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Geometry Based Functions for Stiffness Estimation of Double-D Shaped Caisson Foundations
Pradeep Kumar Dammala1&2, Saleh Jalbi1, Subhamoy Bhattacharya1, Murali Krishna Adapa2, Djillali Amar Bouzid3
1University of Surrey, Guildford, United Kingdom – GU2 7XH, [email protected], [email protected], [email protected] Institute of Technology Guwahati, India – 781 039, [email protected] of Blida, Algeria-09000, [email protected]
ABSTRACT:This article proposes solutions for stiffness estimation of Double-D shaped caisson foundations embedded in three different types of ground profiles (stiffness variation along the depth: homogeneous, linear and parabolic). The approach is based on three dimensional finite element analyses and is in line with the methodology adopted in Eurocode 8-Part 5 (2004)- lumped spring approach. The method of extraction of various stiffness values from the finite element model is described and followed by obtaining the closed form solutions. Parametric study revealed the nominal effect of embedment length of Double-D caisson and hence only the width and diameter effects are included in the suggested formulations. The obtained closed form solutions are presented in terms of multiplication factors for Double-D caissons. Final stiffness terms for a given width and diameter of a Double-D caisson can be conveniently estimated by multiplying the proposed formulations to the circular shaft solutions available in literature. Applicability of the proposed formulations is demonstrated by considering a typical bridge pier supported by Double-D caissons. The proposed formulations requires minimum amount of input parameters and can be used during the tender design to arrive at the required geometry of such foundations.
Keywords: Closed form solutions; Double-D caissons; Plaxis 3D; Lumped springs
1.0 INTRODUCTIONCaisson foundations (often referred as well foundations) are one of the preferred deep foundations for supporting heavy structures such as bridges, offshore wind turbines, etc [1]. Most of the major river bridges in countries such as India, Bangladesh, Italy, Greece, etc. are supported by caissons to protect against scouring. Table 1 enlists some of the major river bridges supported on caisson foundations along with the shape and dimensions of the caissons. Caissons vary in many shapes and the most common types used in India are circular, twin circular and Double-D, see Figure 1. Circular caissons (Figure 1a) are preferred for single lane bridges with single pier and for low discharge rivers (Table 1 and Figure 1a). However, for double lane railway bridges with wide piers, twin-circular (Figure 1b) and double-D caissons (Figure 1c) are used [2]. Compared to twin-circular caissons, Double-D caissons are less prone to scouring due to their intrinsic convex shape, (Figure 1c). The shape of Double-D also facilitates easy casting and sinking due to the presence of dredge holes.
Page 1
Table 1. Abridged list of caisson supported major river bridges in IndiaName of
Bridge/location
Year (opened to
traffic)River Road or Rail
(no of lanes)Shape of caisson
Length of bridge, m
Depth (L), m
Diameter (D), m
Width (B), m B/D Reference
Kolia Bhomora Setu /Assam 1987 Brahmaputra Road (two
lane) Circular 3015 56 12 --- --- Balramamurthi and Kumar [3]
Tista Bridge/West Bengal 1941 Tista Rail (single
lane) Circular 255 17.5 6.6 --- --- Ponnuswamy [2]
Mahatma Gandhi Setu /Bihar 1982 Ganga
(Ganges)Road (two
lane) Circular 5578 55.5 13.5 --- --- Subba Rao [4]
Shastri Bridge/ Uttar Pradesh 1971 Ganga
(Ganges)Road (two
lane) Circular 2083 30 5.5 --- --- Kumar [5]
Second Hoogly Bridge/West
Bengal1992 Hoogly Road (two
lane) Twin-Circular 823 22 8 to 23 --- --- Chatterjee and Dharap [6]
Garhmukteshwar Bridge/Uttar
Pradesh1901 Ganga
(Ganges)Rail (double
lane) Double-D 671 33 9.52 4.88 0.512 Kumar [5]
Saraighat Bridge/Assam 1962 Brahmaputra
Road (double lane) and rail (single lane)
Double-D 1298 55 9.6 9.2 0.958 Dammala et al. [7]
Narayan Setu /Assam 1998 Brahmaputra
Rail (single lane) and road (double lane)
Double-D 2300 66 11.0 6.0 0.545 Mallik and Singh [8]
Bogibeel Bridge/Assam
Under construction Brahmaputra
Rail (double lane) and road
(three lane)Double-D 4940 55.26 10.5 5.7 0.543 Das [9]
