- Email: [email protected]
A long term evaluation of circular mat foundations on clay deposits using fractional derivatives
Computers and Geotechnics xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
A long term evaluation of circular mat foundations on clay deposits using fractional derivatives Cheng-Cheng Zhang a, Hong-Hu Zhu a,b,⇑, Bin Shi a, Behzad Fatahi c a
School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China Nanjing University High-Tech Institute at Suzhou, Suzhou 215123, China c School of Civil and Environmental Engineering, University of Technology Sydney (UTS), Sydney, New South Wales 2007, Australia b
a r t i c l e
i n f o
Article history: Received 7 June 2017 Received in revised form 31 July 2017 Accepted 24 August 2017 Available online xxxx Keywords: Ground settlement Clay deposit Foundation analysis Fractional calculus Time dependent behaviour Settlement prediction
a b s t r a c t This study proposes to use fractional derivatives to evaluate the long term performance of circular mat foundations overlying clays and also predict the associated ground settlement. Closed form solutions for the deflection and bending moment of foundations and the subsequent reaction of subgrade are obtained with the Mittag–Leffler function. Numerical examples are used to determine how the fractional order affects the time dependent properties of the foundation and ground settlement, and to simulate the case history of a large standpipe constructed over Tertiary sediments. New insights into design and prediction of shallow foundations and ground settlement are also discussed. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The performance of foundations constructed over clay deposits and the associated ground settlement are of great concern for geoengineers [1–5] because the interaction between foundations and soil has hitherto been considered as a plate resting on a Winker or an elastic medium with two parameters. These early contributions did provide a sound theoretical basis for later study where complex loading histories and foundations with different shapes are considered [6–10], analytical and numerical methods such as the point collocation method, the Hankel transform, the ring method, the Chebyshev polynomial–Ritz method, and the finite element method have also been proposed and utilised [11–16]. However, since the deformation of clay is time dependent, it should be considered when modelling the interaction between soil and structure. For decades, the performance of a foundation on a viscoelastic medium [17,18] or a poroelastic medium [19,20] has been investigated, which is why the theory of poroelasticity, proposed originally by Terzaghi and further developed by Biot, enables the dissipation of pore water pressure and deformation of the solid skeleton under external loads to be considered. While simple and ⇑ Corresponding author at: School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China. E-mail address: [email protected] (H.-H. Zhu).
convenient simulations of the viscous behaviour of clay using viscoelastic models (e.g. Maxwell model, Kelvin–Voigt model and Merchant model) can graphically visualise materials, they are still limited by the simple constitutive relationship of the Newtonian dashpot [21]. Since fractional calculus is where integrals and derivatives can be of arbitrary non-integer orders and it can simulate hereditary phenomena with a long memory [22], it was introduced to viscoelasticity in the first half of the 20th century [23], and since then, models based on fractional derivatives have been utilised extensively to describe the viscous properties of materials [24–27]. The concept of fractional calculus has been introduced to geotechnics [28–33] where the viscous behaviour of geomaterials such as clays [34], granular soils [35], saturated soils [36], frozen silts [37], and rocks [38], has recently been modelled. Interesting work by Cosenza and Korošak shows that the secondary consolidation of clay can be described using a time-fractional diffusion equation within the framework of Terzaghi’s theory [34], but until now, research on the interaction between foundation and soil while incorporating fractional calculus is still limited [39,40]. While the time dependent response of a circular foundation has been investigated by assuming that the underlying soil is either viscoelastic [41] or poroelastic [42], the interaction between a circular foundation and a fractional type of viscoelastic soil has never been reported in literature.
http://dx.doi.org/10.1016/j.compgeo.2017.08.018 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
2
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Nomenclature Ca Cv D RL D Df EOP E E0 Ef E0 Ea ðÞ Ea;1 ðÞ FDMM HðÞ h J i ðÞ M q0 RMSE R R r r0 r1 SMM
coefficient of secondary consolidation coefficient of primary consolidation derivative Riemann–Liouville fractional derivative bending rigidity of foundation end of primary elastic modulus of fractional derivative element elastic modulus of fractional derivative Maxwell clay elastic modulus of foundation analogous modulus of fractional derivative Maxwell model original Mittag–Leffler function generalised Mittag–Leffler function with one complex parameter equals 1 Fractional derivative Maxwell model Heaviside step function thickness of foundation Bessel function of the first kind of order i non-dimensional moment of foundation load root-mean-square error reaction of subgrade non-dimensional reaction of subgrade radius radius of foundation radius of load standard Maxwell model
In this paper a fractional derivative Maxwell model is used to investigate the interaction between a circular mat foundation and clay deposits, and the associated ground settlement. First, analytical solutions for the deflection and bending moment of foundations and the reaction of subgrade are derived using the corresponding principle and the Laplace transform for an axisymmetric loading history. The solutions are then solved numerically and a comprehensive numerical example is given to show how the fractional derivative order affects the long term performance of foundations and time dependent ground settlement. The model’s ability to simulate the interaction between a circular foundation and clayey ground is demonstrated by analysing the case history of a large water standpipe constructed on a circular mat foundation overlying Tertiary deposits. Finally, the implications of this research into the design of shallow foundations and predicting settlement are discussed.
2.1. Basic concepts in fractional calculus In traditional calculus the nth derivative of a function z(t) is defined as Dnz(t) = dnz(t)/dtn, but this function is extended if the integer order n is substituted by a fraction; this is the starting point for defining fractional calculus. In this study the Riemann–Liouville definitions of fractional integration and derivative are used; the Riemann–Liouville fractional integration of function z(t) is given by [43] RL
Dt v zðtÞ ¼
1 Cðv Þ
Z
t
ðt nÞv 1 zðnÞdn;
t>0
ground settlement ith displacement calculated using FDMM
Smeas i s t t EOP w w w z
ith displacement measured in field parameter of Laplace transform time EOP consolidation time deflection of foundation non-dimensional deflection of foundation Laplace-transformed deflection of foundation depth order of fractional derivative smallest integer exceeding a Euler gamma function non-dimensional parameter strain coefficient of viscosity of fractional derivative element coefficient of viscosity of fractional derivative Maxwell clay Poisson’s ratio of foundation order of fractional integration non-dimensional radius stress creep time of fractional derivative element creep time of fractional derivative Maxwell clay function determining variation of load with time Laplace-transformed function of /ðtÞ
a
[a] CðÞ
c e g g0
l
v
q r s s0
/ðtÞ /ðsÞ
The Riemann–Liouville fractional derivative of order a is formulated as [43] 0
RL
Dat zðtÞ ¼ 0 RL Dt½a ½0 RL Dt v zðtÞ;
ð1Þ
0
where D denotes derivation; v is the order of integration ranging from 0 to 1; and the subscripts 0 and t denote the integration limits.
t>0
ð2Þ
where [a] is the smallest integer exceeding a; and v = [a]a > 0. This definition indicates that the fractional derivative has a strong inherent memory due to its integral form. For simplicity, the superscript ‘‘RL” and the subscripts 0 and t are dropped in the following sections. 2.2. Generalisation of the standard Maxwell model based on fractional derivatives A fractional derivative element known as the ‘‘intermediate model” [25] or ‘‘spring-pot” [27], is introduced first. The constitutive law of this element is defined as
rðtÞ ¼ Esa Da eðtÞ; 0 6 a 6 1
2. Fractional derivative Maxwell model (FDMM)
0
S SFDMM i
ð3Þ
where r and e are stress and strain; E is the elastic modulus; s is the creep time and equals g=E; and g is the coefficient of viscosity. Because the element collapses into a spring and a dashpot when a equals 0 and 1 respectively, the order of the fractional derivative a is a non-dimensional parameter associated with the memory of materials [27]. A standard Maxwell model (SMM) consists of a spring and a dashpot connected in series, but by replacing the dashpot with a fractional derivative element, the standard Maxwell model is upgraded to an FDMM (Fig. 1), whose stress–strain relationship reads
ðDa þ 1=sa0 ÞrðtÞ ¼ E0 Da eðtÞ
ð4Þ
where E0 , s0 and a are three independent parameters of the FDMM.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
3
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Fig. 1. Schematic illustration of a loaded circular mat foundation resting on fractional derivative Maxwell clay deposits; the fractional derivative element is symbolised by a rhombus.
