Alzabeebee1
Received: 10 January 2019 / Accepted: 25 May 2020
© Springer Nature B.V. 2020
Abstract
Seismic settlement of shallow foundations constructed in seismic active areas should be
considered for a reasonable estimation of the total settlement. However, the trend of the
seismic settlement of shallow foundation constructed on a sandy soil is not clearly understood and it is estimated by designer using simple analytical methods. These methods do
not consider the effect of the soil–structure interaction. This research, therefore, reports the
results of 105 robust finite element models developed to investigate the seismic settlement
of a shallow foundation constructed on a dry sand. The influence of the load applied on the
foundation, relative density of sand, foundation embedment, peak ground acceleration of
the earthquake shake, thickness of the sandy soil, and the dominant frequency of the earthquake shake have been examined to provide a comprehensive understanding of the parameters influencing the seismic settlement. The results of the analyses showed that increasing
the load applied on the foundation or the peak ground acceleration remarkably increases
the seismic settlement, while increasing the embedment depth remarkably reduces the seismic settlement. In addition, the relationship between the thickness of the sandy layer and
the seismic settlement is found to be very complex and noticeably influenced by the relative density of the sand. More importantly, it was found that the seismic settlement dramatically increases when the dominant frequency of the earthquake approaches the natural
frequency of the system. Thus, all these parameters are important and should be considered by designers for a reasonable estimation of the seismic settlement. The conclusions
drawn from this paper will aid the development of a good analytical method in future, and
the results reported in this paper also provide useful and novel database to designers and
practitioners.
Keywords Finite element analysis · Earthquake · Strip foundation · Foundation
embedment · Seismic settlement
* Saif Alzabeebee
Saif.Alzabeebee@gmail.com
1
Department of Roads and Transport Engineering, College of Engineering, University of AlQadisiyah, Al‑Qadisiyah, Iraq
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1 Introduction
Foundation constructed in seismic active areas is prone to additional settlement induced by
earthquake shake; this type of settlement is called the seismic settlement. This seismic settlement should be robustly calculated to enable a reasonable estimation of the total settlement in the routine design of shallow foundations. However, the available analytical methods to calculate the seismic settlement of dry sand (Tokimatsu and Seed 1987 method and
Pradel 1998 method) do not consider the effect of the load applied on the foundation. Thus,
these methods cannot be used to understand or calculate the seismic settlement of foundation constructed on a dry sandy soil. In addition, careful examination of the literature has
shown little attention been paid in previous studies to understand or calculate the seismic
settlement of strip foundations resting on sand; basic information on these previous studies
is shown in Table 1.
Kholdebarin et al. (2008, 2016) investigated the seismic bearing capacity and normal
stress developed under a strip foundation constructed on a clayey–sand soil improved by
cement using the two-dimensional finite difference method (FDM). The results showed
that adding 2%, 4%, and 6% of cement increased the seismic bearing capacity by 270%,
140%, and 176%, respectively. Ueng et al. (2010) studied the effect of earthquake characteristics on the liquefaction potential of sand. However, no live load has been considered in
Ueng et al. (2010) research. Ghosh (2011) investigated the interference effect on the seismic response of two square foundations resting on a layered mixed soil. The study mainly
focused on the effect of the spacing ratio between the two foundations. The study showed
that the seismic settlement of the two nearby square foundations is higher than the seismic
settlement of a single square foundation. Vivek (2011) studied the effect of the spacing
ratio between two closely placed strip foundations on the induced seismic settlement. Karamitros et al. (2013a, b, c) studied the seismic settlement, seismic liquefaction, and bearing
capacity degradation of a strip foundation constructed on a clay layer underline by a liquefiable sand. All these studies noticed an improvement in the seismic performance of the
foundation due to the presence of the clay layer. Azzam (2015) investigated the influence
of skirts on the seismic displacement of a foundation constructed on a slope and noticed
that the presence of skirts reduces the seismic displacement. Ghayoomi and Dasht (2015)
examined the effect of the earthquake characteristics on the response of a multi-story
building constructed on a medium–dense dry sand using small scale shaking table tests.
Mansour et al. (2016) studied the effect of the earthquake shake on the bearing capacity of
a shallow foundation resting on a saturated sand. The study focused on the influence of the
relative density of the sand, peak ground acceleration, width of the strip foundation, and
embedment depth. Nguyen et al. (2016) examined the influence of foundation size on the
seismic response of a reinforced concrete (RC) multi-story building resting on soft clayey
soil with an undrained shear strength of 50 kPa.
Ahmadi et al. (2017) tested the efficiency of soil densification, cement grouting, and
drainage in reducing the seismic settlement of a square foundation resting on dry and saturated sands using shaking table experiments. Ahmadi et al. (2017) noticed that all the
aforementioned soil improvement techniques reduce the seismic settlement; however, the
cement grouting was noted to be the most influential soil improvement technique. Dimitriadi et al. (2017) examined the effect of placing a non-liquefiable permeable layer above
a liquefiable sand on the seismic response of a surface strip foundation. Dimitriadi et al.
(2018) extended her wok published in 2017 by examining the effect of the dimensions of
the non-liquefiable permeable layer on the seismic response of the surface strip foundation.
