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Section 4: Soil Mechanics Advances in Boundary Element Techniques V 209
Section 4: Soil Mechanics
Advances in Boundary Element Techniques V 209
210
Dynamic Response of Single Piles Embedded in TransverselyIsotropic Soils to Lateral Loads
Pérsio Leister de Almeida Barros1
1State University of Campinas, Caixa Postal 6021, CEP 13083-852, Campinas SP, [email protected]
Keywords: Piles, Anisotropy, Dynamic Soil-Structure Interaction, Elastodynamics, Boundary Elements Method.
Abstract: The response of vertical piles with circular cross section embedded in a transversely isotropic half-space to time harmonic horizontal loads and bending moments is addressed. The pile is modeled as a seriesof finite beam elements and the soil response is obtained by an indirect formulation of the Boundary ElementMethod. The BEM formulation uses half-space influence functions, which are the medium response to dis-tributed loads, instead of the usual Green’s functions. This approach leads to a non singular formulation, sothe fictitious tractions can be applied on the true pile-soil interface. The coupling between the FEM and BEMmodels is set at the midpoint of the beam elements to obtain a smoother load transfer profile. The influence ofthe soil anisotropy, expressed as anisotropy indices, on the pile impedances is addressed.
Introduction
Pile response to dynamic loads has been studied during the last decades, since the pioneering work of Novak[3], by a number of different methods. These methods are generally classified in three categories: analytical,semi-analytical and numerical methods. The semi-analytical methods are the most widely used today becausethey combine the advantages of the other two types.
The application of semi-analytical methods, specially those based on the Boundary Integral EquationMethod (BIEM) and on the Boundary Element Method (BEM) to the dynamic soil-structure interaction analy-sis has been mostly restricted, however, to the cases where the soil can be treated as an elastic isotropic medium.In some cases, however, the soil medium behaves as an anisotropic material and the hypothesis of isotropy isunrealistic. The most common type of anisotropy found in geomaterials is the transverse isotropy, where thereis an axis of elastic symmetry, generally vertical, and isotropy is found only inside the planes perpendicular tothat axis.
Only a small number of papers on the dynamic interaction between transversely isotropic soils and struc-tures can be found in the literature. Most of them deal with interaction with rigid structures. The case ofthe interaction between flexible piles and transversely isotropic soils is treated by an even smaller group ofresearchers [1]. The present work deals with time harmonic horizontal loads and bending moments applied tothe pile top.
The method of analysis presented herein includes counter actions to avoid the circular distortion of the soilin contact with the pile. In this way real bonding can be achieved along the whole contact circumference.
Problem description
The system analyzed is a vertical pile with circular cross section embedded in an elastic, transversely isotropicsoil. The pile material is elastic and isotropic with Young’s modulus Ep, Poisson’s ratio νp and mass densityρp. The pile length lp is much greater than the pile radius a, so the pile can be accurately represented as aone-dimensional beam.
The surrounding soil is represented by a transversely isotropic half space with elastic symmetry axis co-incident with the pile axis. The soil density is ρ and the stress-strain relationship for that material is givenby:
σxx = c11εxx + c12εyy + c13εzz (1)
σyy = c12εxx + c11εyy + c13εzz (2)
σzz = c13εxx + c13εyy + c33εzz (3)
Advances in Boundary Element Techniques V 211
σyz = 2c44εyz (4)
σxz = 2c44εxz (5)
σxy =12
(c11 − c12) 2εxy (6)
where cij are the elastic constants of the medium.The isotropic case is a special one where c11 = c33 = λ + 2µ, c12 = c13 = λ and c44 = µ, where λ and
µ are the Lamè’s constants. The anisotropic character of the material may be expressed by three anisotropyindices ni given by:
n1 = c33/c11, n2 = (c11 − c12) /2c44, n3 = (c11 − 2c44) /c13 (7)
The pile is perfectly bonded to the soil, so no slippery at the pile-soil interface is allowed. On the pile topan horizontal force Fx and a bending moment My are applied. These are time harmonic external actions withcircular frequency ω.
The horizontal displacement ux and rotation φ measured at the pile top are related to the external actionsby: {
Fx
My/a
}= ac44
[Khh Khm
Kmh Kmm
]{ux
aφ
}(8)
where Kij are the pile-soil system non dimensional impedances.