Z
3. Derivation of analytical solutions
wðrÞ ¼ q0 r 1
1
0
J 0 ðmrÞJ 1 ðmr1 Þ dm Df m4 þ E0 r 0 m
ð13Þ
3.1. Basic equations and elastic solutions Fig. 1 shows a circular mat foundation resting on a layer of clay of the fractional derivative Maxwell type, and also defines the coordinates. The classic equation from plate theory for deflection wðrÞ is
Df r2 r2 wðrÞ þ RðrÞ ¼ qðrÞ
ð5Þ
2
d 1 1 r ¼ 2þ r dr dr 2
ð6Þ
where RðrÞ is the reaction of the subgrade; qðrÞ is the applied load; and Df is the bending rigidity of a foundation defined as 3
Df ¼
Ef h 12ð1 l2 Þ
ð7Þ
where Ef , h and l are the elastic modulus, thickness and Poisson’s ratio of the foundation, respectively. First, a simple scenario is considered where for a Winkler type soil the reaction can be expressed as
@Sðr; zÞ RðrÞ ¼ E0 r 0 @z
ð8Þ
3.2. Fractional viscoelastic solution Considering that fractional models can be analysed using Laplace and Fourier transforms [28], here the fractional viscoelastic problem is treated in terms of the analogous elastic problem. Therefore, applying the Laplace transform to Eq. (13) yields
sÞ ¼ q0 r 1 wðr;
Z
1
0
J 0 ðmrÞJ 1 ðmr 1 Þ Df m4 þ E0 ðsÞr 0 m
/ðsÞdm
ð14Þ
where s is the parameter of the Laplace transform; and /ðsÞ is the Laplace-transformed function of /ðtÞ which determines the variation of surface load with time. Because the circular mat foundation and underlying clay interact according to FDMM, an analogous modulus is given with Eq. (4), by the expression
E0 ðsÞ ¼ E0
s0 sa 1 þ s0 sa
ð15Þ
Moreover, if the load q0 is applied quasi-statically then
/ðtÞ ¼ HðtÞ ¼
z¼0
d maxft; 0g dt
ð16Þ
where r 0 is the radius of the foundation, and Sðr; zÞ is the ground settlement which satisfies
/ðsÞ ¼
@ 2 S 1 @S @ 2 S þ þ ¼0 @r2 r @r @z2
By substituting Eqs. (15) and (17) into Eq. (14) and applying the inverse Laplace transform, the following equation is obtained
Sðr; zÞjz¼0 ¼ wðrÞ
ð9Þ ð10Þ
1 s
wðr; tÞ ¼
By considering a surface loading expressed in the form of a Bessel function of the first kind qðrÞ ¼ J 0 ðmrÞ; the solutions for reaction and deflection are derived as [41]
RðrÞ ¼
wðrÞ ¼
E0 r 0 mJ0 ðmrÞ Df m4 þ E0 r 0 m
ð11Þ
J 0 ðmrÞ Df m4 þ E0 r0 m
ð12Þ
Z q0 r1 1 J 0 ðmrÞJ 1 ðmr1 Þ m4 Df 0 a Df m3 =E0 r 0 1 t dm 1 E a;1 1 þ Df m3 =E0 r 0 1 þ Df m3 =E0 r0 s0 ð18Þ
where Ea;1 ðÞ is the generalised Mittag–Leffler function defined by
Ea;b ðtÞ ¼
Specifically, if the circular mat foundation is subjected to a uniformly distributed load q0 of radius r1 ðr 1 6 r0 Þ and is expressed in R1 the form qðrÞ ¼ q0 r1 0 J 0 ðmrÞJ 1 ðmr 1 Þdm; then Eq. (12) becomes [41]
ð17Þ
1 X
tn Cðan þ bÞ n¼0
ð19Þ
with Ea ðÞ ¼ Ea;1 ðÞ being the original Mittag–Leffler function. For convenience, the following non-dimensional parameters are introduced
q¼
r r1
ð20Þ
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
4
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Table 1 Parameters of the circular mat foundation in the numerical example.
R ðq; tÞ ¼
Radius r 0 (m)
Thickness h (m)
Poisson’s ratio
l
Bending rigidity Df (MPam3)
10
0.5
0.3
100
wðq; tÞ q0
M ðq; tÞ ¼
ð23Þ
wðq; tÞ q0 r 21
ð24Þ
So at this point, Eq. (18) becomes Table 2 The parameters of the clay and the load applied in the numerical example.
w ðq; tÞ ¼ c3
Z
1
0
Clay
(
Applied load
Elastic modulus E0 (MPam1)
Creep time (d)
10
3
r
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c¼p 3
Df =E0 r 0
s0
Intensity q0 (kPa)
Radius r 1 (m)
100
1
wðq; tÞ q0 r 1 =ðE0 r0 Þ
1
1
Ea;1 1 þ ðm=cÞ3
"
ðm=cÞ3 1 þ ðm=cÞ3
!
ta
s0
#) dm ð25Þ
ð21Þ
and the bending moment and the reaction are
M ðq; tÞ ¼
and variables
w ðq; tÞ ¼
J 0 ðmqÞJ 1 ðmÞ m4
ð22Þ
Z 1þl 1 1l J 1 ðmÞ J 0 ðmqÞ J 2 ðmqÞ 2 1 þ l m2 ½1 þ ðm=cÞ3 0 " ! # ðm=cÞ3 ta Ea;1 dm ð26Þ 3 1 þ ðm=cÞ s0
Fig. 2. Influence of fractional derivative order a on the deflection of the foundation.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
R ðq; tÞ ¼ c3
Z 0
1
J 0 ðmqÞJ 1 ðmÞ 1 þ ðm=cÞ3
" Ea;1
ðm=cÞ3 1 þ ðm=cÞ3
!
ta
s0
# dm
ð27Þ
With this procedure, an arbitrary load scheme /ðtÞ (e.g. load applied gradually over time and cyclic load) can be considered, so by applying the Laplace transform, the time dependency of the problem is removed (Eq. (14)) and a simple derivation procedure can be developed. 4. Study on fractional derivative order a through a numerical example 4.1. Numerical evaluation of analytical solutions The integral in Eq. (25) is evaluated numerically using an adaptive Gauss-Kronrod quadrature method, while the Mittag–Leffler function is computed numerically based on an algorithm reported by Gorenflo et al. [44] where arguments of different magnitudes
5
are evaluated using different techniques. The inverse power 1=m4 presented in the integrand helps the whole integral to converge rapidly. 4.2. Influence of the fractional derivative order a on the distributions of deflection and bending moment of foundations, and the reaction of the subgrade A numerical example is presented to elucidate the influence of the fractional derivative order a on the distributions of deflection, reaction, and the bending moment. The parameters of the circular mat foundation, the clay, and the load applied for the numerical example are summarised in Tables 1 and 2. In this case the nondimensional parameter c is 1, and the fractional derivative order a varies between 0 and 1. The computed results are depicted in Figs. 2–4. The time dependent response of a circular foundation is often solved using the Hankel transform which involves numerical inversion schemes [42], whereas Nassar’s method is based on the corresponding principle [41] so the derivation is more concise. The
Fig. 3. Influence of fractional derivative order a on the bending moment of the foundation.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
6
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
solutions are expressed in a compact form which is easy to calculate and implement, therefore Nassar’s solutions are used here to compare and validate the fractional derivative solutions. The results show that the two solutions are identical for a = 1 (Figs. 2 (a–c), 3(a–c) and 4(a–c)), which indicates that Nassar’s solutions should be a special case for the proposed solutions. Fig. 2 shows that deflection peaks at the centre of the foundation and then decreases towards the edge; this deflection is larger for high a values where the difference becomes more pronounced with elapsed time (Fig. 2(a–c)). For example, the peak deflection for a = 1 is 2.18 times larger than for a = 0.2 at 10 d, but the ratio increases to 8.25 at 100 d. However, the variation of deflection over time is not as evident at low a values than at high a values, as shown in Fig. 2(d–f); specifically, peak deflection increases by only 29.32% from 10 d to 100 d for a = 0.3, but the increment is as high as 151.88% for a = 0.7. These observations indicate that the deflection of a circular mat foundation and the corresponding ground settlement are more significant under high values of the fractional derivative order a, which implies that a is inherently linked to the time dependency
of ground deformation and is therefore consistent with the nature of the fractional derivative element introduced in Section 2.2. These trends generally hold for the bending moment of foundations and the reaction of the subgrade, except that the values of the two parameters decrease as a or time increases near the centre of the foundation (Figs. 3 and 4). 4.3. Influence of fractional derivative order a on the time dependent ground settlement Ground settlement induced by external loads is a primary concern in geotechnical engineering. According to Eq. (10), the influence of the fractional derivative order a on time dependent ground settlement at the centre of the foundation, denoted as S , is investigated in this section. Fig. 5 shows that different patterns of ground settlement can be simulated accurately by varying the values of a. One may argue that the decay shown in Fig. 5 can also be achieved by altering the coefficient of viscosity and the curves may be divided into two stages by the end of primary (EOP) consolidation time t EOP , so as a increases, the settlement in Stage I
Fig. 4. Influence of fractional derivative order a on the reaction of subgrade.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
decreases and increases in Stage II. For small a values, primary consolidation is quick and secondary consolidation is slow, whereas primary consolidation is slow and secondary consolidation is quick for large a values. These observations enable a to be introduced over a single coefficient of viscosity when dealing with the time dependency of ground settlement.
7
5. Analysis of a case history Common examples of a circular foundation and a clay interaction model are the foundations of cylindrical tanks and towers, so the ability of the proposed approach to simulate long term ground settlement is assessed here using field data. A case history documented by Law and Kasprzak [45] is simulated and analysed; it is concerned with a large water standpipe, known as the Accokeek standpipe, constructed on a circular mat foundation overlying Tertiary sediments on the east coast of the United States. This standpipe was located at 15580 Livingston Road, Accokeek, Prince George’s County, Maryland, and was one of the largest water tanks within Maryland’s Washington Suburban Sanitary District.
5.1. Background
Fig. 5. Influence of fractional derivative order a on time dependent ground settlement.
The layouts of the water standpipe and bronze bolts to measure settlement, as well as a simplified soil profile of the site, are shown in Fig. 6. The steel water standpipe is 36.58 m high by 22.86 in diameter, and can store about 14,000 m3 of water. The tank was constructed on a 25.91 m-diameter by 1.12 m-thick concrete mat foundation. The subsurface soil is about 31 m thick and consists of four generalised strata with a large amount of clay deposits. Details of the physical and mechanical properties of the strata are provided in Table 3. Stratum III contains medium to stiff silty clay, and has the lowest standard penetration test (SPT) blow counts (N-values) of the four strata. This indicates that Stratum III is relatively more compressible, and is the key stratum inducing long term settlement of the standpipe. The standpipe took almost 11 months to construct. Several bronze bolts were installed near the edge of the foundation to monitor long term settlement; 12 settlement data were obtained for a period of 480 d, and an additional measurement was collected at t = 7,975 d (about 22 yr).
Fig. 6. Layout of the Accokeek standpipe and settlement measurement points, and a simplified soil profile at this site (unit: m); the soil here contains large clay deposits which induced long term settlement of the standpipe.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
8
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Table 3 Physical and mechanical properties of four generalised soil strata underlying the Accokeek standpipe (after Ref. [45]). Stratum
Description
Thickness (m)
Natural water content (%)
N-valuea
I II III IV
Stiff clayey silt to hard clay Compacted silty sand, sand and gravel Medium to stiff silty clay Stiff silty clay to sandy clay
3.66 5.49 3.96 —
9–22 — 37–55 28–92
26–77b >30c 2–8 34d
Note: — indicates data not available. a Blow counts (reported in blows per foot) obtained from the standard penetration test (SPT). b Except for surface samples with lower N-values. c Except for silty samples. d Except for surface samples with a low N-value of 3.