13
Ghosh (2011)
Vivek (2011)
Karamitros et al. (2013a)
Karamitros et al. (2013b)
Karamitros et al. (2013c)
Azzam (2015)
Ghayoomi and Dashti
(2015)
Kholdebarin et al. (2016)
Mansour et al. (2016)
Nguyen et al. (2016)
Ahmadi et al. (2017)
Dimitriadi et al. (2017)
Dimitriadi et al. (2018)
Forcellini (2018)
Zhang and Chen (2018)
Chaloulos et al. (2019)
Far (2019)
Fatahi et al. (2019)
Forcellini (2019)
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
2D FDM
3D FEM
3D FEM
3D FEM
Experimental study using the shaking
table tests
2D FDM
2D FDM
3D FEM
Experimental study using shaking table
tests
2D FEM
Experimental study using shaking table
tests
3D FDM
2D FEM
2D FDM
2D FDM
2D FDM
2D FDM
Experimental study using centrifuge
tests
2D FDM
2D FDM
Kholdebarin et al. (2008) 2D FDM
Ueng et al. (2010)
1
Study type
2
Nos. References
PGA (g)
Type of soil
N/A
2.50
N/A
N/A
N/A
2.80
1.00–4.25
1.00–8.00
N/A
0.49
0.05–0.36
0.05–0.36
0.10–0.35
0.23
0.31–0.77
Layered mixed soil
Layered mixed soil
Loose and medium–dense sands
Medium–dense sand
Loose and medium–dense sands
Loose, medium and dense sands
Medium–dense sand
0.01–0.20 Loose, medium and dense sands
1.25–20.00 0.13–0.29 Clayey sand
fp (Hz)
Equivalent LE
Hs small
VMMSKP
PM4Sand
NTUA Sand
NTUA Sand
VMMSKP
N/A
N/A
0.73
5.00
N/A
N/A
N/A
N/A
10.00
Loose, medium and medium–dense sands
Loose and medium–dense sands
Soft, medium and stiff clays
Medium and dense sands
0.06–0.17 Layered sandy soil deposit with a relative
density range of 35% to 75%
0.23–0.84 Soft clay
0.84
Clay
0.20
Layered soil profile of sand, silty sand
and sandy sil
0.10–0.35
0.15–0.30
0.35–0.89
0.30
MC
1.00–20.00 0.05–0.90 Clayey sand
Nonlinear dynamic soil model N/A
0.10–0.20 Loose, medium–dense, dense and very
dense sands
Equivalent LE*
N/A
0.23–0.84 Soft clay
N/A
0.10–4.00 0.24
Loose, medium and dense sands
MC
MC
NTUA Sand and MC
NTUA Sand and MC
NTUA Sand and MC
LE
N/A
N/A
MC
Soil model
Table 1 Details of the previous studies on the response of soil or shallow foundation subjected to earthquake shake
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Study type
3D FEM
22
13
MC
Soil model
N/A
fp (Hz)
PGA (g)
0.16
Type of soil
Medium–dense sand
*This model is linear elastic, but the hysteretic damping and shear modulus degradation is considered
fp dominant frequency, PGA peak ground acceleration, MC elastic perfectly plastic model with Mohr–Coulomb failure criteria, LE linear elastic model, FDM finite difference
method, FEM finite element method, 3D three-dimensional analysis, 2D two-dimensional analysis, N/A not applicable, VMMSKP Von Mises multi-surface kinematic plasticity model
Kumar et al. (2019)
Nos. References
Table 1 (continued)
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Zhang and Chen (2018) conduct shaking table experiments to examine the benefit of using
soil mix wall and layer of non-liquefiable layer underneath the foundation in reducing the
seismic settlement of strip foundation subjected to centric and eccentric load and resting
on a saturated sand. Fatahi et al. (2019) examined the seismic response of a multi-story
building constructed near a slope of a dry clayey soil. Chaloulos et al. (2019) investigated
the seismic settlement of a building constructed on a liquefiable saturated sand. Far (2019)
used two-dimensional finite difference method to study the seismic response of a steel
frame resting on soft clayey soil. The study focused on the influence of the number of stories and base flexibility on the seismic response of the steel frame. Forcellini (2018) studied the efficiency of base isolation on the seismic response of RC multi-story building resting on clay. Forcellini (2019) examined the seismic response of RC multi-story building
resting on a liquefiable saturated sand. Kumar et al. (2019) examined the seismic response
of a shallow square foundation resting on saturated sand. The study focused on the effect of
the seismic shake on the acceleration induced below the foundation, the settlement of the
foundation, and the excess pore water pressure developed below the foundation.
Based on this review, it is evident that very little attention has been paid in previous
studies to the seismic settlement of foundation constructed on a dry sand, although the
seismic settlement is very important from a practical point of view. In addition, these
previous studies paid little efforts to the effect of many parameters, which are likely to
impact the developed seismic settlement. These parameters are the relative density of the
sandy soil, the surface load carried by the foundation, the foundation embedment, the peak
ground acceleration (PGA) of the earthquake shake, the thickness of the sandy layer, and
the dominant frequency of the earthquake shake. Hence, it appears that the seismic settlement of shallow foundations is poorly understood and gaps in knowledge are exist. Thus, it
is required to identify the parameters that influence the seismic settlement to aid the development of a robust analytical method to calculate it. Thus, the study objectives are:
1. Using a robust soil model to study the seismic settlement of a strip foundation accurately.
2. Providing a comprehensive understanding of the sensitivity of the seismic settlement to
the sand relative density, the load carried by the foundation, the foundation embedment,
the PGA of the earthquake shake, the thickness of the sandy layer, and the dominant
frequency of the earthquake shake.
The importance of this paper is that it considers the parameters which have been poorly
studied in previous studies and shows the influence of these parameters on the seismic settlement of shallow foundations. This has been done as an attempt to provide an enhanced
understating of the seismic settlement to aid future studies on the development of an equation that capable of robustly estimating the seismic settlement.
2 Statement of the problem of the study
A strip foundation with a width of 1.0 m and resting on a dry sandy soil has been considered to address the objectives of the study. Three relative densities (RD) have been considered: these relative densities are 50%, 80%, and 94%, respectively. This wide range of
relative densities have been simulated to provide a comprehensive understanding of the
seismic settlement under all the expected scenarios. The foundation has been assumed to
be subjected to an earthquake shake, and the problem has been analysed using the time
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Fig. 1 Acceleration records of
earthquake shakes used in this
study
Acceleration (g)
history finite element analysis. The 1990 Upland earthquake record has been considered as
the reference earthquake record in this research. This earthquake has been used to investigate the sensitivity of the seismic settlement to the surface load applied on the foundation,
foundation embedment, peak ground acceleration (PGA) of the earthquake, and soil layer
thickness. Figure 1a shows the acceleration–time relationship of this earthquake record and
Fig. 2a shows the Fourier transformation of the acceleration–time relationship. Figure 1a
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
Upland 1990
0
10
Time (Sec)
20
30
(a) Acceleration record of the Upland 1990 earthquake
Acceleration (g)
(Brinkgreve 2006)
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
Northridge 1994
0
10
Time (Sec)
20
30
(b)
Acceleration (g)
Scaled Acceleration record of the Northridge 1994
earthquake (Xu and Fatahi 2019)
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
Loma Prieta 1989
0
10
Time (Sec)
20
(c) Scaled acceleration record of the Loma Prieta 1989
earthquake (Bakr 2018)
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30
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0.3
Acceleration (g)
Fig. 2 Fourier transformation
analysis of earthquake shakes
used in this study (Alzabeebee
2019a)
0.25
Upland 1990
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
14
16
18
20
Frequency (Hz)
(a) Fourier transformation of the Upland
1990 earthquake record
Acceleration (g)
0.3
0.25
Northridge 1994
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
14
16
18
20
Frequency (Hz)
(b) Fourier transformation of the Northridge
1994 earthquake record
Acceleration (g)
0.3
0.25
Loma Prieta 1989
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
14
16
18
20
Frequency (Hz)
(c) Fourier transformation of the Loma Prieta
1989 earthquake record
shows that this earthquake has an absolute peak ground acceleration (PGA) of 0.24 g. Figure 2a shows that the dominant frequency of this earthquake record is 2.90 Hz.