Soil modeling
The unbounded medium surrounding the pile is modeled by a non singular formulation of the indirect BEM.This works proposes the use of half-space solutions of distributed loads called influence functions, instead ofconcentrated point loads. Since the solutions for distributed loads are not singular, the load can be applied onthe real soil-structure interface.
The Green’s functions for a load applied inside a transversely isotropic half space have the general form, incylindrical coordinates [4]:
uikm (r, z) =∫ ∞
0uikm (ζ) pkm (ζ) dζ (9)
σijkm (r, z) =∫ ∞
0σijkm (ζ) pkm (ζ) dζ, i, j, k = r, θ, z (10)
were uikm and σijkm are respectively the m-th component of Fourier expansions of the displacements ui (r, θ, z)and stress components σij (r, θ, z) in the θ direction due to a load in k direction [2].
The terms uikm and σijkm are functions of the material elastic constants, the load depth and frequency andof the receiver coordinates r, z. The terms pkm are the Hankel transform of the m-th component of the Fourierexpansion of the load pk (r, θ) applied in k direction.
For the present application, half space influence functions for loads distributed along cylindrical shaftsand horizontal disks were obtained. The disk influence functions were obtained directly by applying Henkeltransform on the load distribution and the influence functions for cylindrical loads were derived through theanalytical integration of the half-space Green’s functions for rings along the vertical surface. In both case,however, the wave number integration has to be performed numerically.
The influence functions necessary for the present analysis are obtained by setting m = 1 and taking thesymmetrical components of the loads, displacements and stresses. The pr1 and pθ1 loads may be combined toobtain a horizontal px load. This is done by taking pr1 = −pθ1 = px. The displacement components ur1 anduθ1 due to this horizontal load can also be combined to get the horizontal displacement ux = (ur1 − uθ1) /2.
As, in general, ur1 and uθ1 are not equal in modulus and opposed in sign, the horizontal displacement willnot be constant along the circumferential θ direction. There will be a circular distortion ud = (ur1 + uθ1) /2.
Along the soil-structure interface such distortion does not occur since the pile section is supposed to remaincircular. So, a distorting load pd must be applied to impose this condition. This loading is obtained by settingpr1 = pθ1 = pd.
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p p px d m
Figure 1: Horizontal, distorting and moment loads
t
M
M
x
Fx
m y(i)
y
z
Fx
2le
a
y
(i)
(i)
i
i( +1)
( +1)
(i)
Figure 2: Actions on the i-th pile element.
Finally, the pz1 load results in a moment pm applied to the elastic medium and the uz1 displacement resultsin a rotation um/r. The three types of loads are shown schematically in Fig. 1.
The soil-pile interface is discretized in N cylindrical elements along the lateral face and one disk elementat the pile tip. Distributed along each one of these N + 1 elements, fictitious horizontal qx, distortion qd andmoment qm are applied. The displacements and tractions along the interface soil-pile are due to the fictitiousloads are:
u(i)α =
N+1∑j=1
(U (ij)
αx q(j)x + U
(ij)αd q
(j)d + U (ij)
αm q(j)m
)(11)
t(i)α =N+1∑j=1
(T (ij)
αx q(j)x + T
(ij)αd q
(j)d + T (ij)
αm q(j)m
)(12)
were U(ij)αβ and T
(ij)αβ are, respectively, the displacement and traction in α = x, d, m direction at the center of
the i-th element due to a distributed unitary load applied at the j-th element in β direction.
Pile modeling
The pile is modeled as a series of beam elements with assumed cubic displacement shape functions. Theelements have one node at each end and to each node two degrees of freedom (horizontal displacement u
(i)x and
rotation φ(i)y ) are assigned. Horizontal forces F
(i)x and moments M
(i)y act at each node and uniformly distributed
horizontal tractions t(i)x and moments m
(i)y act along each element (see Fig. 2).