5.2. Simulation The scenario described in Section 5.1 is treated as a problem between foundation and soil, where the circular mat foundation is underlain by ground soil and is subjected to loads from the water standpipe; this means the proposed model can deal with this problem. The radius r0 and the thickness h of the circular foundation, and the radius r 1 of the load are set as equal to the real dimensions, and since the foundation is made of concrete, Poisson’s ratio l and the elastic modulus E are estimated to be 0.2 and 40 GPa, respectively, and hence a bending rigidity Df of 4.85 GPam3 is obtained. The load q0 applied onto the circular foundation is assumed to be 300 kPa by considering the construction process and working condition of the water standpipe. The elastic modulus E0 of the FDMM controls the initial ground settlement and is therefore determined using the initial field measurements. A creep time s0 and a fractional derivative order a are obtained by fitting the FDMM to the former 12 settlement data points using the least squares method, i.e.
Table 4 Parameters used to simulate the Accokeek standpipe constructed on a circular mat foundation overlying Tertiary sediments. Parameter
Unit
Value
r0 h
m m — GPa GPam3 kPa m MPam1 d —
12.95 1.12 0.2 40 4.85 300 11.43 30.23 0.40 0.169
l Ef Df q0 r1 E0
s0 a Note: — indicates data not available.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 FDMM Smeas Þ i¼1 ðSi i RMSE ¼ n
ð28Þ
where RMSE is short for root-mean-square error; n is the number of is the total settlement data points (in this case n equals 12); SFDMM i is the ith ith displacement calculated using the FDMM; and Smeas i displacement measured in the field. Here, Eq. (28) is minimised with respect to s0 and a using the Levenberg-Marquardt method in MATLABÒ. The resulting RMSE value is 6.002, which corresponds to an R-squared value of 0.841. Simulations are also performed to examine the distributions of deflection, the bending moment of the foundation, and the reaction of the subgrade. All the parameters used in the simulations are summarised in Table 4. Settlement measured at the edge of the foundation and simulations from the proposed model are shown in Fig. 7. The foundation had a highly time dependent settlement due to the clay strata, so overall, the proposed model fit the measured data quite well. Moreover, the settlement simulated at t = 7,975 d is 61.88 mm, which only deviates by 3.27% from the field measurement (59.92 mm); this result verifies our approach to model the interaction between the circular foundation and the clayey ground. The simulation also indicates that settlement in the first year accounts for over 60% of the settlement at t = 7,975 d, due to the sharp decrease in the increasing rate of ground settlement (Table 5, the rate of increase is less than 0.010 mm/d after 630 days). The simulated distributions of deflection, the bending moment of the foundation, and the reaction of the subgrade are shown in Fig. 8. The foundation deflected more over time but at a decreasing rate due to creep in the ground soil (Fig. 8(a)). At each moment, deflection peaks at the centre of the foundation and decreases towards the side at the maximum rate seen at the edge of the standpipe (Fig. 8(b)). The subgrade is strongly influenced by the creep of ground soil, but it maintains an approximately parabolic shape (Fig. 8(d)). Near the edge of the foundation the reaction remains constant, but it decreases within the foundation at a decreasing rate over time and increases with time outside the foundation. This trend holds for the bending moment of the foundation (Fig. 8(c)). 6. Discussion 6.1. Implications for designing shallow foundations A mat or raft is a typical shallow foundation that supports tower shaped structures or storage tanks (e.g. the Accokeek standpipe
Table 5 The simulated increasing rate of ground settlement of the Accokeek standpipe. Fig. 7. Measured and simulated settlement at the edge of the Accokeek standpipe foundation.
Day (d) Value (mm/d)
5 3.373
10 0.414
30 0.136
90 0.052
630 0.010
1450 0.005
6160 0.001
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
9
Fig. 8. Simulated deflections and bending moments of the foundation, and the reaction of the subgrade of the Accokeek standpipe.
described in this study). Mat foundations are designed to be rigid and non-rigid; in the former the subgrade is assumed to be distributed uniformly while the latter considers a non-uniform pattern by analysing the foundation and soil interaction [46]. Unlike traditional non-rigid methods such as the Winkler and coupled methods [47], the time dependent interaction in this model between the soil and the structure is based on fractional derivatives. The nonlinear and time dependent distribution of deflected foundations and subgrade reaction can be simulated using this model (Figs. 2, 4 and 8(a, d)), due to the finite rigidity of the foundation and the creep of ground soil. The structural continuity and flexural strength of mat foundations needs special attention, especially those constructed on clayey soils undergoing long term deformation. Foundation settlement due to creep in the underlying clay deposits is of concern in foundation design, so to avoid catastrophic failure its deflection should not exceed the maximum allowable value [46]. Since the foundation of the Accokeek standpipe was partially loaded the simulation revealed some differential that would become more pronounced over time (Fig. 8(a)), and might threaten its serviceability over time. The simulation also shows that if the thickness, Poisson’s ratio or elastic modulus of the foundation had decreased, settlement could also have been decreased (Fig. 9). In fact after 20 years’ service the settlement of this would have decreased by 49.53% or 21.15%, respectively if the thickness or elastic modulus had increased by 25% (Fig. 9(a, c)). Comparatively, the influence of Poisson’s ratio on settlement is less evident (Fig. 9(b)).
6.2. Towards predicting ground settlement Fractional derivative models can describe the time dependent deformation of geomaterials accurately, as shown in Fig. 7 and as demonstrated by some recent studies on various types of soil [32,34,37,48]; this is attributed to introducing the fractional derivative order a, to traditional models. The numerical simulation by Wang et al. [36] shows that increasing the fractional derivative order hastens the settlement of a saturated soil layer; the coefficient of viscosity appears to have the same effect. However, our simulation shows that the fractional derivative order a is associated with the decay of settlement curves and it also reflects the primary and secondary consolidation processes of ground soil (Fig. 5, Stages I and II). With large a values, primary consolidation is slow (corresponding to a small coefficient of primary consolidation C v ), while secondary consolidation is quick (corresponding to a large coefficient of secondary consolidation C a ). This observation is consistent with the one dimensional oedometer tests on clays reported by Cosenza and Korošak [34] and Zhu et al. [48], which were carried out within the framework of Terzaghi’s theory and viscoelasticity. In this context the fractional derivative order a is related to the consolidation properties of clay and it is therefore a potential quantitative estimator of the amplitude of consolidation. Unlike previous studies on sampled soils which incorporated the concept of fractional calculus [32,34,37,48], this study is a first attempt to model the time dependent settlement of ground soil in the field (Section 5, the Accokeek standpipe case study). This
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
10
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Fig. 9. Influence of thickness h, Poisson’s ratio l, and the elastic modulus E on the simulated settlement of the Accokeek standpipe.
approach can simulate ground settlement accurately by visualising the soil in graphical forms and also consider the interaction between foundations and soil. Ground soil only has three parameters: E0 controls instantaneous settlement, s0 (or g0 ) affects the viscous response, and a is concerned with primary and secondary consolidation, but there are still several limitations. First, since formulation and calculation is complex, we used a two-component model (i.e. the fractional derivative Maxwell model) when modelling the interaction between a foundation and clay deposits within the framework of viscoelasticity, but since soil plasticity is not considered this model may be better for over consolidated rather than normally consolidated clay. This model is applicable to the Accokeek standpipe case study because the sediments are primarily stiff clays (Fig. 6) and the fitted results are satisfactory (Fig. 7). However, more refined constitutive models such as the Bingham model where plastic strains are considered, and general stress–strain–time models may be needed for soft clays and to deal with complex boundary value problems. Although the former two parameters can be determined using laboratory or in-situ tests [49,50], the fractional derivative order a cannot be obtained directly at this stage, but since this parameter is correlated with C v , C a and tEOP , it may be determined through ordinary oedometer tests. This means that further theoretical and experimental studies are needed to elucidate this point.
ing the reaction of the subgrade, was obtained in terms of the Mittag–Leffler function for an axisymmetric vertical load scheme. Analytical solutions involving integrals of the Mittag–Leffler function and the Bessel function were evaluated numerically. Comparisons with Nassar’s solutions show that the proposed solutions are more general and Nassar’s solutions are a special case when a = 1. Therefore, a comprehensive numerical example is used to examine how the fractional derivative order a influences the long term performance of foundations and time dependent ground settlement. There are two distinct stages in the primary and secondary consolidation of clay with regards to the settlement curves, both of which indicate that ground settlement can be simulated more accurately by introducing a rather than a single coefficient of viscosity. A case history of a large water standpipe constructed on a circular mat foundation overlying Tertiary sediments was simulated and analysed by the proposed model. This proposed model fitted the field measurements very well and successfully modelled the interaction between a circular foundation and clayey ground. The distributions of foundation deflection, bending moment, and subgrade reaction, which were not measured and documented, were also simulated. The implications resulting from this research for designing shallow foundations and predicting ground settlement were discussed, with the conclusion being that the model based on fractional derivatives has enormous potential for modelling and predicting time dependent ground settlement.
7. Summary and conclusions
Acknowledgments
In this study the interaction between a circular mat foundation and clay, and the associated ground settlement were investigated using a fractional derivative Maxwell model. By using the correspondence principle and the Laplace transform, the fractional viscoelastic problem is treated as an analogous elastic problem where a complex load history can be considered. A closed form solution for the deflection and bending moment of a foundation, includ-
The authors would like to thank Su-Ping Liu (Nanjing University, China) for his assistance in preparing the manuscript. This work was supported by the National Natural Science Foundation of China (Grant Nos. 41672277 and 41722209), the Natural Science Foundation of Jiangsu Province (Grant No. BK20161238), and the Fundamental Research Funds for the Central Universities (Grant No. 020614380050).