Furthermore, two more records have also been utilized to study the effect of the dominant frequency on the seismic settlement of shallow foundations; these records are the
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Northridge (1994) earthquake and the Loma Prieta (1989) earthquake. The time–acceleration records of these earthquakes have been taken from the literature (Bakr 2018; Xu and
Fatahi 2019); however, it is important to state that the PGA of these earthquakes has been
scaled to a value of 0.24 g to allow a direct comparisons with the results of the Upland
earthquake. This approach has also been considered in many studies in the literature (Abuhajar et al. 2015a, b; Alzabeebee 2019a, b). In addition, a Fourier transformation analysis
has been conducted in this research to find the dominant frequency for these earthquakes.
Figure 1b, c shows the acceleration–time relationship of these earthquakes, and Fig. 2b, c
presents the results of the Fourier transformation analysis of these records. It can be seen
from Fig. 2b, c that the dominant frequency of the Northridge and Loma Prieta earthquakes
is 0.93 Hz and 0.66 Hz, respectively.
3 Soil model
The hardening small strain stiffness model (Hs small) has been utilized to simulate the
behaviour of the soil. This model has been considered as it can accurately model the seismic soil–structure interaction problems as has been demonstrate in many studies in the
literature (Al-Defae et al. 2013; Amorosi et al. 2014; Knappett et al. 2015; Liang et al.
2015; Fabozzi and Bilotta 2016; Amorosi et al. 2017; Bakr and Ahmad 2018; Liang et al.
2019). This model captures the stiffness degradation at a very low strain level, effect of
stress level on soil stiffness, stiffness degradation, and volume change (Kampas et al. 2019,
2020; Bakr and Ahmed 2018; Bakr et al. 2019). The model has relatively large number
ref
), the oedometer
of input parameters (11 parameters). The triaxial reference stiffness ( E50
ref
ref
reference stiffness ( Eoed ), and the unloading reference stiffness ( Eur ) describe the soil stiffness at loading and reloading. The Mohr–Coulomb shear strength parameters (the angle of
internal friction (ϕ′) and the cohesion of the soil (c′)) are used in this model to define the
failure conditions, and the volume change is defined by the dilatancy angle (𝜓 ′). 𝜐u is used
to define the unloading and reloading Poisson’s ratio. Furthermore, this model considers
the influence of the stress state on the soil stiffness by using the parameter m (Schanz et al.
1999) and the parameter Rf specifies the strain level at failure. Also, the lateral earth pressure is calculated using the lateral earth pressure coefficient (parameter Konc). The shear
). Pref is the refermodulus at small strain level is considered using the parameter (Gref
o
ence soil stress. Importantly, the model considers the effect of shear modulus degradation,
which is a very important feature for correct modelling of seismic soil–structure interaction
problems, using the parameter 𝛾0.7. 𝛾0.7 is the shear strain level at which the secant shear
modulus (Gs) becomes equal to 70% of the initial shear modulus (Go). It should be noted
that the Hs small model uses the famous Hardin and Drnevich (1972) equation to simulate the degradation of the shear modulus with the increase of the strain level as shown in
Eq. 1. The accuracy of this equation in the modelling stiffness–strain relationship has been
demonstrated by many studies in the literature (e.g. Tsinidis et al. 2016). Also, the model
implicitly considers the hysteretic damping and this hysteretic damping can be expressed
mathematically using Eq. 2 (Brinkgreve et al. 2007).
Gs
1
=
Go
1 + 0.35 𝛾𝜀
0.7
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(1)
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𝜉=
ED =
ED
4𝜋Es
(2)
�
�
2𝛾0.7
4Go 𝛾0.7 ⎛⎜
𝛾c
a𝛾c ⎞⎟
−
ln
2𝛾c −
1
+
𝛾
a ⎜
a
𝛾0.7 ⎟
1 + a𝛾0.7
⎝
⎠
c
(3)
1
G 𝜀2
2 s 0.7
(4)
Es =
where 𝜀 is the shear strain, 𝜉 is the hysteretic damping ratio, 𝛾c is the shear strain developed
due to cyclic load, and Es is the cyclic maximum strain energy.
More information on this model can be found in the doctoral thesis of Benz (2007) and
in Benz et al. (2009), who developed this model based on the original hardening soil model
developed by Schanz et al. (1999).
4 Concrete model
The concrete material of the foundation has been simulated using the viscos–linear elastic
model. The model requires four input parameters; these parameters are: the modulus of elasticity of the concrete, the Poisson’s ratio of the concrete, and the viscous damping parameters
𝛼 and 𝛽.
5 Analysis parameters
The material properties of the soils have been taken from the database of the sand properties
published and validated by Brinkgreve et al. (2010). In addition, the shear wave velocity (Vs),
compression wave velocity (VP), and shortest wavelength (LR) have been determined using
Eqs. 5, 6, and 7, respectively (Brinkgreve et al. 2006; Saikia 2014). The initial modulus of
elasticity of the soil (Eo) has been calculated using Eq. 8. However, it should be mentioned
that the average value of the initial modulus of elasticity of the soil (Eo) has been utilized to
calculate the shear and compression wave velocities. This average Eo value has been determined by calculating the average of the maximum and minimum initial shear modulus (Go)
values utilizing Eq. 9. In addition, the first and second natural frequencies ( f1 and f2) have
been calculated using Eq. 10 (Roesset 1977; Gazetas 1982; Vivek 2011).
Tables 2 and 3 present the Hs small material properties and the ground motion properties
for the considered soils, respectively. In addition, Fig. 3a–c presents the stiffness degradation
and damping ratio curves of the soils used in the finite element analyses.
The E and 𝜐 of the concrete have been taken equal to 24,000,000 kPa and 0.15, respectively.
In addition, the damping ratio of the concrete has been considered equal to 3.0% (Bakr 2018).
√
Eo
Vs =
(
)
(5)
2 1+ur 𝜌
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Table 2 Parameters of the
soils used in the finite element
analyses (Brinkgreve et al. 2010)
Parameter
RD 50%
RD 80%
RD 94%
𝛾 (kN/m3)
17.0
18.2
18.76
ref
(kPa)
E50
30,000
48,000
5600
ref
Eoed
(kPa)
ref
Eur
(kPa)
Gref
o . (kPa)
m
𝛾0.7
𝜐ur
𝜙′
c
𝜓′
Rf
30,000
48,000
56,400
90,000
144,000
169,200
94,000
114,000
12,320
0.544
1.5E − 4
0.2
34.3
0.1
4.3
0.938
0.450
1.2E − 4
0.2
38.0
0.1
8.0
0.900
0.406
1.06E − 4
0.2
39.75
0.1
9.75
0.883
𝛾 is the soil unit weight
Table 3 Ground motion
parameters
Parameter
RD 50%
RD 80%
RD 94%
Vs (m/s)
223.6
224.2
224.9
VP (m/s)
LR (m)*
f1 (Hz)
f2 (Hz)
365.1
11.18
1.39
4.19
366.2
11.21
1.40
4.20
367.3
11.25
1.41
4.21
*Determined based on a frequency of 20 Hz, which is the highest frequency based on the Fourier transformation analysis (Fig. 2)
√
)
(
√
√
1 − 𝜐ur Eo
√
VP =
(
)(
)
1 + 𝜐ur 1 − 2𝜐ur 𝜌
LR =
Vs
highest frequency
(7)
)
(
Eo = 2 1 + 𝜐ur Go
( )
Go 𝜎3� = Gref
o
fn =
(
c� cot 𝜙� + 𝜎3�
c� cot 𝜙� + Pref
Vs
(2n − 1)
4H
(6)
(8)
)m
(9)
(10)
where 𝜌 is the density of the soil; 𝜎3′ is the lateral soil stress; and H is the thickness of the
soil layer.