Godavari Bridge/Andhra
Pradesh1974 Godavari
Road (two lane) and rail (single lane)
Double-D 4135 33 7.01 7.0 1.00 Ponnuswamy [2]
Page 2
Fig. 1 Typical railway bridge supported by caissons of varying geometry (a) Circular, (b) twin circular and (c) Double-D
Predicting the stiffness of such massive foundations is one of the important design considerations as these values are necessary for Serviceability Limit State (SLS) and Fatigue Limit State (FLS) design calculations. There are numerous approaches available in literature to idealize foundation stiffness- the distributed springs approach [10-12], lumped spring approach [13] and numerical finite/boundary element analysis [14], where each method has a certain level of applicability and complexity. Figure 2 schematically illustrates the approaches available in modeling the soil caisson interaction. The lumped spring approach has been extensively used in literature for different types of foundations [15-18] in estimating the foundation response to dynamic loading. In simple terms, the foundation and surrounding soil are replaced by a set of elastic springs at the ground level (see Fig. 1c) namely a lateral spring KL, a rotational spring KR, a coupling spring KLR, and a vertical spring KV.
Gazetas [13] proposed impedance functions for arbitrarily shaped surface and embedded footings. Researchers idealize arbitrary shaped deep foundations with an equivalent circular foundation, often resulting in an inappropriate design due to the geometric effects caused by the unaccounted lateral earth pressure [19, 20]. Therefore, geometry based impedance functions for deep foundations are a necessity for a thorough analysis and this article proposes such stiffness functions for rigid Double-D caisson foundations supporting bridge piers. The article is organized in the following order to achieve the objectives.
a. An analytical method of extracting the stiffness values from finite element program is described.
b. Series of finite element analyses are performed to arrive at the targeted closed form solutions for rigid Double-D caissons embedded in three different soil profiles with varying stiffness along the depth.
c. An example application of the proposed solutions is demonstrated for a bridge pier supported on a Double-D foundation.
Page 3
Fig. 2. Idealization of the caisson-soil interaction
2.0 METHOD OF EXTRACTION OF STIFFNESS VALUES FROM FEAA method has been described by Jalbi et al. [17] to compute the three stiffness terms (KL, KLR, and KR) from advanced three-dimensional Finite Element Analysis (FEA). The vertical stiffness (KV) is not expected to play a significant role as the caisson foundations are vertically stable. Hence, KV is neglected in the present study. A schematic view of the loading conditions along with the idealization of soil-caisson interaction is schematically presented in Figure 3. The linear range of a load-deformation curve can be used to estimate foundation rotations (θ) and deflections (ρ) based on Eq. 1.
RLR
LRL
KKKK
MH
(1)where H and M are the lateral load and moment at the pile head respectively. Equation 1 can be re-written (Eq. 2) where I (Flexibility Matrix) is a 2x2 matrix given by Eq. 3
MH
I
(2)
RRL
LRL
IIII
I(3)
To obtain the equation unknowns, one can run a numerical model for a lateral load (say H=H1) with zero moment (M=0) and obtain values of deflection and rotation (ρ1 and θ1). The results can be expressed through Eqs. 4~6.
0
1
1
1 HIIII
RRL
LRL
(4)
1
111 H
IIH LL
(5)
Page 4
1
111 H
IIH RLRL
(6)
Fig. 3 Schematic representation of (a) embedded shaft (b) rigid body rotation with translation and (c) idealization as lumped springs
Similarly, another numerical analysis can be performed for a defined moment (M=M1) and zero lateral load (H=0) and the results are shown in. Eqs. 7 -9.
12
2 0MII
II
RRL
LRL
(7)
1
212 M
IIM LRLR
(8)
1
212 M
IIM RR
(9)
From the above analysis (Eqs. 4~9), terms for the I matrix (Eq. 3) can be obtained. Equation 2 can be rewritten as Eq. 10 through matrix operation.
MH
I1
(10)Comparing Eqs. (1 and 10), the relation between the stiffness matrix and the inverse of flexibility matrix (I) can be given by Eq. 11. Equation 12 is a matrix operation which can be carried out easily to obtain KL, KR and KLR.