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
References [1] Jahangir E, Deck O, Masrouri F. Estimation of ground settlement beneath foundations due to shrinkage of clayey soils. Can Geotech J 2012;49:835–52. http://dx.doi.org/10.1139/t2012-042. [2] Naseri M, Hosseininia ES. Elastic settlement of ring foundations. Soils Found 2015;55:284–95. http://dx.doi.org/10.1016/j.sandf.2015.02.005. [3] Campolunghi MP, Capelli G, Funiciello R, Lanzini M. Geotechnical studies for foundation settlement in Holocenic alluvial deposits in the City of Rome (Italy). Eng Geol 2007;89:9–35. http://dx.doi.org/10.1016/j.enggeo.2006.08.003. [4] Zhao Z, Li D, Zhang F, Qiu Y. Ultimate lateral bearing capacity of tetrapod jacket foundation in clay. Comput Geotech 2017;84:164–73. http://dx.doi.org/ 10.1016/j.compgeo.2016.12.005. [5] Fu D, Gaudin C, Tian Y, Cassidy MJ, Bienen B. Uniaxial capacities of skirted circular foundations in clay. J Geotech Geoenviron Eng 2017;143:4017022. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001685. [6] Mullapudi R, Ayoub A. Nonlinear finite element modeling of beams on twoparameter foundations. Comput Geotech 2010;37:334–42. http://dx.doi.org/ 10.1016/j.compgeo.2009.11.006. [7] Hossain MS, Hu Y, Ekaputra D. Skirted foundation to mitigate spudcan punchthrough on sand-over-clay. Géotechnique 2014;64:333–40. http://dx.doi.org/ 10.1680/geot.13.T.027. [8] Chen SS, Liao KH, Shi JY. A dimensionless parametric study for forced vibrations of foundation–soil systems. Comput Geotech 2016;76:184–93. http://dx.doi.org/10.1016/j.compgeo. 2016.03.012. [9] Dunne HP, Martin CM. Capacity of rectangular mudmat foundations on clay under combined loading. Géotechnique 2017;67:168–80. http://dx.doi.org/ 10.1680/jgeot.16.P.079. [10] Yu H, Cai C, Yuan Y, Jia M. Analytical solutions for Euler-Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads. Int J Numer Anal Methods Geomech 2017;41:1125–37. http://dx.doi.org/10.1002/nag.2672. [11] Dumir PC. Circular plates on Pasternak elastic foundations. Int J Numer Anal Methods Geomech 1987;11:51–60. http://dx.doi.org/ 10.1002/nag.1610110105. [12] Gucunski N. Rocking response of flexible circular foundations on layered media. Soil Dyn Earthq Eng 1996;15:485–97. http://dx.doi.org/10.1016/ S0267-7261(96)00029-2. [13] Buczkowski R, Torbacki W. Finite element modelling of thick plates on twoparameter elastic foundation. Int J Numer Anal Methods Geomech 2001;25:1409–27. http://dx.doi.org/10.1002/nag.187. [14] Wang YH, Cheung YK. Plate on cross-anisotropic foundation analyzed by the finite element method. Comput Geotech 2001;28:37–54. http://dx.doi.org/ 10.1016/S0266-352X(00)00018-5. [15] Wang YH, Tham LG, Tsui Y, Yue ZQ. Plate on layered foundation analyzed by a semi-analytical and semi-numerical method. Comput Geotech 2003;30:409–18. http://dx.doi.org/10.1016/S0266-352X(03)00014-4. [16] Zhou D, Lo SH, Au FTK, Cheung YK. Three-dimensional free vibration of thick circular plates on Pasternak foundation. J Sound Vib 2006;292:726–41. http:// dx.doi.org/10.1016/j.jsv.2005.08.028. [17] Maheshwari P, Khatri S. Response of infinite beams on geosynthetic-reinforced granular bed over soft soil with stone columns under moving loads. Int J Geomech 2013;13:713–28. http://dx.doi.org/10.1061/(ASCE)GM.19435622.0000269. [18] Qin A, Sun D, Zhang J. Semi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soils. Comput Geotech 2014;62:110–7. http://dx.doi.org/10.1016/j.compgeo.2014.06.014. [19] Ai ZY, Cheng YC, Cao GJ. A quasistatic analysis of a plate on consolidating layered soils by analytical layer-element/finite element method coupling. Int J Numer Anal Methods Geomech 2014;38:1362–80. http://dx.doi.org/ 10.1002/nag.2261. [20] Kim J, Selvadurai APS. A note on the consolidation settlement of a rigid circular foundation on a poroelastic halfspace. Int J Numer Anal Methods Geomech 2016;40:2003–16. http://dx.doi.org/10.1002/nag.2519. [21] Liingaard M, Augustesen A, Lade PV. Characterization of models for timedependent behavior of soils. Int J Geomech 2004;4:157–77. http://dx.doi.org/ 10.1061/(ASCE)1532-3641(2004) 4:3(157). [22] Mainardi F. An historical perspective on fractional calculus in linear viscoelasticity. Fract Calc Appl Anal 2012;15:712–7. http://dx.doi.org/ 10.2478/s13540-012-0048-6. [23] Gemant A. A method of analyzing experimental results obtained from elastoviscous bodies. J Appl Phys 1936;7:311–7. http://dx.doi.org/10.1063/ 1.1745400.
11
[24] Smit W, de Vries H. Rheological models containing fractional derivatives. Rheol Acta 1970;9:525–34. http://dx.doi.org/10.1007/BF01985463. [25] Bagley RL, Torvik PJ. A generalized derivative model for an elastomer damper. Shock Vib Bull 1979;49:135–43. [26] Koeller RC. Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 1984;51:299–307. http://dx.doi.org/10.1115/1.3167616. [27] Welch SwJ, Rorrer RAL, Duren RG. Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech Time-Dependent Mater 1999;3:279–303. http://dx.doi.org/10.1023/ A:1009834317545. [28] Atanackovic TM, Stankovic B. Stability of an elastic rod on a fractional derivative type of foundation. J Sound Vib 2004;277:149–61. http://dx.doi.org/ 10.1016/j.jsv.2003.08.050. [29] Dikmen Ü. Modeling of seismic wave attenuation in soil structures using fractional derivative scheme. J Balk Geophys Soc 2005;8:175–88. [30] Yin DS, Ren JJ, He CL, Chen W. A new rheological model element for geomaterials. Chinese J Rock Mech Eng 2007;26:1899–903 [in Chinese with English abstract]. [31] Zhu H-H, Liu L-C, Pei H-F, Shi B. Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin-Voigt model. Geomech Eng 2012;4:67–78. [32] Yin D, Wu H, Cheng C, Chen Y. Fractional order constitutive model of geomaterials under the condition of triaxial test. Int J Numer Anal Methods Geomech 2013;37:961–72. http://dx.doi.org/10.1002/nag.2139. [33] Sun Y, Indraratna B, Carter JP, Marchant T, Nimbalkar S. Application of fractional calculus in modelling ballast deformation under cyclic loading. Comput Geotech 2017;82:16–30. http://dx.doi.org/10.1016/ j.compgeo.2016.09.010. [34] Cosenza P, Korošak D. Secondary consolidation of clay as an anomalous diffusion process. Int J Numer Anal Methods Geomech 2014;38:1231–46. http://dx.doi.org/10.1002/nag.2256. [35] Sun Y, Shen Y. Constitutive model of granular soils using fractional-order plastic-flow rule. Int J Geomech 2017;17:4017025. http://dx.doi.org/10.1061/ (ASCE)GM.1943-5622.0000904. [36] Wang L, Sun D, Li P, Xie Y. Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils. Comput Geotech 2017;83:30–9. http://dx.doi.org/10.1016/j.compgeo.2016.10.020. [37] Liao M, Lai Y, Liu E, Wan X. A fractional order creep constitutive model of warm frozen silt. Acta Geotech 2017;12:377–89. http://dx.doi.org/10.1007/s11440016-0466-4. [38] Zhou HW, Wang CP, Han BB, Duan ZQ. A creep constitutive model for salt rock based on fractional derivatives. Int J Rock Mech Min Sci 2011;48:116–21. http://dx.doi.org/10.1016/j.ijrmms.2010.11.004. [39] Zhang C, Zhu H, Shi B, Liu L. Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation. J Rock Mech Geotech Eng 2014;6:373–9. http://dx.doi.org/10.1016/j.jrmge.2014.04.007. [40] Cai W, Chen W, Xu W. Fractional modeling of Pasternak-type viscoelastic foundation. Mech Time-Dependent Mater 2017;21:119–31. http://dx.doi.org/ 10.1007/s11043-016-9321-0. [41] Nassar M. Bending of a circular plate on a linear viscoelastic foundation. Appl Math Model 1981;5:60–2. http://dx.doi.org/10.1016/0307-904X(81)90062-7. [42] Senjuntichai T, Sapsathiarn Y. Time-dependent response of circular plate in multi-layered poroelastic medium. Comput Geotech 2006;33:155–66. http:// dx.doi.org/10.1016/j.compgeo.2006. 03.005. [43] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, USA: Wiley-Interscience; 1993. [44] Gorenflo R, Loutchko J, Luchko Y. Computation of the Mittag-Leffler function Ea, b (z) and its derivative. Fract Calc Appl Anal 2002;5:491–518. [45] Law S, Kasprzak J. Long-term settlement of a large water standpipe. Water Eng Manage 2001:28–32. [46] Coduto DP. Foundation Design: Principles and Practices. second ed. New Jersey, USA: Prentice Hall; 2001. [47] ACI. Suggested analysis and design procedures for combined footings and mats. ACI 336.2R-88, American Concrete Institute; 2002. [48] Zhu HH, Zhang CC, Mei GX, Shi B, Gao L. Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives. Mar Georesources Geotechnol 2017;35:688–97. http://dx.doi.org/10.1080/ 1064119X.2016.1217958. [49] Schmertmann JH. Static cone to compute static settlement over sand. J Soil Mech Found Div 1970;96:1011–43. [50] Wong RCK, Varatharajan S. Viscous behaviour of clays in one-dimensional compression. Can Geotech J 2014;51:795–809. http://dx.doi.org/10.1139/cgj2013-0198.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
A long term evaluation of circular mat foundations on clay deposits using fractional derivatives Cheng-Cheng Zhang a, Hong-Hu Zhu a,b,⇑, Bin Shi a, Behzad Fatahi c a
School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China Nanjing University High-Tech Institute at Suzhou, Suzhou 215123, China c School of Civil and Environmental Engineering, University of Technology Sydney (UTS), Sydney, New South Wales 2007, Australia b
a r t i c l e
i n f o
Article history: Received 7 June 2017 Received in revised form 31 July 2017 Accepted 24 August 2017 Available online xxxx Keywords: Ground settlement Clay deposit Foundation analysis Fractional calculus Time dependent behaviour Settlement prediction
a b s t r a c t This study proposes to use fractional derivatives to evaluate the long term performance of circular mat foundations overlying clays and also predict the associated ground settlement. Closed form solutions for the deflection and bending moment of foundations and the subsequent reaction of subgrade are obtained with the Mittag–Leffler function. Numerical examples are used to determine how the fractional order affects the time dependent properties of the foundation and ground settlement, and to simulate the case history of a large standpipe constructed over Tertiary sediments. New insights into design and prediction of shallow foundations and ground settlement are also discussed. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The performance of foundations constructed over clay deposits and the associated ground settlement are of great concern for geoengineers [1–5] because the interaction between foundations and soil has hitherto been considered as a plate resting on a Winker or an elastic medium with two parameters. These early contributions did provide a sound theoretical basis for later study where complex loading histories and foundations with different shapes are considered [6–10], analytical and numerical methods such as the point collocation method, the Hankel transform, the ring method, the Chebyshev polynomial–Ritz method, and the finite element method have also been proposed and utilised [11–16]. However, since the deformation of clay is time dependent, it should be considered when modelling the interaction between soil and structure. For decades, the performance of a foundation on a viscoelastic medium [17,18] or a poroelastic medium [19,20] has been investigated, which is why the theory of poroelasticity, proposed originally by Terzaghi and further developed by Biot, enables the dissipation of pore water pressure and deformation of the solid skeleton under external loads to be considered. While simple and ⇑ Corresponding author at: School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China. E-mail address: [email protected] (H.-H. Zhu).