13
G/Go
Fig. 3 Stiffness degradation and
damping ratio curves for: a RD
50%; b RD 80%; and c RD 94%
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0001
Gs/Go
Gt/Go
Damping
0.001
γ (%)
0.01
0.1
10
9
8
7
6
5
4
3
2
1
0
Damping ratio (%)
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7
6
5
4
3
2
Gs/Go
Gt/Go
Damping
0.001
γ (%)
0.01
Damping ratio (%)
G/Go
(a)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0001
1
0.1
0
7
6
5
4
3
2
Gs/Go
Gt/Go
Damping
0.001
γ (%)
0.01
Damping ratio (%)
G/Go
(b)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0001
1
0.1
0
(c)
6 Finite element modelling
A finite element model has been developed using PLAXIS 2D (Brinkgreve et al. 2006) to
model the problem of this research. The two-dimensional plane strain assumption has been
used. Figure 4 shows the developed model. The width and the height of the model are considered equal to 80 m and 40 m, respectively. The width of the model has been determined
after a sensitivity analysis with different widths to determine the width beyond which there
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Fig. 4 The finite element mesh of the problem
Fig. 5 Effect of model width on
the obtained seismic settlement
13
Seismic settlement (mm)
is no impact of the model extend on the results of the finite element analysis as shown in
Fig. 5 as an example. Figure 5 is for the case of a foundation resting on a sand with RD of
50% and subjected to a surface load of 25 kPa. It is also useful to mention that the model
width considered in this study is also larger than those considered in previous numerical
studies of the seismic response of foundations (Karamitros et al. 2013a, b, c; Kholdebarin
et al. 2008; Azzam 2015; Mansour et al. 2016). The bed rock has been assumed to be 40 m
beneath the natural ground level, and this is why the depth of the model is considered equal
to 40 m; similar approach has also been utilized in previous studies (Zhang and Liu 2018,
2020; Alzabeebee 2019b; Meena and Nimbalkar 2019; Alzabeebee 2020a, b).
The soil and the foundation have been modelled using 15-node triangular elements.
Interface elements have been used in the contact area between the soil and the foundation; the coefficient of reduction at the interface interaction was considered equal to 0.7
(Tsinidis 2018). As shown in Fig. 4, the model has been divided into two zones; the
zone near the foundation, with a length of 20 m and a width of 10 m, has been discretized with very fine elements (the average size of the element was 0.25 m), while the
zone far from the foundation has been meshed using relatively coarser mesh having an
average size of the element of 0.8 m. These sizes produced a total number of elements
of 2568 element for the whole model. This technique has been done to reduce the computational time by using an overall coarse mesh, but also keeping the quality of the
results very high by using very fine mesh near the zone of interest (i.e. near the foundation). However, the average size of the element of both zones is less than one-fifth of the
0
Model width = 10
Model width = 20
Model width = 30
Model width = 40
Model width = 80
-5
-10
m
m
m
m
m
-15
-20
-25
0
2
4
6
Time (s)
8
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shortest wavelength to account for the conditions of wave propagation (Kramer 1996;
Bakr and Ahmad 2018; Bakr et al. 2019). It should also be noted that this mesh configuration has been selected after a sensitivity analysis that has been undertaken to find the
mesh beyond which there is no influence of the mesh size (i.e. number of elements) on
the results. Figure 6 shows an example of the effect of the number of elements on the
settlement response of the foundation, which is for the case of a foundation resting on
sand with relative density of 50% and subjected to a surface load of 25 kPa. It is clear
from the figure that the effect of mesh size is marginal. In addition, it is also obvious
that increasing the number of elements beyond 2568 does not influence the obtained
settlement, and therefore, the model of this study has been built with total number of
elements of 2568 element.
The sides of the model have been prevented from movement in the horizontal direction, while the rock layer at the base of the model has been modelled by restraining the
movement in both the horizontal and vertical directions. The earthquake effect has been
simulated by adding a prescribed acceleration at the bottom of the model. The accelerogram records shown in Fig. 1a–c have been fed to the bottom of the model using
the prescribed acceleration technique. This modelling technique has been proposed by
PLAXIS, and the robustness of this methodology has been demonstrated by many studies in the literature against centrifuge and field results (e.g. Al-Defae et al. 2013; Knappett et al. 2015; Liang et al. 2015; Fabozzi and Bilotta 2016; Bakr and Ahmad 2018;
Nasiri et al. 2020). The effect of waves reflection, which is a phenomenon happens in the
dynamic analysis, has been eliminated by using absorbent dampers as proposed by Lysmer and Kuhlemeyer (1969) (Alzabeebee 2014; Fattah et al. 2014, 2015a, b; Alzabeebee
2017; Ghosh and Kumar 2017; Singh et al. 2017; Alzabeebee et al. 2018a; Azzam et al.
2018; Kumar and Ghosh 2020). These absorbent boundaries have been used at the sides
of the numerical model during the time history earthquake analysis. In addition, the
stability of the analysis is further ensured by using numerical damping (Forcellini 2018)
The analysis has been conducted in three steps to allow a realistic modelling of the
problem:
Seismic settlement (mm)
Step 1: in this step, the initial stresses of the soil mass have been calculated. The at-rest
earth pressure coefficient has been calculated using Jacky’s equation (Brinkgreve et al.
2006).
0
Number of elements = 1143
Number of elements = 1507
Number of elements = 2223
Number of elements = 2568
Number of elements = 6505
-5
-10
-15
-20
-25
0
2
4
6
Time (s)
8
10
Fig. 6 Effect of number of elements on the obtained seismic settlement
13
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Step 2: in this step, the structure load has been modelled by applying a surface load on
top of the strip foundation and conducting a static analysis. The load has been applied in
steps in this static analysis.