11
RRL
LRL
RRL
LRL
IIII
IKKKK
K(11)
1
2
2
1
1
2
2
1
1
1
MH
MHIK
(12)Therefore, mathematically, two FEA analyses are required to obtain the three spring stiffness terms. It is important to note that the above methodology is only applicable in the linear range and it is advisable only to use the obtained stiffness
Page 5
values for Eigen frequency analysis or a first estimate of the deformations.
3.0 3D FINITE ELEMENT MODELLING
3.1 Details of the modelA series of three dimensional (3D) finite element analyses were carried out using commercially available finite element software - PLAXIS 3D. To save computational power and operational time cost, only half of the system was modelled due to symmetry.
3.1.1 Soil modellingThe soil continuum has been modelled as a linear elastic material as that would provide a reasonable estimate of the maximum bending moment [21]. Three types of soil idealizations (stiffness variation with depth) are considered in this study – (a) Homogeneous (b) linear heterogeneity and (c) parabolic heterogeneity. In simple terms, homogeneous soil represents the uniform soil stiffness along the depth such as over-consolidated (OC) clay. Linear distribution is expected in sandy soils while normally consolidated (NC) clays exhibit parabolic distribution of stiffness with depth. A realistic multi-layered soil stratum may not exactly hold either of the chosen profiles, however, one can approximate the stiffness using selected profiles reasonably. This type of representation is well accepted in the Eurocode 8-Part 5 [22] and in many engineering applications [17, 18, 23, 24].
Stiffness variation along the depth (Esz ) can be represented by the following formulation (Eqn. 13) according to Eurocode 8-Part 5 [22] and is described in Fig. 4.
E sz=E s@1 D( ZD )
α
(13)
whereEs@1 D
represents the stiffness of soil at one diameter (D) depth, α
represents the degree of inhomogeneity, Z is the depth and D represents the diameter of the shaft, see Figure 4 (b) for details. Homogeneous soils will have an α value of zero, linear inhomogeneity has a value of one while α varies from 0 to 1 for parabolic variation (0.5 has been chosen in the present study).
PLAXIS 3D allows the user to define either a constant stiffness or stiffness increasing linearly with depth (normalized with shaft diameter) which requires the datum stiffness and the slope of it with depth. These were used to model homogeneous and linear variation. For parabolic variation of stiffness, the soil stratum was discretized in to multiple layers, each with a thickness calculated based on the linear variation in the layer (Fig. 4a). An initial stiffness value and the linear slope was provided as an input to each layer to represent a parabolic stiffness variation. Figure 4(b) illustrates the stiffness variation considered for one-meter diameter shaft.
Page 6
Fig. 4(a) Soil profiles considered for the study
10
9
8
7
6
5
4
3
2
1
00 1 2 3 4 5 6 7 8 9 10
Parabolic (0<<1)
Linear (=1)
E sz/E s@1D
Z/D
Stiffness at 1 D depth (E s@1D)
Hom ogeneous (=0)
Fig. 4(b) Stiffness profile for a rigid circular caisson of 1m diameter
3.1.2 Soil extentA schematic representation of soil extent, shaft with interface, Double-D geometry and predominant loading conditions on a typical Double-D caisson supported bridge are shown in Figure 5. As the elastic stiffness values are sensitive to the soil extent and boundaries, it is therefore required to understand the influence zone from the applied loading. The objective was to ensure the least stress disturbance by the vicinity of the translational boundary conditions at the sides and fixed boundary at the bottom face. Kim & Jeong [25] modelled the soil stratum width (on either side) and depth to be 11D and 1.7L respectively, while Murphy et al. [26] used 20D and 2L respectively. Similarly, Zhang & Andersen [27] modeled the stratum with 20D width, whilst Shadlou & Bhattacharya [24] used 80 D, and Higgins et al. [23] implemented 1.5 L as stratum depth, which shows a wide range of selected soil extents.