convenient simulations of the viscous behaviour of clay using viscoelastic models (e.g. Maxwell model, Kelvin–Voigt model and Merchant model) can graphically visualise materials, they are still limited by the simple constitutive relationship of the Newtonian dashpot [21]. Since fractional calculus is where integrals and derivatives can be of arbitrary non-integer orders and it can simulate hereditary phenomena with a long memory [22], it was introduced to viscoelasticity in the first half of the 20th century [23], and since then, models based on fractional derivatives have been utilised extensively to describe the viscous properties of materials [24–27]. The concept of fractional calculus has been introduced to geotechnics [28–33] where the viscous behaviour of geomaterials such as clays [34], granular soils [35], saturated soils [36], frozen silts [37], and rocks [38], has recently been modelled. Interesting work by Cosenza and Korošak shows that the secondary consolidation of clay can be described using a time-fractional diffusion equation within the framework of Terzaghi’s theory [34], but until now, research on the interaction between foundation and soil while incorporating fractional calculus is still limited [39,40]. While the time dependent response of a circular foundation has been investigated by assuming that the underlying soil is either viscoelastic [41] or poroelastic [42], the interaction between a circular foundation and a fractional type of viscoelastic soil has never been reported in literature.
http://dx.doi.org/10.1016/j.compgeo.2017.08.018 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
2
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Nomenclature Ca Cv D RL D Df EOP E E0 Ef E0 Ea ðÞ Ea;1 ðÞ FDMM HðÞ h J i ðÞ M q0 RMSE R R r r0 r1 SMM
coefficient of secondary consolidation coefficient of primary consolidation derivative Riemann–Liouville fractional derivative bending rigidity of foundation end of primary elastic modulus of fractional derivative element elastic modulus of fractional derivative Maxwell clay elastic modulus of foundation analogous modulus of fractional derivative Maxwell model original Mittag–Leffler function generalised Mittag–Leffler function with one complex parameter equals 1 Fractional derivative Maxwell model Heaviside step function thickness of foundation Bessel function of the first kind of order i non-dimensional moment of foundation load root-mean-square error reaction of subgrade non-dimensional reaction of subgrade radius radius of foundation radius of load standard Maxwell model
In this paper a fractional derivative Maxwell model is used to investigate the interaction between a circular mat foundation and clay deposits, and the associated ground settlement. First, analytical solutions for the deflection and bending moment of foundations and the reaction of subgrade are derived using the corresponding principle and the Laplace transform for an axisymmetric loading history. The solutions are then solved numerically and a comprehensive numerical example is given to show how the fractional derivative order affects the long term performance of foundations and time dependent ground settlement. The model’s ability to simulate the interaction between a circular foundation and clayey ground is demonstrated by analysing the case history of a large water standpipe constructed on a circular mat foundation overlying Tertiary deposits. Finally, the implications of this research into the design of shallow foundations and predicting settlement are discussed.
2.1. Basic concepts in fractional calculus In traditional calculus the nth derivative of a function z(t) is defined as Dnz(t) = dnz(t)/dtn, but this function is extended if the integer order n is substituted by a fraction; this is the starting point for defining fractional calculus. In this study the Riemann–Liouville definitions of fractional integration and derivative are used; the Riemann–Liouville fractional integration of function z(t) is given by [43] RL
Dt v zðtÞ ¼
1 Cðv Þ
Z
t
ðt nÞv 1 zðnÞdn;
t>0
ground settlement ith displacement calculated using FDMM
Smeas i s t t EOP w w w z
ith displacement measured in field parameter of Laplace transform time EOP consolidation time deflection of foundation non-dimensional deflection of foundation Laplace-transformed deflection of foundation depth order of fractional derivative smallest integer exceeding a Euler gamma function non-dimensional parameter strain coefficient of viscosity of fractional derivative element coefficient of viscosity of fractional derivative Maxwell clay Poisson’s ratio of foundation order of fractional integration non-dimensional radius stress creep time of fractional derivative element creep time of fractional derivative Maxwell clay function determining variation of load with time Laplace-transformed function of /ðtÞ
a
[a] CðÞ
c e g g0
l
v
q r s s0
/ðtÞ /ðsÞ
The Riemann–Liouville fractional derivative of order a is formulated as [43] 0
RL
Dat zðtÞ ¼ 0 RL Dt½a ½0 RL Dt v zðtÞ;
ð1Þ
0
where D denotes derivation; v is the order of integration ranging from 0 to 1; and the subscripts 0 and t denote the integration limits.
t>0
ð2Þ
where [a] is the smallest integer exceeding a; and v = [a]a > 0. This definition indicates that the fractional derivative has a strong inherent memory due to its integral form. For simplicity, the superscript ‘‘RL” and the subscripts 0 and t are dropped in the following sections. 2.2. Generalisation of the standard Maxwell model based on fractional derivatives A fractional derivative element known as the ‘‘intermediate model” [25] or ‘‘spring-pot” [27], is introduced first. The constitutive law of this element is defined as
rðtÞ ¼ Esa Da eðtÞ; 0 6 a 6 1
2. Fractional derivative Maxwell model (FDMM)
0
S SFDMM i
ð3Þ
where r and e are stress and strain; E is the elastic modulus; s is the creep time and equals g=E; and g is the coefficient of viscosity. Because the element collapses into a spring and a dashpot when a equals 0 and 1 respectively, the order of the fractional derivative a is a non-dimensional parameter associated with the memory of materials [27]. A standard Maxwell model (SMM) consists of a spring and a dashpot connected in series, but by replacing the dashpot with a fractional derivative element, the standard Maxwell model is upgraded to an FDMM (Fig. 1), whose stress–strain relationship reads
ðDa þ 1=sa0 ÞrðtÞ ¼ E0 Da eðtÞ
ð4Þ
where E0 , s0 and a are three independent parameters of the FDMM.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
3
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Fig. 1. Schematic illustration of a loaded circular mat foundation resting on fractional derivative Maxwell clay deposits; the fractional derivative element is symbolised by a rhombus.
Z
3. Derivation of analytical solutions
wðrÞ ¼ q0 r 1
1
0
J 0 ðmrÞJ 1 ðmr1 Þ dm Df m4 þ E0 r 0 m
ð13Þ
3.1. Basic equations and elastic solutions Fig. 1 shows a circular mat foundation resting on a layer of clay of the fractional derivative Maxwell type, and also defines the coordinates. The classic equation from plate theory for deflection wðrÞ is
Df r2 r2 wðrÞ þ RðrÞ ¼ qðrÞ
ð5Þ
2
d 1 1 r ¼ 2þ r dr dr 2
ð6Þ
where RðrÞ is the reaction of the subgrade; qðrÞ is the applied load; and Df is the bending rigidity of a foundation defined as 3
Df ¼
Ef h 12ð1 l2 Þ
ð7Þ
where Ef , h and l are the elastic modulus, thickness and Poisson’s ratio of the foundation, respectively. First, a simple scenario is considered where for a Winkler type soil the reaction can be expressed as
@Sðr; zÞ RðrÞ ¼ E0 r 0 @z
ð8Þ
3.2. Fractional viscoelastic solution Considering that fractional models can be analysed using Laplace and Fourier transforms [28], here the fractional viscoelastic problem is treated in terms of the analogous elastic problem. Therefore, applying the Laplace transform to Eq. (13) yields
sÞ ¼ q0 r 1 wðr;
Z
1
0
J 0 ðmrÞJ 1 ðmr 1 Þ Df m4 þ E0 ðsÞr 0 m
/ðsÞdm
ð14Þ
where s is the parameter of the Laplace transform; and /ðsÞ is the Laplace-transformed function of /ðtÞ which determines the variation of surface load with time. Because the circular mat foundation and underlying clay interact according to FDMM, an analogous modulus is given with Eq. (4), by the expression
E0 ðsÞ ¼ E0
s0 sa 1 þ s0 sa
ð15Þ
Moreover, if the load q0 is applied quasi-statically then
/ðtÞ ¼ HðtÞ ¼
z¼0
d maxft; 0g dt
ð16Þ
where r 0 is the radius of the foundation, and Sðr; zÞ is the ground settlement which satisfies
/ðsÞ ¼
@ 2 S 1 @S @ 2 S þ þ ¼0 @r2 r @r @z2
By substituting Eqs. (15) and (17) into Eq. (14) and applying the inverse Laplace transform, the following equation is obtained
Sðr; zÞjz¼0 ¼ wðrÞ
ð9Þ ð10Þ
1 s
wðr; tÞ ¼
By considering a surface loading expressed in the form of a Bessel function of the first kind qðrÞ ¼ J 0 ðmrÞ; the solutions for reaction and deflection are derived as [41]
RðrÞ ¼
wðrÞ ¼
E0 r 0 mJ0 ðmrÞ Df m4 þ E0 r 0 m
ð11Þ
J 0 ðmrÞ Df m4 þ E0 r0 m
ð12Þ
Z q0 r1 1 J 0 ðmrÞJ 1 ðmr1 Þ m4 Df 0 a Df m3 =E0 r 0 1 t dm 1 E a;1 1 þ Df m3 =E0 r 0 1 þ Df m3 =E0 r0 s0 ð18Þ
where Ea;1 ðÞ is the generalised Mittag–Leffler function defined by
Ea;b ðtÞ ¼
Specifically, if the circular mat foundation is subjected to a uniformly distributed load q0 of radius r1 ðr 1 6 r0 Þ and is expressed in R1 the form qðrÞ ¼ q0 r1 0 J 0 ðmrÞJ 1 ðmr 1 Þdm; then Eq. (12) becomes [41]
ð17Þ
1 X
tn Cðan þ bÞ n¼0
ð19Þ
with Ea ðÞ ¼ Ea;1 ðÞ being the original Mittag–Leffler function. For convenience, the following non-dimensional parameters are introduced
q¼
r r1
ð20Þ
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
4
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Table 1 Parameters of the circular mat foundation in the numerical example.