Step 3: the effect of the earthquake has been modelled in this step by conducting a time
history analysis (i.e. dynamic analysis). A total time of 10 s has been considered in the
time history analysis. It is very important to note that the analysis time has been limited to 10 s because it has been noticed in the early stages of the analysis trials that the
seismic settlement stabilizes after approximately 5 s for the Upland earthquake, 9 s for
the Northridge earthquake, and 7.5 s for the Loma Prieta earthquake. Hence, it is not
necessary to consider the whole time of the earthquake shake as it adds significant computational time and therefore increases the cost of the analysis. The dynamic time step
(Δt ) used in the analysis was equal to 0.001 s to ensure a robust and accurate modelling.
This time step has been determined using Eq. 11, to meet the requirement of the wave
propagation in the dynamic analysis (Majumder et al. 2017; Alzabeebee 2017; Alzabeebee et al. 2018a).
Δt ≤
Smallest average element size
Velocity of smallest propagating wave
(11)
It is worthy to state that the PLAXIS dynamic module has been verified by Azzam
et al. (2018). Also, the Hs small model in PLAXIS 2D has been intensively verified by
many studies in the literature (Al-Defae et al. 2013; Knappett et al. 2015; Liang et al.
2015; Fabozzi and Bilotta 2016; Bakr and Ahmad 2018; Liang et al. 2019). In addition, the
methodology of the current study follows these previous methodologies exactly. Thus, the
model developed in this paper is robust as the model has been developed considering the
followings:
1. Effect of model boundaries.
2. Effect of mesh size.
3. Using a numerical model capable of capturing the behaviour of the soil when subjected
to seismic loads.
4. Using a verified methodology based on previous studies in the literature.
Therefore, and based on the aforementioned discussion, the author is confident that the
model produces robust results that can help to advance the knowledge on the subject. It is
also worthy to note that the same methodology has been used by the author to study the
seismic response of buried rigid and flexible pipes (refer to Alzabeebee 2019a, b for further
details).
7 Static analysis
A static analysis has been conducted in initial stage of this study to find the static ultimate
and allowable bearing capacities of the foundation and to validate the developed numerical model. The static allowable bearing capacity is very important because it will be used
in the subsequent seismic analyses to understand the relationship between the surface load
applied on the foundation and the additional settlement due to the effect of the earthquake
shake. The FEM model developed in the previous section has been used to find the ultimate
13
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bearing capacity by loading the foundation up to the failure. The Hs small model has been
used with the materials properties shown in Table 2. Results of the applied stress–settlement behaviour for the three considered relative densities are reported in Fig. 7; the influence of the soil nonlinearity and soil relative density is very obvious in the results of this
figure. Furthermore, the ultimate bearing capacity of the foundation is equal to 250 kPa,
500 kPa, and 865 kPa, for RD of 50%, 80%, and 94%, respectively. Hence and by considering a factor of safety of 2.5, the allowable bearing capacity is equal to 100 kPa, 200 kPa,
and 346, kPa for a relative density of 50%, 80%, and 94%, respectively.
The obtained numerical bearing capacity values have been compared with the computed
bearing capacity using the Terzaghi (1943) closed from solution (Bowles 1996). Terzaghi,
Meyerhof, and Hansen bearing capacity factors have been used in the calculation of the
analytical bearing capacity values. Table 4 compares the numerical and analytical ultimate bearing capacity values and reports the percentage difference between the numerical
and analytical values. The table clearly shows that the analytical solutions overestimate
the bearing capacity with an overestimation percentage ranging from 2 to 29%. Only the
Hansen bearing capacity factors underestimate the bearing capacity for an angle of internal
friction of 40°. It is worthy to state that Griffiths (1982) also noted an overestimation of the
bearing capacity calculated using the closed from solutions compared to the finite element
analysis and justified this overestimation by the nonlinearity of the bearing capacity factor
N𝛾 (Griffiths 1982). It is also necessary to mention that Chavda and Dodagoudar (2018)
also noted that Terzaghi equation and Terzaghi bearing capacity factors overestimate the
bearing capacity compared to the finite element analysis. Chavda and Dodagoudar (2018)
noted that the overestimation percentage ranged between 7 and 20%. In conclusions, the
percentages stated in Table 4 and the agreement with previous observations in the literature
illustrate the robustness of the developed finite element model.
8 Parametric study
The parametric study aims to quantify the sensitivity of the seismic settlement to the
change of the soil density, surface load, foundation embedment, earthquake intensity, thickness of the soil layer, and dominant frequency of the earthquake. Quantifying the influence of these parameters helps to develop further understanding of the seismic settlement
and hence enables the development of a robust analytical model to accurately predict the
Fig. 7 The results of the applied
stress–settlement relationship
obtained from static analyses
Applied stress (kPa)
Settlement (mm)
0
-10
-20
-30
0
100
200
300
400
500
600
700
800
900
RD = 50%
RD = 80%
RD = 94%
-40
-50
-60
-70
-80
13
13
This study (kPa)
250
500
865
RD (%)
50
80
94
648
942
306
Terzaghi (kPa)
29
8
22
Difference (%)
589
885
269
Meyerhof (kPa)
18
2
7
Difference (%)
517
752
293
Hansen (kPa)
3
− 13
17
Difference (%)
Table 4 Comparison of the ultimate bearing capacity obtained using the finite element analyses and closed-form solution of Terzaghi utilizing Terzaghi, Meyerhof, and
Hansen bearing capacity factors
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Fig. 9 Influence of the surface
load (LL) on the seismic settlement of a foundation constructed
on sand with RD of 80%
Seismic settlement (mm)
Fig. 8 Influence of the surface
load (LL) on the seismic settlement of a foundation constructed
on sand with RD of 50%
Seismic settlement (mm)
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0
-5
-10
-15
-20
-25
-30
LL= 0% of Qa
LL= 25% of Qa
LL= 50% of Qa
LL= 100% of Qa
-35
-40
-45
0
2
4
6
Time (s)
8
10
0
-2
-4
-6
-8
LL= 0% of Qa
LL= 25% of Qa
LL= 50% of Qa
LL= 100% of Qa
-10
-12
-14
0
2
4
6
Time (s)
8
10
seismic settlement. It is worth stating that the seismic settlement has been measured at the
centre of the foundation. Also, it is necessary to state that the earthquake also induces rocking and translation movement (Kim et al. 2015; Ko et al. 2018; Sharma and Deng 2019).
However, this paper only focuses on the seismic settlement for the sake of briefing and due
to pages/words limitations.
8.1 Effect of the magnitude of surface load
It is well known that the surface load carried by the foundation is transmitted to the soil
and hence, induces strain in the soil. However, it is not known to what extend this strain
affects the seismic settlement of the foundation. Therefore, numerical models with different surface load magnitudes (25%, 50%, and 100% of the allowable bearing capacity of the
soil (Qa)) have been developed and analysed to study the impact of the surface load carried
by the foundation on the seismic settlement caused by the earthquake shake. In addition
to these, the case of no load on the surface of the foundation has also been considered to
allow clear understanding of the effect of the surface load and to show the settlement produced solely due the earthquake shake.