The displacement and force vectors for the i-th element are, respectively, ue and pe. The distributed
Advances in Boundary Element Techniques V 213
moment m(i)y is due to the t
(i)m component of the traction along the pile-soil interface. It is given by:
m(i)y = πa2t(i)m , 1 ≤ i ≤ N (13)
So, the vector of nodal equivalent actions is:
be =πa
3
{6l(i)e t(i)x + 3at(i)m , 2l(i)e l(i)e t(i)x , 6l(i)e t(i)x − 3at(i)m ,−2l(i)e l(i)e t(i)x
}(14)
At the top node of the first element the external loads are applied and at the bottom node of the N -th elementthe tractions at the pile tip result in:
F (N+1)x = πa2t(N+1)
x , M (N+1)y =
13πa3t(N+1)
m (15)
The equation of movement of the element is:(Ke − ω2Me
)ue + be = pe (16)
where Ke and Me are, respectively, the element stiffness and mass matrices.The equation of movement for the entire pile is, similarly:(
K − ω2M)u + b = p (17)
were K and M are the global stiffness and mass matrices, respectively, u is the vector of nodal displacementsand rotations, b is the vector of nodal loads equivalent to the interface tractions and p is the vector of externalloads. The K and M matrices are obtained by the standard FEM assemblage procedures, while b = Bt is thevector of the equivalent nodal actions which correspond to the tractions t.
Pile-soil coupling
The coupling between the soil model and the pile model is accomplished by imposing displacement compati-bility and equilibrium along the pile-soil interface. The displacement compatibility can only be enforced at afinite number of points (rings) along the pile. These points are chosen at the center of each boundary element(the BEM nodes) and not at the nodes of the pile finite element model (the FEM nodes). This approach hasproven to lead to a much more stable numerical solution and also to a much smoother traction profile along theinterface.
Since the displacements in the pile model are calculated primarily at the FEM nodes, it is necessary to relatethese displacements to the displacements at the BEM nodes. This relationship can be expressed by:
u = Cu (18)
were u and u are the displacement vectors at the center and at the end of the elements, respectively.From the soil model equations (11) to (12), the displacements u and tractions t can be expressed by:
u = Uq, t = Tq (19)
Also, the distortion displacement vector ud can be expressed by:
ud = Udq (20)
The following system of equations is then obtained:{ (KC−1U + BT
)q = p
Udq = 0(21)
wereK = K − ω2M (22)
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Table 1: Test materials elastic constants.
Soil c11/c44 c12/c44 c13/c44 c33/c44 n1 n2 n3
Isotropic 3 1 1 3 1 1 1(a) 3 1 1 6 2 1 1(b) 3 1 1 1.5 0.5 1 1(c) 3 -1 1 3 1 2 1(d) 3 2 1 3 1 0.5 1(e) 3 1 0.5 3 1 1 2(f) 3 1 2 3 1 1 0.5
The solution of the system (21) gives the fictitious tractions q which can be used to calculate the dis-placements and tractions along the interface. Unfortunately, the system (21) is very ill conditioned due to thepresence of the inverse matrix C−1.
In order to avoid the problems with the system of equations (21), its first line is multiplied by CK−1
resulting: { (U + CK−1BT
)q = CK−1p
Udq = 0(23)
The system (23) is much better conditioned. The only drawback here is the presence of the inverse K−1.But K is symmetric and real, so the evaluation of the inverse requires comparatively less effort then the solutionof the whole system. It must be pointed out, however, that K may be singular. This occur when ω = 0 (thestatic case) and for the natural frequencies of the isolated pile. For these cases a limiting procedure may beadopted.
The solution of (23) gives the fictitious loads q which are used to calculate the interface tractions t. Thesetractions are, in turn, used directly to solve the FEM model, since K−1 is already calculated. This gives thedisplacements and rotations at the nodes of the FEM model.
Numerical results
In order to evaluate the influence of the medium anisotropy on the pile response, a series of numerical studieswere performed. For these studies a set of different test materials was constructed. The test materials arebased on an initial isotropic material with Poisson’s ratio of ν = 0.25. From this initial isotropic material,transversely isotropic materials are obtained by varying separately the n1, n2 and n3 anisotropy indices. Theseelastic constants for these materials are listed in Table 1.
The response of a pile with slenderness ratio le/a = 50, density ratio ρp/ρ = 1.5 and Young’s modulusratio Ep/Eh = 1000 was then obtained. The soil horizontal Young’s modulus is obtained from:
Eh =(c11 − c12)
[c33 (c11 + c12) − 2c2
13
]c11c33 − c2
13
(24)
After a convergence study, a minimum number of elements N = 20 was established for sufficient accuracyof the results over the frequency range used in the analyses. The plots in Fig. 3 show the numerical results ofthe non dimensional impedances Kij as functions of the non dimensional frequency a0 = ωa
√ρ/c44.