Page 7
Fig. 5 (a) Numerical model domain (b) Double-D caisson with interaction elements (c) Double-D caisson geometric representation and (c) typical
loading direction on a bridge
For this purpose, a “trial and error” study was performed, where a circular shaft of 5 m diameter with 50 m in depth, embedded in a homogeneous soil stratum (100 MPa of young’s modulus) was chosen. The lateral soil extents (both x and y directions, see Fig 5a) were varied from 10D to 100D and the depth of stratum has been kept constant at 2L, based on the literature suggested range. A very fine mesh was chosen to discretize the soil and shaft. A lateral load of 100 kN and a moment of 100 kN.m were applied at the shaft head and the corresponding lateral (δx) and axial (δz) displacements are monitored. Figure 6 presents the effect of soil extent on δx and δz. It must be noted that the increase in soil extent (10D to 100D) increases the displacements up to a certain extent beyond which no significant alteration has been recorded. In this case, soil extent of 50D looks reasonable for nominal change in recorded displacements and, therefore, the same extent was chosen for the final model in both x and y directions. The depth of the stratum was considered as 2L.
Page 8
0 20 40 60 80 1000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
x_H
z_H
x_M
z_M
Lx/D iameter
Late
ral d
ispl
acem
ent (
x), m
m
0.000
0.002
0.004
0.006
0.008
0.010
Axi
al d
ispl
acem
ent (
z), m
m
Fig. 6 Variation of δx and δz with soil extent
3.1.3 Foundation modellingA “Rigid Body” option has been set to the foundation where it is restricted from axial deformation or in bending. This assumption is valid since caisson foundations have low aspect ratio (due to the high diameter/width) and also concrete has higher flexural and shear stiffness than soil. The interface was chosen as rigid (Rint=1.0) between the soil and foundation, with the same stiffness properties as the surrounding soil, see Fig. 5b. This assumption is valid since the surrounding soil is modeled as linear elastic material with no gapping and slippage considered in between. Also, as the main purpose of impedance functions is to identify the Eigen frequency of the system and so elasticity must be maintained and in effect this is elasto-dynamic solution [17]. Some gapping may exist between the caisson and surrounding soil with high strains, however, the fundamental mode of vibration takes place in the small amplitude vibrations and hence, gapping or slip may be ignored for preliminary analysis [28].
A very fine mesh was implemented for enhanced accuracy. Foundation was modeled as a hollow rigid body without any consideration given to the internal geometry of the foundation. In a FE analysis, stiffness estimation for a rigid pile and a caisson is similar [29]. A recent study by Alder et al. [30] shows that for “rigid” foundations (high foundation stiffness/soil stiffness), solid and hollow caissons exhibit similar lateral and rotational stiffness values. Similar observations were also made by Gelagoti et al. [31] and Latini and Zania [18].
3.1.4 Loading conditionsThe stiffness of foundation is determined along the longitudinal axis of the bridge as the predominant loads are expected to act in that direction. Figure 5 (c and d) schematically illustrates the loading idealization. The generated model with caisson embedded in soil stratum, is brought to equilibrium by applying hydrostatic stresses in the first phase followed by the subsequent loading phases. The displacements were set to zero prior to application of the loads and the loading is applied in two phases. As described in the section 2.0, two independent loading conditions are required for stiffness estimation-one with only lateral loading (H) and the latter with only moment (M) acting exactly at the reference point (0, 0, 0) of caisson head, see Fig. 5b for details. As the soil is a linear elastic material, it is expected that any magnitude of loading would yield linear stress-strain behavior. However, for confirmation, the lateral loads are increased from 50 kN to 1000 kN
Page 9
while moments applied are varied from 500 kN.m to 10000 kN.m. Figures 7 (a and b) present the variation of lateral displacement (δx) and rotation (θ) at the caisson head, due to independent lateral and moment application respectively. An obvious linear slope with lateral load and moment can be noted, which would yield similar linear stiffness values at any magnitude of loading. Once the displacements and rotations at the caisson head are recorded for both the loading conditions, stiffness terms (KL, KLR, KR) can be evaluated as explained in Section 2.0.