Figures 8, 9, and 10 depict the results of the analyses for the three relative densities.
The results of all models show a gradual increase of the seismic settlement until reaching a stabilized value at a time of about 5 s. In addition, these figures show that the
13
Fig. 10 Influence of the surface
load (LL) on the seismic settlement of a foundation constructed
on sand with RD of 94%
Seismic settlement (mm)
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0
-2
-4
-6
LL = 0% OF Qa
LL= 25% of Qa
LL= 50% of Qa
LL= 100% of Qa
-8
-10
-12
0
2
4
6
8
10
Time (s)
seismic settlement–time relationship has the same trend regardless of the magnitude of
surface load or the relative density of the soil. It is also clear from the figures that there
is a considerable settlement for the case of a relative density of 50% even when there is
no surface load (i.e. 0% of Qa), where the maximum settlement produced for this case
is 8 mm. However, for other relative densities, the settlement produced for the case of
no surface load is very marginal with a maximum value of 1.0 mm and 0.9 mm for RD
of 80% and 94%, respectively. It is also clear from the figures that the surface load carried by the foundation has a dramatic impact on the developed seismic settlement. This
behaviour is due to the increase of the strain in the soil as the surface load increases;
this makes the soil responds more dramatically to the effect of the seismic shake.
For a relative density of 50% (Fig. 8), increasing the surface load from 25% of Qa
to 50% and 100% of Qa increases the seismic settlement by 28% and 75%, respectively.
However, for a relative density of 80% (Fig. 9), the percentage increase of the seismic settlement is 46% and 128% as the surface load increases from 25% of Qa to 50%
and 100% of Qa, respectively. The percentage increase of the seismic settlement for a
sandy soil having a relative density of 94% is equal to 61% and 179% as the surface
load increases from 25% of Qa to 50% and 100% of Qa, respectively. Based on these
calculated percentages, it can be concluded that the effect of the surface load on the
seismic settlement rises as the RD increases. It is also worthy to note that similar effect
of the surface load on the seismic settlement has also been observed for saturated sand
by Karamitros et al. (2013a, b, c) and Dimitriadi et al. (2017, 2018).
It is also clear from Figs. 8, 9, and 10 that as the relative density increases the seismic settlement decreases; this behaviour is due to the increase of the stiffness and shear
strength of the soil as the relative density increases. Similar behaviour has also been
noted by Ahmadi et al. (2017). Another justification for this behaviour is that the settlement induced due to earthquake shake causes a densification of the soil, and thus, the
soil with a less relative density has a higher tendency to compact (i.e. settle). It is important to note that the trend of the seismic settlement obtained using the Hs small model
is similar to the trend of the experimental results of the seismic settlement reported by
Ghayoomi and Dashti (2015) and Ahmadi et al. (2017), who studied the response of
foundation subjected to earthquake excitation using experimental models. These similarities of the response produced using the Hs small model with the experimental results
give additional trust in the quality of the finite element model used in this study.
13
Fig. 12 Influence of depth of
embedment (D) on the seismic
settlement of a foundation resting on sand with RD of 80%
and subjected to surface load of
100% of Qa
Seismic settlement (mm)
Fig. 11 Influence of depth of
embedment (D) on the seismic
settlement of a foundation resting on sand with RD of 50%
and subjected to surface load of
100% of Qa
Seismic settlement (mm)
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0
D = 0.0 m
D = 0.5 m
D = 1.0 m
D = 2.0 m
-5
-10
-15
-20
-25
-30
-35
-40
0
2
4
6
8
10
6
8
10
Time (s)
0
-2
-4
-6
-8
-10
D = 0.0 m
D = 0.5 m
D = 1.0 m
D = 2.0 m
-12
-14
0
2
4
Time (s)
8.2 Effect of foundation embedment
It is widely recognized that increasing the foundation embedment increases both the soil
stiffness and soil strength due to the increase of the confining pressure (Alzabeebee et al.
2017, 2018b, c). Due to this, many studies have examined the influence of embedment
on the settlement of shallow foundations subjected to static loads (Nguyen and Merifield
2012; Benmebarek et al. 2017; Acharyya and Dey 2018; Ghalesari et al. 2019) or harmonic
load produced due to machine vibration (Fattah et al. 2015a, b; Hakhamaneshi and Kutter
2016; Mbawala et al. 2017). However, it is not known if the embedment reduces the seismic settlement and it is not known to what extend that the embedment helps in reducing
the seismic settlement. Thus, finite element models have been built to model the case of
embedded strip foundations with embedment depths of 0.5 m, 1.0 m, and 2.0 m, and for
the three relative densities modelled in this research (i.e. 50%, 80%, and 94%). The results
from these models have been compared with those obtained and reported earlier.
Figures 11, 12, and 13 display the effect of embedment (D) on the seismic settlement
of a strip foundation subjected to surface load equal to 100% of Qa for the cases of RD of
50%, 80%, and 94%, respectively. It is obvious from the figures that the foundation embedment (D) does not affect the trend of the seismic settlement–time relationship. However,
the figures reveal that rising the depth of embedment reduces the seismic settlement; this
is due to the increase of the stiffness and strength of the soil as the depth of embedment
13
Fig. 13 Influence of depth of
embedment (D) on the seismic
settlement of a foundation resting on sand with RD of 94%
and subjected to surface load of
100% of Qa
Seismic settlement (mm)
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0
-2
-4
-6
-8
D = 0.0 m
D = 0.5 m
D = 1.0 m
D = 2.0 m
-10
-12
0
2
4
6
8
10
Time (s)
increases (Mbawala et al. 2017). For a relative density of 50% (Fig. 11), the percentage
decrease of the seismic settlement is equal to 23%, 36%, and 59% as the embedment depth
increases from 0.0 m to 0.5 m, 1.0 m, and 2.0 m, respectively. However, for the same
embedment depths but for the case of a relative density of 80% (Fig. 12), the percentage
decrease is 20%, 48%, and 70%, respectively. Finally, the percentage decrease for the relative density of 94% (Fig. 13) is 19%, 38%, and 59%, respectively. Hence, the results demonstrate the beneficial effect of foundation embedment and it can be suggested, based on
these findings, for the designers to consider placing the foundation at an embedment depth
to reduce the potential additional settlement due to the earthquake effect.