The analysis of the numerical results presented in Fig. 3 reveals:
• The horizontal impedance Khh is much more influenced by the n2 anisotropy index than by the other twoindices. Out of these two, the n1 index shows very small influence on Khh. This behavior is expectedbecause n1 affects the vertical stiffness of the medium. The n3 index have an intermediate influence onKhh, and the variation is much greater when n3 decreases from 1 to 0.5 than when it increases from 1 to2. As a general rule, the horizontal impedance increases monotonically with each of the three anisotropyindices.
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0.0 0.1 0.2 0.3 0.4 0.5
a0
12.0
16.0
20.0
24.0
28.0
32.0
Re(
Khh
)
0.0 0.1 0.2 0.3 0.4 0.5
a0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Im(K
hh)
Isotropic(a)(b)(c)(d)(e)(f)
0.0 0.1 0.2 0.3 0.4 0.5
a0
350
400
450
500
550
600
650
Re(
Km
m)
0.0 0.1 0.2 0.3 0.4 0.5
a0
0
50
100
150
200
Im(K
mm)
Isotropic(a)(b)(c)(d)(e)(f)
Figure 3: Horizontal and rocking impedance of a pile embedded in various soils (le/a = 50, ρp/ρ = 1.5,Ep/Eh = 1000)
• For the rocking impedance Kmm, the n3 index shows the largest influence, specially when n3 decreases.Again, the n1 index shows the smaller influence, although here this influence is larger than on the hor-izontal impedance. The n2 index show a surprising behavior. The Kmm value decreases both when n2
increases and when it decreases. But this behavior can be at least partially explained. The pile’s Young’smodule is not constant for all tests. Since the soil Eh depends on the elastic constants cij according tothe equation (24), Ep is also altered in order to maintain the Ep/Eh constant. When the n2 index varies,Eh reaches a maximum value at approximately n2 = 1.4. For both n2 = 2 and n2 = 0.5, Eh assumesmaller values than for n2 = 1. So, for the two anisotropic cases the pile-soil system stiffness is smallerthan for the isotropic case.
Acknowledgment
The work presented here was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo(FAPESP).
References
[1] Liu, W. and Novak, M. (1994). “Dynamic response of single piles embedded in transversely isotropiclayered media”. Earthquake Engineering and Structural Dynamics. 23, 1239–1257.
[2] Muki, R. (1960) “Asymmetric problems of the theory of elasticity for a semi-infinite solid and a thickplate.” Progress in Solid Mechanics. Vol. 1, North Holland Publishing Company, Amsterdam.
[3] Novak, M. (1977). “Dynamic stiffness and damping of piles.” Canadian Geotechnical Journal. 11(4), 574–598.
[4] Rajapakse, R.K.N.D. and Wang, Y. (1993), “Green’s functions for transversely isotropic elastic half space.”Journal of Engineering Mechanics. ASCE. 119, 1724–1746.
216
Two-dimensional Time-Independent Green’s Functions for Unsaturated Soils
B. Gatmiri 1, E. Jabbari 2
1 Ecole Nationale des Ponts et Chaussées, Paris, France andCivil Engineering Department, University of Tehran, Tehran, Iran
Email: [email protected]
2 Civil Engineering Department, University of Tehran, Tehran, Iran Email: [email protected]
Keywords: Green’s function; Unsaturated soil; Boundary element method.
Abstract. In this article, after a brief discussion about the formulation of unsaturated soils including theequilibrium, air and moisture transfer equations, the closed form Green’s functions of the governing differential equations for an unsaturated two-dimensional deformable porous media with linear elastic behavior for a symmetric polar domain, considering suction effects and dissolved air in water, have beenintroduced. The results have been compared with their corresponding elastostatic Green’s functions.
Introduction
Numerical modeling has been largely developed in soil mechanics behavior by different methods. Amongthem, the development of the boundary element method, which is the most suitable one for domain withinfinite boundary conditions like soils media, has been restricted by the necessity of deriving the Green’sfunctions of the governing differential equations.