0 300 600 900 12000.0
0.3
0.6
0.9
1.2
1.5
x_H
_H
Lateral force (H ), kN
Late
ral d
ispl
acem
ent (
x), m
m
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
Ro
tati
on
()
, rad
s
(a)
0 4000 8000 120000.0
0.1
0.2
0.3
0.4
x_H
_H
(b)
Late
ral d
ispl
acem
ent (
x), m
m
Mom ent (M ), kN.m
0.000
-0.002
-0.004
-0.006
-0.008
-0.010
-0.012
Ro
tati
on
()
, rad
s
Fig. 7 Variation of δx and θ for (a) H and (b) M at the caisson head
3.2 Model ValidationIn order to check the accuracy of the developed numerical model, the reported results on rigid circular foundations from literature were considered due to the lack of Double-D caisson formulations. Higgins et al. [23] proposed non-dimensional formulations to estimate the stiffness of rigid piles in soils of uniform stiffness (homogeneous) and linearly varying stiffness. Similarly, Shadlou & Bhattacharya [24] proposed such stiffness formulations for both rigid and flexible shafts for a wide range of L/D ratio in three soil profiles. Aissa et al. [32] formulated stiffness functions for rigid monopiles in homogeneous soils while Abed et al. [33] proposed formulations for rigid piles in linearly varying and parabolic stiffness varying soils with smooth and rough interface. A rigid circular shaft of diameter 5 m (D) is chosen with varying embedment length (L) from 30 m to 60 m, representing practical L/D ratio of 6 to 12. The elastic modulus of soil at 1
diameter depth (E s@1 D ) was chosen as 100 MPa.
Figure 8 (a-c) present the variation of normalized lateral, rotational and coupled stiffness values with the embedment depth (L/D). It can be inferred that the developed model provides literature consistent results and the nominal differences (± 5 %) are attributed to the possible numerical errors and can be ignored. Hence, it can be concluded that the soil extents selected, the method of stiffness extraction are justifiable and therefore the same approach is used to formulate the stiffness functions of double D shaped foundations.
Page 10
6 9 12 151E-3
0.01
0.1
1 Higgins et al. (2013) Shadlou & Bhattacharya (2016) Abed et al. (2016) Aissa et al. (2017) This study
Homogeneous
Parabolic
KL/E
s@1D
.D
L/D
Linear
(a)
6 9 12 150.1
1
10
100 Higgins et al. (2013) Shadlou & Bhattacharya (2016) Abed et al. (2016) Aissa et al. (2017) This study
(b)
KR/E
s@1D
.D3
L/D
Linear
Parabolic
Homogeneous
6 9 12 150.01
0.1
1
10 Higgins et al. (2013) Shadlou & Bhattacharya (2016) Abed et al. (2016) Aissa et al. (2017) This study
(c)
KL
R/E
s@1DD
2
L/D
Parabolic
Linear
Hom ogeneous
Fig. 8 Variation of (a) normalized lateral stiffness (b) rocking and (c) coupled stiffness
Page 11
4.0 DERIVATION OF CLOSED FORM SOLUTIONSThis section explains the procedure for obtaining closed form solutions from the FE analyses performed. Only the results from linearly varying stiffness soil are presented for brevity. Table 2 summarizes the parametric analysis performed to obtain the stiffness functions.
Table 2. Summary of the analysis performedGround profile Length (m) Diameter (m) Width (m)Homogeneous
30, 40, 50 60 2.5, 5, 10, 15 0, 2.5, 5, 7.5,
10, 12.5, 15
Linear heterogeneous
Parabolic heterogeneous
To simplify the mathematical operations, the final stiffness of a double-D well foundation is expressed as a geometrical factor (α ,β ,ψfor KL, KR and KLR
respectively) multiplied by the circular shaft stiffness (with B=0) of similar diameter.
K L=α (K L( B=0 ))(14)
K R=β ( K R(B=0) )(15)
K LR=ψ (K LR(B=0 ))(16)
The stiffness values of rigid circular foundations can be easily obtained from the solutions suggested in literature. Available closed form solutions for deep rigid circular foundations embedded in different soils are presented in Table 3.