8.3 Effect of the peak ground acceleration of the earthquake
Fig. 14 Effect of the peak ground
acceleration (PGA) on the seismic settlement–time relationship
of a foundation resting on sand
with a RD of 80% and subjected
to a surface load of 100% of Qa
Seismic settlement (mm)
The influence of the peak ground acceleration of the earthquake shake has been investigated by scaling the acceleration–time relationship of the Upland earthquake (shown in
Fig. 1a). Four additional acceleration–time relationships have been developed; these new
records have a peak ground acceleration (PGA) of 0.1 g, 0.2 g, 0.4 g, and 0.6 g. This
approach of scaling is similar to the approach used by Bakr and Ahmad (2018) and Alzabeebee (2019a, b).
Figure 14 presents an example of the effect of the PGA on the seismic settlement–time
relationship for a foundation resting on sand with RD of 80% and subjected to a surface
0
-20
-40
PGA= 0.10 g
PGA= 0.20 g
PGA= 0.24 g
PGA= 0.40 g
PGA= 0.60 g
-60
-80
0
2
4
6
Time (s)
13
8
10
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140
RD= 50%
120
RD= 80%
100
RD= 94%
80
60
40
20
0
0
0.2
PGA (g)
0.4
0.6
0.4
0.6
0.4
0.6
Seismic settlement (mm)
(a)
140
RD= 50%
120
RD= 80%
100
RD= 94%
80
60
40
20
0
0
0.2
PGA (g)
(b)
Seismic settlement (mm)
Fig. 15 Effect of the peak ground
acceleration (PGA) of the earthquake on the maximum seismic
settlement: a surface load = 25%
of Qa; b surface load = 50% of
Qa; and c surface load = 100%
of Qa
Seismic settlement (mm)
load equal to 100% of Qa. It is evident from the figure that increasing the PGA surges the
seismic settlement. However, the figure also shows that the PGA does not influence the
trend of the seismic settlement–time relationship as all the accelerations produce similar
response but with different seismic settlement values, although this is not clear for the case
of a PGA of 0.1 g because the settlement produced is very small as can be seen in the
figure.
140
RD= 50%
120
RD= 80%
100
RD= 94%
80
60
40
20
0
0
0.2
PGA (g)
(c)
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Figure 15a–c shows the relationship of the maximum seismic settlement, relative density, and the PGA for surface foundations subjected to a surface load of 25% of Qa, 50%
of Qa, and 100% of Qa, respectively. The figures show a nonlinear increase of the seismic
settlement as the PGA increases for all the load cases of this study; this indicates that the
relationship trend does not depend on the surface load. The nonlinear trend of the relationship is due to the increase of the acceleration, which has a significant impact on the rate of
densification of the soil.
Calculation of the percentage difference of the maximum seismic settlement indicates
that as the relative density increases the effect of the acceleration also rises. For example,
increasing the PGA from 0.1 g to 0.6 g for the load case of 100% of Qa increases the seismic settlement by 859%, 2534%, and 3141%, for a relative density of 50%, 80%, and 94%,
respectively.
8.4 Effect of the soil layer thickness
The analytical equation proposed by Tokimatsu and Seed (1987) and improved further by
Pradel (1998) correlated the seismic settlement directly to the thickness of the sandy layer.
However, and as discussed in the introduction, no study has investigated the influence of
the soil layer thickness on the seismic settlement, and hence, no study has evaluated the
accuracy of the assumption made in the analytical equations. Therefore, this section covers
this aspect, where further models have been developed. In these models, the thickness of
the soil (H) has been changed to 10 m, 20 m, and 30 m, respectively. In addition, the relative densities of 50%, 80%, and 94% have also been considered for all of these new thicknesses to examine the combined effect of the soil layer thickness and the relative density of
the soil. The results of these models have also been compared with the models of thickness
of 40 m.
Figure 16a–c displays the relationship of the maximum seismic settlement, relative density, and the thickness of the soil for a foundation loaded with a surface load of 25% of Qa,
50% of Qa, and 100% of Qa, respectively. It is obvious from the figures that the maximum
seismic settlement is dramatically affected by the thickness of the soil. However, the trend
of the relationship is very complex, and it depends on the relative density of the soil and
the magnitude of the load applied on the foundation. For a relative density of 50%, the
seismic settlement decreases as the thickness of the soil increases from 10 to 20 m; however, the settlement then dramatically increases as the thickness rises to 30 m and 40 m.
The results of RD 50% also show that the magnitude of the load applied on the foundation
has a remarkable influence on the obtained seismic settlement. For RD 80% and 94%, the
seismic settlement decreases as the thickness rises from 10 to 30 m, but it upsurges as the
thickness rises further to 40 m. These complex trends of the relationships of the seismic
settlement–soil thickness are due to the complex interaction of the followings:
• The decrease of the acceleration amplification due to the attenuation caused by the
increase of the thickness of the soil as it is clearly evident in Fig. 17, which shows, as
an example, the effect of the soil layer thickness on the acceleration developed beneath
the foundation for the case of a relative density of 50% and a surface load of 100%
of Qa. The decrease of the acceleration reaching the foundation reduces the settlement especially in the zone below the foundation. On the other hand, the increase of
soil layer thickness increases the layer which is affected by the earthquake shake (i.e.
13
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Fig. 16 Effect of the soil layer
thickness on the maximum
seismic settlement: a surface
load = 25% of Qa; b surface
load = 50% of Qa; and c surface
load = 100% of Qa
Seismic settlement (mm)
(a)
45
40
RD= 50%
35
RD= 80%
30
RD= 94%
25
20
15
10
5
0
0
10
20
30
50
40
Thickness of sand layer (H) (m)
Seismic settlement (mm)
(b)
45
40
35
30
RD= 50%
25
RD= 80%
20
RD= 94%
15
10
5
0
0
10
20
30
40
50
Thickness of sand layer (H) (m)
(c)
increases the compressible layer thickness). This means that there are opposite factors
working together as the soil thickness increases.
• The influence of the relative density of the soil on the hysteretic damping. This adds
further complications to the problem, and it is one of the reasons that the seismic settlement–soil thickness relationship does not follow the same trend for all of the considered relative densities.
13
Fig. 17 Acceleration underneath
the foundation induced due to
earthquake shake for the case of
RD of 50% and a surface load of
100% of Qa: a thickness of sand
layer (H) = 10 m; b H = 20 m; c
H = 30 m; and d H = 40 m
Acceleration (m/sec²)
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6
4
2
0
-2
-4
-6
0
2
4
6
8
10
8
10
8
10
8
10
Acceleration (m/sec²)
Time (Sec)
(a)
6
4
2
0
-2
-4
-6
0
2
4
6
Time (Sec)
(b)
Acceleration (m/sec²)
6
4
2
0
-2
-4
-6
0
2
4
6
Acceleration (m/sec²)
Time (Sec)
(c)
6
4
2
0
-2
-4
-6
0
2
4
6
Time (Sec)
(d)
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• The initial strain level for the three cases presented in Fig. 16a–c is different as the
applied surface load is different.