The Green’s functions for elastostatic problems have been derived by classical methods. Particularly, theGreen’s functions for poroelastic and thermoelastic problems have been introduced in static and dynamiccases [1]. These Green’s functions have been all derived for fluid saturated soils [2,3]. Although the most ofthe difficulties arises in deriving the Green’s function for time-dependent problems, they have not beenpresented yet for unsaturated case even for time independent problems.
In unsaturated soils the differential equations are different from those of saturated case due to thepresence of one more parameter (air pressure) and one more equation in one hand and the presence ofsuction and dissolved air in water effects in the other hand. This research is an attempt to derive such time-independent Green’s functions for a two-dimensional axisymmetric domain in polar coordinates andspecially consider the mentioned effects. In the following paper the corresponding Green’s functions will be presented for the three-dimensional case.
Governing differential equations
The governing differential equations using the effective stress concept consist of [4]:Solid skeleton. Equilibrium and (linear-elastic) constitutive equations for soil’s solid skeleton including suction effects:
0ii,aj,aijij bpp (1)
wasaijij dpdpDdDpd (2)
considering the strain-deformation relations:
i,jj,iij uu2
1(3)
may be written as:
01 ii,wsi,asjj,iij,j bpDpDuu (4)
Advances in Boundary Element Techniques V 217
where and are Lamé’s coefficients of soil elasticity and is the coefficient of deformations due to
suction effect. Also sD
, , , , and stand for stress, strain, soil’s dispacement in direction , air
and water pressure and body force in direction , respectively.iu ap wp ib i
i
Air
Water
Soil Particles
Fig. 1: Unsaturated soil scheme
Air phase. Time-independent continuity and transfer equations for the air phase, considering dissolved air inwater:
0waa Huudiv (5)
Zp
Kua
aaa (6)
au , , wu H , a and a are air and water velocity, Henry’s coefficient for dissolved air in water and
density and specific weight of air, respectively. The air coefficient of permeability is defined as:
Er
a
aa SeDK 1 (7)
in which a , and are air dynamic viscosity, void ratio and degree of saturation, respectively and and
are constants [5].
e rS D
EIts seems reasonable to dispense of the variations of due to and consequently of for the
simplicity, since deriving the considered Green’s functions will become too difficult, at least with commonmethods, due to the nonlinearity of the governing differential equations. Therefore, the effects of have
been considered in air coefficient of permeability by assuming as a multilinear function of
aK rS wa pp
rS
aK wa pp for
each finite domain. However eq (5) may be written as:
022
ww
waa
a
aa pKH
pK
(8)
Water phase. Time-independent continuity and transfer equations for the water phase:
0wwudiv (9)
Zp
Kuw
www (10)
where the water coefficient of permeability is defined as:
53
0 1
.
ru
rurwzw S
SSKK (11)
0wzK is the intrinsic coefficient of permeability and is the residual degree of saturation. A discussion
similar to that made for shows that it is inevitable to dispense of the variations of in finite domains of
. Assuming constant for the specified regions of is, indeed, assuming it as a multilinear function of
which simply reflects the basic concept of the relation between and . Therefore:
ruS
aK wK
rS wK rS
rS wK rS
218
02
ww
ww pK
(12)
The mentioned differential equations are simplified so that make possible to derive the acceptableGreen’s functions, while the main features of the unsaturated soils such as suction effects and dissolved air inwater have been kept in consideration. One can arrange the governing differential equations (4,8,12) in thematrix form:
fuCij (13)
where and: ijijij dcC
ii uu apu3 wpu4 21,i (14)
ii bf 03f 04f 21,i (15)
11c 12c sDc 113 sDc14
a
aaKc21
w
wa KHc22
w
wwKc31 (16)
ijd are the differential operators.