R ðq; tÞ ¼
Radius r 0 (m)
Thickness h (m)
Poisson’s ratio
l
Bending rigidity Df (MPam3)
10
0.5
0.3
100
wðq; tÞ q0
M ðq; tÞ ¼
ð23Þ
wðq; tÞ q0 r 21
ð24Þ
So at this point, Eq. (18) becomes Table 2 The parameters of the clay and the load applied in the numerical example.
w ðq; tÞ ¼ c3
Z
1
0
Clay
(
Applied load
Elastic modulus E0 (MPam1)
Creep time (d)
10
3
r
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c¼p 3
Df =E0 r 0
s0
Intensity q0 (kPa)
Radius r 1 (m)
100
1
wðq; tÞ q0 r 1 =ðE0 r0 Þ
1
1
Ea;1 1 þ ðm=cÞ3
"
ðm=cÞ3 1 þ ðm=cÞ3
!
ta
s0
#) dm ð25Þ
ð21Þ
and the bending moment and the reaction are
M ðq; tÞ ¼
and variables
w ðq; tÞ ¼
J 0 ðmqÞJ 1 ðmÞ m4
ð22Þ
Z 1þl 1 1l J 1 ðmÞ J 0 ðmqÞ J 2 ðmqÞ 2 1 þ l m2 ½1 þ ðm=cÞ3 0 " ! # ðm=cÞ3 ta Ea;1 dm ð26Þ 3 1 þ ðm=cÞ s0
Fig. 2. Influence of fractional derivative order a on the deflection of the foundation.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
R ðq; tÞ ¼ c3
Z 0
1
J 0 ðmqÞJ 1 ðmÞ 1 þ ðm=cÞ3
" Ea;1
ðm=cÞ3 1 þ ðm=cÞ3
!
ta
s0
# dm
ð27Þ
With this procedure, an arbitrary load scheme /ðtÞ (e.g. load applied gradually over time and cyclic load) can be considered, so by applying the Laplace transform, the time dependency of the problem is removed (Eq. (14)) and a simple derivation procedure can be developed. 4. Study on fractional derivative order a through a numerical example 4.1. Numerical evaluation of analytical solutions The integral in Eq. (25) is evaluated numerically using an adaptive Gauss-Kronrod quadrature method, while the Mittag–Leffler function is computed numerically based on an algorithm reported by Gorenflo et al. [44] where arguments of different magnitudes
5
are evaluated using different techniques. The inverse power 1=m4 presented in the integrand helps the whole integral to converge rapidly. 4.2. Influence of the fractional derivative order a on the distributions of deflection and bending moment of foundations, and the reaction of the subgrade A numerical example is presented to elucidate the influence of the fractional derivative order a on the distributions of deflection, reaction, and the bending moment. The parameters of the circular mat foundation, the clay, and the load applied for the numerical example are summarised in Tables 1 and 2. In this case the nondimensional parameter c is 1, and the fractional derivative order a varies between 0 and 1. The computed results are depicted in Figs. 2–4. The time dependent response of a circular foundation is often solved using the Hankel transform which involves numerical inversion schemes [42], whereas Nassar’s method is based on the corresponding principle [41] so the derivation is more concise. The
Fig. 3. Influence of fractional derivative order a on the bending moment of the foundation.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
6
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
solutions are expressed in a compact form which is easy to calculate and implement, therefore Nassar’s solutions are used here to compare and validate the fractional derivative solutions. The results show that the two solutions are identical for a = 1 (Figs. 2 (a–c), 3(a–c) and 4(a–c)), which indicates that Nassar’s solutions should be a special case for the proposed solutions. Fig. 2 shows that deflection peaks at the centre of the foundation and then decreases towards the edge; this deflection is larger for high a values where the difference becomes more pronounced with elapsed time (Fig. 2(a–c)). For example, the peak deflection for a = 1 is 2.18 times larger than for a = 0.2 at 10 d, but the ratio increases to 8.25 at 100 d. However, the variation of deflection over time is not as evident at low a values than at high a values, as shown in Fig. 2(d–f); specifically, peak deflection increases by only 29.32% from 10 d to 100 d for a = 0.3, but the increment is as high as 151.88% for a = 0.7. These observations indicate that the deflection of a circular mat foundation and the corresponding ground settlement are more significant under high values of the fractional derivative order a, which implies that a is inherently linked to the time dependency
of ground deformation and is therefore consistent with the nature of the fractional derivative element introduced in Section 2.2. These trends generally hold for the bending moment of foundations and the reaction of the subgrade, except that the values of the two parameters decrease as a or time increases near the centre of the foundation (Figs. 3 and 4). 4.3. Influence of fractional derivative order a on the time dependent ground settlement Ground settlement induced by external loads is a primary concern in geotechnical engineering. According to Eq. (10), the influence of the fractional derivative order a on time dependent ground settlement at the centre of the foundation, denoted as S , is investigated in this section. Fig. 5 shows that different patterns of ground settlement can be simulated accurately by varying the values of a. One may argue that the decay shown in Fig. 5 can also be achieved by altering the coefficient of viscosity and the curves may be divided into two stages by the end of primary (EOP) consolidation time t EOP , so as a increases, the settlement in Stage I
Fig. 4. Influence of fractional derivative order a on the reaction of subgrade.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
decreases and increases in Stage II. For small a values, primary consolidation is quick and secondary consolidation is slow, whereas primary consolidation is slow and secondary consolidation is quick for large a values. These observations enable a to be introduced over a single coefficient of viscosity when dealing with the time dependency of ground settlement.
7
5. Analysis of a case history Common examples of a circular foundation and a clay interaction model are the foundations of cylindrical tanks and towers, so the ability of the proposed approach to simulate long term ground settlement is assessed here using field data. A case history documented by Law and Kasprzak [45] is simulated and analysed; it is concerned with a large water standpipe, known as the Accokeek standpipe, constructed on a circular mat foundation overlying Tertiary sediments on the east coast of the United States. This standpipe was located at 15580 Livingston Road, Accokeek, Prince George’s County, Maryland, and was one of the largest water tanks within Maryland’s Washington Suburban Sanitary District.
5.1. Background
Fig. 5. Influence of fractional derivative order a on time dependent ground settlement.
The layouts of the water standpipe and bronze bolts to measure settlement, as well as a simplified soil profile of the site, are shown in Fig. 6. The steel water standpipe is 36.58 m high by 22.86 in diameter, and can store about 14,000 m3 of water. The tank was constructed on a 25.91 m-diameter by 1.12 m-thick concrete mat foundation. The subsurface soil is about 31 m thick and consists of four generalised strata with a large amount of clay deposits. Details of the physical and mechanical properties of the strata are provided in Table 3. Stratum III contains medium to stiff silty clay, and has the lowest standard penetration test (SPT) blow counts (N-values) of the four strata. This indicates that Stratum III is relatively more compressible, and is the key stratum inducing long term settlement of the standpipe. The standpipe took almost 11 months to construct. Several bronze bolts were installed near the edge of the foundation to monitor long term settlement; 12 settlement data were obtained for a period of 480 d, and an additional measurement was collected at t = 7,975 d (about 22 yr).
Fig. 6. Layout of the Accokeek standpipe and settlement measurement points, and a simplified soil profile at this site (unit: m); the soil here contains large clay deposits which induced long term settlement of the standpipe.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
8
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Table 3 Physical and mechanical properties of four generalised soil strata underlying the Accokeek standpipe (after Ref. [45]). Stratum
Description
Thickness (m)
Natural water content (%)
N-valuea
I II III IV
Stiff clayey silt to hard clay Compacted silty sand, sand and gravel Medium to stiff silty clay Stiff silty clay to sandy clay
3.66 5.49 3.96 —
9–22 — 37–55 28–92
26–77b >30c 2–8 34d
Note: — indicates data not available. a Blow counts (reported in blows per foot) obtained from the standard penetration test (SPT). b Except for surface samples with lower N-values. c Except for silty samples. d Except for surface samples with a low N-value of 3.