Thus, the initial strain of the soil beneath the foundation, the interaction of the acceleration amplification and soil thickness, and the dependency of the damping on the relative
density of the soil create this complex decrease–increase trend shown in Fig. 16a–c.
8.5 Effect of the dominant frequency of the earthquake shake
Fig. 18 Effect of dominant
frequency on seismic settlement–
time relationship of a foundation
resting on sand with a RD of
80% and subjected to a surface
load of 100% of Qa
Seismic settlement (mm)
The effect of the dominant frequency has been examined by repeating the analyses using
the Loma Prieta (1989) and Northridge (1994) earthquake records. As has been previously mentioned, the dominant frequency of the Northridge and Loma Prieta earthquakes
is 0.93 Hz and 0.66 Hz, respectively. The cases considered in this section are for a surface
strip foundation subjected to three levels of surface loads (25% of Qa, 50% of Qa, and
100% of Qa).
Figure 18 presents an example of effect of the dominant frequency on the obtained
time–seismic settlement relationship, which is for the case of a foundation resting on a
sand with RD of 80% and subjected to a surface load of 100% of Qa. The figure shows the
remarkable influence of the dominant frequency on the obtained results and on the trend of
the time–seismic settlement relationship. For sake of clarity, the obtained maximum settlement for the aforementioned cases has been compared with that obtained for the Upland
earthquake. Figure 19a–c shows the relationship between the maximum seismic settlement
and the dominant frequency for the cases of RD of 50%, 80%, and 94%, respectively. The
figures also show the natural frequency of the system for each relative density to show the
impact of the natural frequency on the developed seismic settlement.
It is clear from Fig. 19a–c that the dominant frequency has a remarkable effect on the
developed seismic settlement regardless of the soil relative density or the loading condition, although all of the records have the same peak ground acceleration (0.24 g). The
seismic settlement significantly increases as the dominant frequency becomes closer to the
natural frequency of the system due to resonant effect. However, it should be stated that
the percentage increase of the seismic settlement as the dominant frequency approaches
the natural frequency is not constant and depends on the load applied on the foundation
and the relative density of the sand. This percentage increase rises as the load applied on
the foundation or the relative density of the soil increases. For example, the percentage
increase for the case of RD 50% is equal to 6%, 15%, and 16% for the case of surface load
0
-50
-100
-150
Upland
Loma Prieta
Northridge
-200
-250
0
2
4
6
8
10
Time (s)
13
Fig. 19 Effect of the dominant
frequency on the maximum
seismic settlement: a RD 50%; b
RD 80%; and c RD 94%
Seismic settlement (mm)
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300
LL = 25% Qa
250
LL = 50% Qa
200
LL = 100% Qa
150
100
50
0
0
0.5
1
1.5
2
Dominant frequency (Hz)
2.5
3
Seismic settlement (mm)
(a)
300
LL = 25% Qa
250
LL = 50% Qa
200
LL = 100% Qa
150
100
50
0
0
0.5
1
1.5
2
Dominant frequency (Hz)
2.5
3
Seismic settlement (mm)
(b)
300
LL = 25% Qa
250
LL = 50% Qa
200
LL = 100% Qa
150
100
50
0
0
0.5
1
1.5
2
2.5
3
Dominant frequency (Hz)
(c)
of 25% of Qa, 50% of Qa, and 100% of Qa, respectively. For the case of RD 94%, the percentage increase is equal to 71%, 75%, and 77% for the case of 25% of Qa, 50% of Qa and
100% of Qa, respectively.
Figure 19a–c also shows that the seismic settlement remarkably declines as the
dominant frequency becomes larger than the natural frequency as can be clearly seen
for the results of the dominant frequency of 2.90 Hz (i.e. the Upland earthquake). In
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conclusion, the results obtained in this section highlight the importance of calculating
the natural frequency of the system and the expected dominant frequency as these two
factors have a remarkable impact on the developed seismic settlement.
9 Summary and conclusions
A sophisticated two-dimensional finite element model has been built and utilized to
study the seismic settlement of a strip foundation resting on a dry sand with the aim to
improve the state of the art of the soil–foundation interaction related to the effect of the
earthquake shake. The influence of the relative density of the sand, the surface load carried by the foundation, the embedment depth, the peak ground acceleration of the earthquake shake, the thickness of the soil, and dominant frequency of the earthquake shake
have been systematically investigated to provide a useful insight into the mechanism of
the seismic settlement of a shallow foundation under different scenarios. In addition,
static analyses have been conducted to find the static bearing capacity and to validate
the finite element model. The following conclusions can be stressed, justified by the
observations discussed earlier in this research:
• Comparing the results of the bearing capacity calculated based on the finite element
analysis with the bearing capacity calculated using the Terzaghi closed-form solution utilizing Terzaghi, Meyerhof, and Hansen bearing capacity factors showed that
the closed-form solution mostly provides an overestimation with the exception of
Hansen bearing capacity factors for the case of a relative density of 94%. This agrees
well with the observation of Griffiths (1982) and Chavda and Dodagoudar (2018).
• The surface load carried by the foundation has been shown to have an important
impact on the developed seismic settlement, where it increases the seismic settlement with a percentage increase ranged from 29% to 179%, depending on the relative density of the soil and the magnitude of the surface load being considered. Furthermore, the effect of the surface load rises as the relative density increases.
• The embedment depth noticeably reduces the seismic settlement. The numerical
simulation results showed that the percentage decrease of the maximum seismic settlement ranged from 19 to 70%, depending on the relative density and the depth of
the embedment. Hence, it is suggested for the designer to consider placing the foundation below the natural ground surface to reduce the expected seismic settlement.
• Increasing the peak ground acceleration (PGA) of the earthquake shake nonlinearly
rises the seismic settlement for all of the relative densities modelled in this research.
Also, the effect of the PGA rises as the relative density increases.
• The seismic settlement is remarkably influenced by the soil thickness. However,
the relationship between the soil thickness and the seismic settlement is very complex and does not follow a specific trend for all the relative densities. This complex
relationship is due to the attenuation of acceleration amplification as the soil thickness increases, the increase of the compressible layer as the sand layer thickness
increases, and the dependency of the hysteretic damping on the relative density of
the soil.
• It has been shown that the natural frequency and the dominant frequency are very
important parameters when calculating the seismic settlement as the seismic settlement dramatically increases when the dominant frequency approaches the
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Natural Hazards
natural frequency of the system due to the resonant effect. In addition, it has also
been noticed that the increase of the seismic settlement as the dominant frequency
approaches the resonant frequency is also affected by the load applied on the foundation and the relative density of the sand. Thus, the designer should calculate the
natural frequency of the system and carefully considers the loading condition of the
foundation and the expected dominant frequency of future earthquakes to enable
accurate estimation of seismic settlement.
Compliance with ethical standards
Conflict of interest The author declares no conflict of interest.
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