Page 12
Table 3. Impedance functions for circular shafts (B=0) for L/D>2Ground profile
(see Fig.4 for definition) ResearchersK L (B=0)
f υs DES @1 D
K LR (B=0 )
f υs D2 ES @1 D
K R (B=0)
f υs D3 ES @1 D
Homogeneous
Carter & Kulhawy [16] a 1 .35( LD )
0.81−1 .8 ( L
D )1. 6
7 .135( LD )
2. 15
Higgins et al. [23]a 2( LD )
0 . 81−2 .7 ( L
D )1. 70
7 . 125( LD )
2. 47
Shadlou & Bhattacharya [24]b 3 .2( LD )
0. 62−3 .55 ( L
D )1. 56
6 . 6( LD )
2 .5
Aissa et al. [32]c 2 .756 ( LD )
0. 668−1 .595( L
D )1 . 636
1 .731( LD )
2 .495
Linear heterogeneity
Higgins et al. [23]a 2( LD )
1 .66−3 .35 ( L
D )2. 60
7 . 7( LD )
3 . 45
Shadlou & Bhattacharya [24]b 2 .35( LD )
1. 53−3 .55 ( L
D )2. 50
7 . 35( LD )
4 . 5
Abed et al. [33]c 1 .708( LD )
1 . 661−1 .233( L
D )2. 655
1 .153( LD )
3 .605
Parabolic heterogeneity
Shadlou & Bhattacharya [24]b 2 .665 ( LD )
1. 07−3 . 60( L
D )2
6 . 5( LD )
3
Abed et al. [33]c 2 .841( LD )
0. 977−2 .933( L
D )1. 767
3 .894( LD )
2 .562
Page 13
a(Carter and Kulhawy [16]; Higgins et al. [23]) defined Poisson ratio function, f υs=
1+υs
1+0 . 75υs
b[16]**f υs=1+0. 6|υs−0 .75| c(Abed et al. [33;] Aissa et al. [32]) formulations are valid for a poisson ratio (υs ) of 0.40 with a rough soil-shaft interface and f υs
need not to be considered
Page 14
For a double- D caisson (as shown in Fig. 5c), geometry, namely length of embedment (L), width (B) and diameter (D) may effect the stiffness. Therefore, stiffness values obtained from the parametric study are normalized against the circular foundation (B=0) case.
4.1 Embedment depth (L) effectsFor a fixed diameter, the normalized values of KL, KLR, KR at different embedment lengths were plotted against the normalized width (B/D) and are shown in Figures 9 a, b and c, respectively. It can be observed that the embedment length may not significantly affect the stiffness, especially in the low B/D ratio, considered as the practical range for design (Table 1). Therefore, for the simplified formulations, the effect of embedment depth on the stiffness factors was not taken in to account in the FE models as they are already considered in the stiffness formulations of circular shaft (Table 3).
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
L=30 m L=40 m L=50 m L=60 mK
L/KL(
B=0
)
B/D
D=5 m
Circular caisson (B=0)
(a)
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5(b)
D=5 m L=30 m L=40 m L=50 m L=60 m
KR/K
R(B
=0)
B/D
Circular caisson (B=0)
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
(c)
KLR
/KLR
(B=0
)
D=5 m L=30 m L=40 mL=50 mL=60 m
B/D
Circular caisson (B=0)
Fig. 9 Embedment depth effects on normalized (a) lateral, (b) rocking and (c) coupled stiffness for a homogeneous soil profile
Page 15
4.2 Width (B) effectsOn the other hand, the variation of normalized stiffness with B/D for a fixed embedment length (L=30 m) was plotted for different diameters (Figure 10). A significant increase of stiffness with caisson width can be observed. Primarily, one must note that “linear” variation of the normalized stiffness can be a better fit with high correlation coefficient (R2) values and the equation can be
m( BD )+c
(17)
Secondly, for the same length and B/D ratios, different multiplication factors are expected for different diameters (different m values in equation 17). The linear trends, together with the linear fit values (referred as m1 and c1 here after) are shown in Figure 10.