5.2. Simulation The scenario described in Section 5.1 is treated as a problem between foundation and soil, where the circular mat foundation is underlain by ground soil and is subjected to loads from the water standpipe; this means the proposed model can deal with this problem. The radius r0 and the thickness h of the circular foundation, and the radius r 1 of the load are set as equal to the real dimensions, and since the foundation is made of concrete, Poisson’s ratio l and the elastic modulus E are estimated to be 0.2 and 40 GPa, respectively, and hence a bending rigidity Df of 4.85 GPam3 is obtained. The load q0 applied onto the circular foundation is assumed to be 300 kPa by considering the construction process and working condition of the water standpipe. The elastic modulus E0 of the FDMM controls the initial ground settlement and is therefore determined using the initial field measurements. A creep time s0 and a fractional derivative order a are obtained by fitting the FDMM to the former 12 settlement data points using the least squares method, i.e.
Table 4 Parameters used to simulate the Accokeek standpipe constructed on a circular mat foundation overlying Tertiary sediments. Parameter
Unit
Value
r0 h
m m — GPa GPam3 kPa m MPam1 d —
12.95 1.12 0.2 40 4.85 300 11.43 30.23 0.40 0.169
l Ef Df q0 r1 E0
s0 a Note: — indicates data not available.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 FDMM Smeas Þ i¼1 ðSi i RMSE ¼ n
ð28Þ
where RMSE is short for root-mean-square error; n is the number of is the total settlement data points (in this case n equals 12); SFDMM i is the ith ith displacement calculated using the FDMM; and Smeas i displacement measured in the field. Here, Eq. (28) is minimised with respect to s0 and a using the Levenberg-Marquardt method in MATLABÒ. The resulting RMSE value is 6.002, which corresponds to an R-squared value of 0.841. Simulations are also performed to examine the distributions of deflection, the bending moment of the foundation, and the reaction of the subgrade. All the parameters used in the simulations are summarised in Table 4. Settlement measured at the edge of the foundation and simulations from the proposed model are shown in Fig. 7. The foundation had a highly time dependent settlement due to the clay strata, so overall, the proposed model fit the measured data quite well. Moreover, the settlement simulated at t = 7,975 d is 61.88 mm, which only deviates by 3.27% from the field measurement (59.92 mm); this result verifies our approach to model the interaction between the circular foundation and the clayey ground. The simulation also indicates that settlement in the first year accounts for over 60% of the settlement at t = 7,975 d, due to the sharp decrease in the increasing rate of ground settlement (Table 5, the rate of increase is less than 0.010 mm/d after 630 days). The simulated distributions of deflection, the bending moment of the foundation, and the reaction of the subgrade are shown in Fig. 8. The foundation deflected more over time but at a decreasing rate due to creep in the ground soil (Fig. 8(a)). At each moment, deflection peaks at the centre of the foundation and decreases towards the side at the maximum rate seen at the edge of the standpipe (Fig. 8(b)). The subgrade is strongly influenced by the creep of ground soil, but it maintains an approximately parabolic shape (Fig. 8(d)). Near the edge of the foundation the reaction remains constant, but it decreases within the foundation at a decreasing rate over time and increases with time outside the foundation. This trend holds for the bending moment of the foundation (Fig. 8(c)). 6. Discussion 6.1. Implications for designing shallow foundations A mat or raft is a typical shallow foundation that supports tower shaped structures or storage tanks (e.g. the Accokeek standpipe
Table 5 The simulated increasing rate of ground settlement of the Accokeek standpipe. Fig. 7. Measured and simulated settlement at the edge of the Accokeek standpipe foundation.
Day (d) Value (mm/d)
5 3.373
10 0.414
30 0.136
90 0.052
630 0.010
1450 0.005
6160 0.001
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
9
Fig. 8. Simulated deflections and bending moments of the foundation, and the reaction of the subgrade of the Accokeek standpipe.
described in this study). Mat foundations are designed to be rigid and non-rigid; in the former the subgrade is assumed to be distributed uniformly while the latter considers a non-uniform pattern by analysing the foundation and soil interaction [46]. Unlike traditional non-rigid methods such as the Winkler and coupled methods [47], the time dependent interaction in this model between the soil and the structure is based on fractional derivatives. The nonlinear and time dependent distribution of deflected foundations and subgrade reaction can be simulated using this model (Figs. 2, 4 and 8(a, d)), due to the finite rigidity of the foundation and the creep of ground soil. The structural continuity and flexural strength of mat foundations needs special attention, especially those constructed on clayey soils undergoing long term deformation. Foundation settlement due to creep in the underlying clay deposits is of concern in foundation design, so to avoid catastrophic failure its deflection should not exceed the maximum allowable value [46]. Since the foundation of the Accokeek standpipe was partially loaded the simulation revealed some differential that would become more pronounced over time (Fig. 8(a)), and might threaten its serviceability over time. The simulation also shows that if the thickness, Poisson’s ratio or elastic modulus of the foundation had decreased, settlement could also have been decreased (Fig. 9). In fact after 20 years’ service the settlement of this would have decreased by 49.53% or 21.15%, respectively if the thickness or elastic modulus had increased by 25% (Fig. 9(a, c)). Comparatively, the influence of Poisson’s ratio on settlement is less evident (Fig. 9(b)).
6.2. Towards predicting ground settlement Fractional derivative models can describe the time dependent deformation of geomaterials accurately, as shown in Fig. 7 and as demonstrated by some recent studies on various types of soil [32,34,37,48]; this is attributed to introducing the fractional derivative order a, to traditional models. The numerical simulation by Wang et al. [36] shows that increasing the fractional derivative order hastens the settlement of a saturated soil layer; the coefficient of viscosity appears to have the same effect. However, our simulation shows that the fractional derivative order a is associated with the decay of settlement curves and it also reflects the primary and secondary consolidation processes of ground soil (Fig. 5, Stages I and II). With large a values, primary consolidation is slow (corresponding to a small coefficient of primary consolidation C v ), while secondary consolidation is quick (corresponding to a large coefficient of secondary consolidation C a ). This observation is consistent with the one dimensional oedometer tests on clays reported by Cosenza and Korošak [34] and Zhu et al. [48], which were carried out within the framework of Terzaghi’s theory and viscoelasticity. In this context the fractional derivative order a is related to the consolidation properties of clay and it is therefore a potential quantitative estimator of the amplitude of consolidation. Unlike previous studies on sampled soils which incorporated the concept of fractional calculus [32,34,37,48], this study is a first attempt to model the time dependent settlement of ground soil in the field (Section 5, the Accokeek standpipe case study). This
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
10
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
Fig. 9. Influence of thickness h, Poisson’s ratio l, and the elastic modulus E on the simulated settlement of the Accokeek standpipe.
approach can simulate ground settlement accurately by visualising the soil in graphical forms and also consider the interaction between foundations and soil. Ground soil only has three parameters: E0 controls instantaneous settlement, s0 (or g0 ) affects the viscous response, and a is concerned with primary and secondary consolidation, but there are still several limitations. First, since formulation and calculation is complex, we used a two-component model (i.e. the fractional derivative Maxwell model) when modelling the interaction between a foundation and clay deposits within the framework of viscoelasticity, but since soil plasticity is not considered this model may be better for over consolidated rather than normally consolidated clay. This model is applicable to the Accokeek standpipe case study because the sediments are primarily stiff clays (Fig. 6) and the fitted results are satisfactory (Fig. 7). However, more refined constitutive models such as the Bingham model where plastic strains are considered, and general stress–strain–time models may be needed for soft clays and to deal with complex boundary value problems. Although the former two parameters can be determined using laboratory or in-situ tests [49,50], the fractional derivative order a cannot be obtained directly at this stage, but since this parameter is correlated with C v , C a and tEOP , it may be determined through ordinary oedometer tests. This means that further theoretical and experimental studies are needed to elucidate this point.
ing the reaction of the subgrade, was obtained in terms of the Mittag–Leffler function for an axisymmetric vertical load scheme. Analytical solutions involving integrals of the Mittag–Leffler function and the Bessel function were evaluated numerically. Comparisons with Nassar’s solutions show that the proposed solutions are more general and Nassar’s solutions are a special case when a = 1. Therefore, a comprehensive numerical example is used to examine how the fractional derivative order a influences the long term performance of foundations and time dependent ground settlement. There are two distinct stages in the primary and secondary consolidation of clay with regards to the settlement curves, both of which indicate that ground settlement can be simulated more accurately by introducing a rather than a single coefficient of viscosity. A case history of a large water standpipe constructed on a circular mat foundation overlying Tertiary sediments was simulated and analysed by the proposed model. This proposed model fitted the field measurements very well and successfully modelled the interaction between a circular foundation and clayey ground. The distributions of foundation deflection, bending moment, and subgrade reaction, which were not measured and documented, were also simulated. The implications resulting from this research for designing shallow foundations and predicting ground settlement were discussed, with the conclusion being that the model based on fractional derivatives has enormous potential for modelling and predicting time dependent ground settlement.
7. Summary and conclusions
Acknowledgments
In this study the interaction between a circular mat foundation and clay, and the associated ground settlement were investigated using a fractional derivative Maxwell model. By using the correspondence principle and the Laplace transform, the fractional viscoelastic problem is treated as an analogous elastic problem where a complex load history can be considered. A closed form solution for the deflection and bending moment of a foundation, includ-
The authors would like to thank Su-Ping Liu (Nanjing University, China) for his assistance in preparing the manuscript. This work was supported by the National Natural Science Foundation of China (Grant Nos. 41672277 and 41722209), the Natural Science Foundation of Jiangsu Province (Grant No. BK20161238), and the Fundamental Research Funds for the Central Universities (Grant No. 020614380050).