0 2 4 6 8 10 12 140
1
2
3
4
D=2.5 m D=5 m D=10 m D=15 m
KL/K
L(B
/D=0
)
B/D
(a)
Diameter, m m1 C1 R2 2.5 0.17 1.21 0.97 5 0.26 1.06 0.99
10 0.37 1.02 0.99 15 0.43 1.00 0.99
y=mx+C
L=30 m
0 2 4 6 8 10 12 140
1
2
3
4
L=30 m
Diameter, m m1 C1 R2 2.5 0.20 1.27 0.96 5 0.31 1.07 0.99
10 0.47 1.02 0.99 15 0.56 1.00 0.99
D=2.5 m D=5 m D=10 m D=15 m
KR/K
R(B
/D=
0)
B/D
(b)
y=mx+C
Page 16
0 2 4 6 8 10 12 140
1
2
3
4
L=30 m D=2.5 m D=5 m D=10 m D=15 m
KL
R/K
LR
(B/D
=0)
B/D
(c)
y=m x+C
Diameter, m m1 C1 R2 2.5 0.18 1.24 0.96 5 0.28 1.07 0.99
10 0.40 1.02 0.99 15 0.46 1.00 0.99
Fig. 10 Caisson’s width effects on (a) lateral, (b) rocking and (c) coupled stiffness for a homogeneous soil profile
4.3 Diameter EffectsTo include the effect of diameter on the closed form solutions, the slope (m1) values are normalized against the slope obtained from the smallest diameter considered (mD=2.5), see Fig.11. A logarithmic variation gives a better fit (high R2) and the slope of this variation is referred as m2 hereafter. It is important to note that the increase in diameter of caisson has significant affect than the length of caisson, due to the fact that the relative soil caisson contact surface area with increase in diameter may be high enough to affect the stiffness.
0 1 2 3 4 5 6 70.5
1.0
1.5
2.0
2.5
3.0
KLR
KL
KR
m/m
D=2
.5m
D/D2.5m
y=m.ln(x)+C
Stiffness function
m2 C2 R2
KL 0.858 0.966 0.9945 KR 0.986 0.945 0.9995 KLR 0.852 0.967 0.9998
practical range
Fig. 11 Diameter effects for a linearly varying stiffness
The achieved stiffness multiplication constants (α ,β andψ ) are provided below.For homogeneous soils,
α=0 . 15[{0. 17( D2 .5 )}+0. 84 ]( B
D )+1(18)
Page 17
β=0 .22[{0.22( D2. 5 )}+0 .83]( B
D )+1(19)
ψ=0.17[{0 .17 ( D2.5 )}+0 .85 ]( B
D )+1(20)
For linear variation of stiffness,
α=0 . 17[{0 .86 ln( D2. 5 )}+1]( B
D )+1(21)
β=0 . 20[{0 .98 ln ( D2. 5 )}+1]( B
D )+1(22)
ψ=0. 18[{0 .85 ln( D2 .5 )}+1]( B
D )+1(23)
For parabolic variation of stiffness,
α=0 . 19[{0.80 ln( D2 .5 )}+1]( B
D )+1(24)
β=0 . 22[{0. 85 ln( D2 .5 )}+1]( B
D )+1(25)
ψ=0. 18[{0 .88 ln ( D2 .5 )}+1]( B
D )+1(26)
It must be mentioned that these formulations are based on analysis run for L/D>2 and D≥2.5 m (typical geometry of Double-D caissons range). Hence, the applicability of the formulations can be considered appropriate for practical range of Double-D caisson foundations.
5.0 EXAMPLE APPLICATION OF PROPOSED FORMULATIONSAn example of a multi-layered soil profile with varying soil stiffness along the depth is chosen to demonstrate the applicability of proposed equations. The considered soil profile is near a bridge site in India, Saraighat Bridge [7]. The bridge is supported on Double-D caisson foundations. Figure 12 details the soil stratigraphy in terms of standard penetration resistance (SPT N value), correlated shear wave velocity (Vs) and the soil stiffness (Esz) variation with depth. Further details about the location of the borehole and the soil profile can be found in Dammala et al. [34]. Realistic soil profiles do not usually follow an idealized stiffness variation, however, a representative stiffness curve may be selected to properly suit the realistic profile’s stiffness. For the chosen profile, the stiffness can be better represented with a linear variation using the diameter of the caisson, Figure 12 c.
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0 100 200 300 400 5000 10 20 30 40 50 60
60
55
50
45
40
35
30
25
20
15
10
5
00 100k 200k 300k 400k 500k 600k 700k 800k
(c)
Vs, m/s
Vs
(b)
Very hard gravelmixed with clay
Highly dense siltyclayey sand (15m)
Dense sandy silt mixed with clay (14m)
SPT N
SPT N
Dep
th, m
(a)
Loose silty sand (11m)
Eso, kPa
Chosen Real field
Fig. 12 Soil profile near Saraighat Bridge (a) SPT N value (b) Shear wave velocity (Vs) and (c) realistic soil stiffness with the considered linear variation
The stiffness terms of a Double-D caisson (KL, KR and KLR) can be estimated using the proposed formulations (Eqns. 21-23) in a simple three step procedure:
Step 1: Calculation of circular caisson stiffness values using Abed et al. [33]formulations presented in Table 3.