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018
C.-C. Zhang et al. / Computers and Geotechnics xxx (2017) xxx–xxx
References [1] Jahangir E, Deck O, Masrouri F. Estimation of ground settlement beneath foundations due to shrinkage of clayey soils. Can Geotech J 2012;49:835–52. http://dx.doi.org/10.1139/t2012-042. [2] Naseri M, Hosseininia ES. Elastic settlement of ring foundations. Soils Found 2015;55:284–95. http://dx.doi.org/10.1016/j.sandf.2015.02.005. [3] Campolunghi MP, Capelli G, Funiciello R, Lanzini M. Geotechnical studies for foundation settlement in Holocenic alluvial deposits in the City of Rome (Italy). Eng Geol 2007;89:9–35. http://dx.doi.org/10.1016/j.enggeo.2006.08.003. [4] Zhao Z, Li D, Zhang F, Qiu Y. Ultimate lateral bearing capacity of tetrapod jacket foundation in clay. Comput Geotech 2017;84:164–73. http://dx.doi.org/ 10.1016/j.compgeo.2016.12.005. [5] Fu D, Gaudin C, Tian Y, Cassidy MJ, Bienen B. Uniaxial capacities of skirted circular foundations in clay. J Geotech Geoenviron Eng 2017;143:4017022. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001685. [6] Mullapudi R, Ayoub A. Nonlinear finite element modeling of beams on twoparameter foundations. Comput Geotech 2010;37:334–42. http://dx.doi.org/ 10.1016/j.compgeo.2009.11.006. [7] Hossain MS, Hu Y, Ekaputra D. Skirted foundation to mitigate spudcan punchthrough on sand-over-clay. Géotechnique 2014;64:333–40. http://dx.doi.org/ 10.1680/geot.13.T.027. [8] Chen SS, Liao KH, Shi JY. A dimensionless parametric study for forced vibrations of foundation–soil systems. Comput Geotech 2016;76:184–93. http://dx.doi.org/10.1016/j.compgeo. 2016.03.012. [9] Dunne HP, Martin CM. Capacity of rectangular mudmat foundations on clay under combined loading. Géotechnique 2017;67:168–80. http://dx.doi.org/ 10.1680/jgeot.16.P.079. [10] Yu H, Cai C, Yuan Y, Jia M. Analytical solutions for Euler-Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads. Int J Numer Anal Methods Geomech 2017;41:1125–37. http://dx.doi.org/10.1002/nag.2672. [11] Dumir PC. Circular plates on Pasternak elastic foundations. Int J Numer Anal Methods Geomech 1987;11:51–60. http://dx.doi.org/ 10.1002/nag.1610110105. [12] Gucunski N. Rocking response of flexible circular foundations on layered media. Soil Dyn Earthq Eng 1996;15:485–97. http://dx.doi.org/10.1016/ S0267-7261(96)00029-2. [13] Buczkowski R, Torbacki W. Finite element modelling of thick plates on twoparameter elastic foundation. Int J Numer Anal Methods Geomech 2001;25:1409–27. http://dx.doi.org/10.1002/nag.187. [14] Wang YH, Cheung YK. Plate on cross-anisotropic foundation analyzed by the finite element method. Comput Geotech 2001;28:37–54. http://dx.doi.org/ 10.1016/S0266-352X(00)00018-5. [15] Wang YH, Tham LG, Tsui Y, Yue ZQ. Plate on layered foundation analyzed by a semi-analytical and semi-numerical method. Comput Geotech 2003;30:409–18. http://dx.doi.org/10.1016/S0266-352X(03)00014-4. [16] Zhou D, Lo SH, Au FTK, Cheung YK. Three-dimensional free vibration of thick circular plates on Pasternak foundation. J Sound Vib 2006;292:726–41. http:// dx.doi.org/10.1016/j.jsv.2005.08.028. [17] Maheshwari P, Khatri S. Response of infinite beams on geosynthetic-reinforced granular bed over soft soil with stone columns under moving loads. Int J Geomech 2013;13:713–28. http://dx.doi.org/10.1061/(ASCE)GM.19435622.0000269. [18] Qin A, Sun D, Zhang J. Semi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soils. Comput Geotech 2014;62:110–7. http://dx.doi.org/10.1016/j.compgeo.2014.06.014. [19] Ai ZY, Cheng YC, Cao GJ. A quasistatic analysis of a plate on consolidating layered soils by analytical layer-element/finite element method coupling. Int J Numer Anal Methods Geomech 2014;38:1362–80. http://dx.doi.org/ 10.1002/nag.2261. [20] Kim J, Selvadurai APS. A note on the consolidation settlement of a rigid circular foundation on a poroelastic halfspace. Int J Numer Anal Methods Geomech 2016;40:2003–16. http://dx.doi.org/10.1002/nag.2519. [21] Liingaard M, Augustesen A, Lade PV. Characterization of models for timedependent behavior of soils. Int J Geomech 2004;4:157–77. http://dx.doi.org/ 10.1061/(ASCE)1532-3641(2004) 4:3(157). [22] Mainardi F. An historical perspective on fractional calculus in linear viscoelasticity. Fract Calc Appl Anal 2012;15:712–7. http://dx.doi.org/ 10.2478/s13540-012-0048-6. [23] Gemant A. A method of analyzing experimental results obtained from elastoviscous bodies. J Appl Phys 1936;7:311–7. http://dx.doi.org/10.1063/ 1.1745400.
11
[24] Smit W, de Vries H. Rheological models containing fractional derivatives. Rheol Acta 1970;9:525–34. http://dx.doi.org/10.1007/BF01985463. [25] Bagley RL, Torvik PJ. A generalized derivative model for an elastomer damper. Shock Vib Bull 1979;49:135–43. [26] Koeller RC. Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 1984;51:299–307. http://dx.doi.org/10.1115/1.3167616. [27] Welch SwJ, Rorrer RAL, Duren RG. Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech Time-Dependent Mater 1999;3:279–303. http://dx.doi.org/10.1023/ A:1009834317545. [28] Atanackovic TM, Stankovic B. Stability of an elastic rod on a fractional derivative type of foundation. J Sound Vib 2004;277:149–61. http://dx.doi.org/ 10.1016/j.jsv.2003.08.050. [29] Dikmen Ü. Modeling of seismic wave attenuation in soil structures using fractional derivative scheme. J Balk Geophys Soc 2005;8:175–88. [30] Yin DS, Ren JJ, He CL, Chen W. A new rheological model element for geomaterials. Chinese J Rock Mech Eng 2007;26:1899–903 [in Chinese with English abstract]. [31] Zhu H-H, Liu L-C, Pei H-F, Shi B. Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin-Voigt model. Geomech Eng 2012;4:67–78. [32] Yin D, Wu H, Cheng C, Chen Y. Fractional order constitutive model of geomaterials under the condition of triaxial test. Int J Numer Anal Methods Geomech 2013;37:961–72. http://dx.doi.org/10.1002/nag.2139. [33] Sun Y, Indraratna B, Carter JP, Marchant T, Nimbalkar S. Application of fractional calculus in modelling ballast deformation under cyclic loading. Comput Geotech 2017;82:16–30. http://dx.doi.org/10.1016/ j.compgeo.2016.09.010. [34] Cosenza P, Korošak D. Secondary consolidation of clay as an anomalous diffusion process. Int J Numer Anal Methods Geomech 2014;38:1231–46. http://dx.doi.org/10.1002/nag.2256. [35] Sun Y, Shen Y. Constitutive model of granular soils using fractional-order plastic-flow rule. Int J Geomech 2017;17:4017025. http://dx.doi.org/10.1061/ (ASCE)GM.1943-5622.0000904. [36] Wang L, Sun D, Li P, Xie Y. Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils. Comput Geotech 2017;83:30–9. http://dx.doi.org/10.1016/j.compgeo.2016.10.020. [37] Liao M, Lai Y, Liu E, Wan X. A fractional order creep constitutive model of warm frozen silt. Acta Geotech 2017;12:377–89. http://dx.doi.org/10.1007/s11440016-0466-4. [38] Zhou HW, Wang CP, Han BB, Duan ZQ. A creep constitutive model for salt rock based on fractional derivatives. Int J Rock Mech Min Sci 2011;48:116–21. http://dx.doi.org/10.1016/j.ijrmms.2010.11.004. [39] Zhang C, Zhu H, Shi B, Liu L. Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation. J Rock Mech Geotech Eng 2014;6:373–9. http://dx.doi.org/10.1016/j.jrmge.2014.04.007. [40] Cai W, Chen W, Xu W. Fractional modeling of Pasternak-type viscoelastic foundation. Mech Time-Dependent Mater 2017;21:119–31. http://dx.doi.org/ 10.1007/s11043-016-9321-0. [41] Nassar M. Bending of a circular plate on a linear viscoelastic foundation. Appl Math Model 1981;5:60–2. http://dx.doi.org/10.1016/0307-904X(81)90062-7. [42] Senjuntichai T, Sapsathiarn Y. Time-dependent response of circular plate in multi-layered poroelastic medium. Comput Geotech 2006;33:155–66. http:// dx.doi.org/10.1016/j.compgeo.2006. 03.005. [43] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, USA: Wiley-Interscience; 1993. [44] Gorenflo R, Loutchko J, Luchko Y. Computation of the Mittag-Leffler function Ea, b (z) and its derivative. Fract Calc Appl Anal 2002;5:491–518. [45] Law S, Kasprzak J. Long-term settlement of a large water standpipe. Water Eng Manage 2001:28–32. [46] Coduto DP. Foundation Design: Principles and Practices. second ed. New Jersey, USA: Prentice Hall; 2001. [47] ACI. Suggested analysis and design procedures for combined footings and mats. ACI 336.2R-88, American Concrete Institute; 2002. [48] Zhu HH, Zhang CC, Mei GX, Shi B, Gao L. Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives. Mar Georesources Geotechnol 2017;35:688–97. http://dx.doi.org/10.1080/ 1064119X.2016.1217958. [49] Schmertmann JH. Static cone to compute static settlement over sand. J Soil Mech Found Div 1970;96:1011–43. [50] Wong RCK, Varatharajan S. Viscous behaviour of clays in one-dimensional compression. Can Geotech J 2014;51:795–809. http://dx.doi.org/10.1139/cgj2013-0198.
Please cite this article in press as: Zhang C-C et al. A long term evaluation of circular mat foundations on clay deposits using fractional derivatives. Comput Geotech (2017), http://dx.doi.org/10.1016/j.compgeo.2017.08.018