Step 1(a): The lateral stiffness of a rigid circular shaft (K L( B=0 )) with D=9.6 m, ES@1 D =130 MPa, and length of 40 m (L=40 m) is given asK L( B=0 )
D .ES @1 D=1 .708 [40
9.6 ]1. 661
=18 .279
K L( B=0 )=18. 279×9. 6×130000=22 .812×106 (kN /m )K L( B=0 )=22. 8 GN/m
Step 1(b): The rocking stiffness of circular shaft (K R(B=0 )) according to Abed et al. [33] is given as (Table 3)
K R(B=0 )=22687 GN.m/rad
Step 1(c): Similarly, coupled stiffness of circular shaft (K LR(B=0 )) (Table 3)K LR(B=0 )
D2 . ES @1 D
=−1.233[ 409 .6 ]
2.665
=−55 . 296
K LR(B=0 )=−55 . 296×9 .62×130000=−662 .5×106(kN )K LR(B=0 )=−662. 5 GNStep 2: Determination of stiffness multiplication constants (α, β, ψ) using the
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proposed formulations.
The value of α according to Equation 17 is given as
α=0 . 170[(0. 86×ln( 9 .62.5 ))+1]( 9. 2
9 . 6 )+1=1. 351
The value of β according to Equation 18 is given as
β=0 . 20[(0. 98×ln ( 9 . 62. 5 ))+1]( 9. 2
9 . 6 )+1=1. 444
Similarly, the coupled stiffness constant (ψ
) according to Equation 19 is given as
ψ=0. 18[(0 .85×ln ( 9 . 62. 5 ))+1]( 9. 2
9 . 6 )+1=1. 369
Step 3: Final step involves the multiplication of the stiffness constants with the circular stiffness values in order to obtain the geometry based stiffness of Double-D caissons.
The lateral stiffness of double-D caisson (K L
) according to Equation 14 is given as
K L=1 .351×22. 812=30 . 82 GN/m
The rocking stiffness of double-D caisson (K R
) according to Equation 15 is given as
K R=1 . 444×22687=32760 GN.m/rad
Similarly, the coupled stiffness of double-D caisson (K LR
) according to Equation 16 is obtained as
K LR=1 .369×−662.5=−906 .96 GN
For comparison purpose, the stiffness values for the Double-D caisson were also evaluated using finite element modelling (with the realistic ground profile as shown in Figure 12c). Table 4 summarizes the stiffness values estimated by the proposed equations and finite element model. A close match of the results can be seen, justifying the appropriateness of stiffness curve selection. Therefore, one can be confident that the proposed relationships can predict the stiffness, even for a multi-layered soil profile with proper selection of stiffness variation.
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Table 4. Comparison of stiffnessStiffness Proposed formulation 3D FEA (PLAXIS)
KL, GN/m 30.82 27.54KLR, GN -906.96 -823KR, GN m/rad 32760 30461
6.0 CONCLUSIONS
Caisson foundations are traditional deep foundations supporting major structures such as bridges and offshore wind turbines. Design engineer requires the stiffness of foundation in order to do a performance based design of a structure. This article proposes the stiffness functions in terms of factors based on geometry for Double-D caisson foundations embedded in three different types of soil profiles. The analysis is based on three-dimensional finite element modelling and is in line with the methodology adopted in Eurocode 8 –Part 5 (2004). Initially, the method of extraction of stiffness terms from the FEA is described. A parametric study is performed to identify the sensitive parameters affecting the stiffness and finally the impedance functions are presented in the form of multiplying constants. The final stiffness of a Double-D caisson can be achieved by multiplying the proposed constants with the rigid circular shaft formulations available in the literature. The applicability of the proposed solutions is demonstrated by considering a bridge pier supported on Double-D caisson foundations. The proposed formulations with the minimum amount of input parameters can be used during the tender design.
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