Green’s functions
One of the few methods for deriving the Green’s functions matrix of a system of differential equations is themethod of Kupradze [6], which is a straightforward mathematical method. Based on this method the Green’sfunctions are the cofactors of : ijC
*ijij Cg (17)
in which is a potential function and satisfies the equation:
0xCdet ij (18)
in which x is the Dirac delta function in two-dimensional space. By definition of the potential function
, a set of fundamental solutions will be achieved. This leads to such equation:
08 xD 3121121112 cccccD (19)
where is occurrence of the Laplacian operator. The solution of eq (19) in an axisymmetricdomain is:
nn 22 n
D
rLogr
27648
656
(20)
and the or Green’s functions are: ijg
28
232
2
2
r
xxrrLogg jiij
ij
aa
siai K
DrLogxg
28
1213
wwa
awswasii KK
KDHKDrLogxg
28
1214
aa
a
K
rLogg
233
ww
w
K
rLogg
244
Advances in Boundary Element Techniques V 219
04334 g~g 043 ii gg 21,j,i (21)
It is evident that while H and approach to zero, the Green’s functions in eq (21) approach toelastostatic Green’s functions [1,7]:
sD
28
232
2
2
r
xxrrLogg jiij
ij
aa
iai K
rLogxg
24
213 04ig 043 ii gg
aa
a
K
rLogg
233
ww
w
K
rLogg
244
04334 gg 043 ii gg 21,j,i (22)
For instance the derived Green's functions are drawn through Figs. 2 to 5 with the following parameters:
kPaE 4103 350. 020.H 2sD
28069 sm.g 1 13101 kPa
32931 mkg.a31000 mkgw smkg.a
510851
sm.aKw91021 5Kw 50.Sr 050.Sru
24101 mDKa 62.EKa 7500 .e (23)
10
5
05
10
X
10
5
0
5
10
Y
1.5 10 71 10 75 10 8
0
0
5
05X
105
0
5
10
X
10
5
0
5
10
Y
2 10 8
0
2 10 8
05
0
5X
Fig. 2: Green’s function Fig. 3: Green’s functionSolid skeleton displacement in direction one Solid skeleton displacement in direction one
11g 12g
due to Hevisaide point load in direction one. due to Hevisaide point load in direction two.
0.50.25
00.25
0.5
X
0.5
0.25
0
0.25
0.5
Y
5 10 9
0
5 10 9
50.25
00.25X
0.50.25
0
0.250.5
X 0.5
0.25
0
0.25
0.5
Y
0.1
0
0.1
0.50.25
0
0.25X
Fig. 4: Green’s function Fig. 5: Green’s function
Solid skeleton displacement in direction one Solid skeleton displacement in direction one 13g 14g
due to injection of air unit volume. due to injection of water unit volume.
220
Conclusion
In this research the closed form Green’s functions of two-dimensional governing differential equations of unsaturated soils, including equilibrium equations with linear elastic constitutive equations and two equations of air and water transfer, considering suction effects and dissolved air in water, have been derived. For verification of the results, it has been demonstrated that if the conditions approach to elastostatic case, the Green’s functions will approach to elastostatic Green’s functions exactly. Although the mathematical procedure is not very complicated, it seems to be a new experience to introduce a set of fundamental solutions for the unsaturated case, probably for the first time. The derived Green's functions may be used to develop a boundary element computer program for modeling unsaturated soil's problems in steady state.
References
[1] P.K.Banerjee The Boundary Element Methods in Engineering, McGraw-Hill Book Company, England (1994).
[2] J.Chen International Journal of Solids and Structures, 31 (10), 147-1490 (1994).
[3] B.Gatmiri, M.Kamalian International Journal of Geomechanics, 2 (4), 381-398 (2002).
[4] B.Gatmiri, P.Delage, M.Cerrolaza Advances in Engineering Software, 29 (1), 29-43 (1998).
[5] T.W.Lambe, R.V.Whitman Soil Mechanics, John Wiley and Sons, New York (1969).
[6] V.D.Kupradze et al. Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Netherlands (1979).
[7] G.Beer Programming the boundary element method, John Wiley and Sons, England (2001).
Advances in Boundary Element Techniques V 221
222
Three-dimensional Time-Independent Green’s Functions for Unsaturated Soils
B. Gatmiri 1, E. Jabbari 2
1 Ecole Nationale des Ponts et Chaussées, Paris, France andCivil Engineering Department, University of Tehran, Tehran, Iran
Email: [email protected]
2 Civil Engineering Department, University of Tehran, Tehran, Iran Email: [email protected]
Keywords: Green’s function; Unsaturated soil; Boundary element method.
Abstract. The Three-dimensional Green’s functions of the governing differential equations of unsaturated soils have been presented in this paper. The governing equations consist of equilibrium, air and moisturemass conservation and transfer equations. The Green’s functions have been derived for three-dimensionaldeformable porous media with linear-elastic behavior for soil skeleton in a symmetric spherical domain.Some of the special unsaturated soil properties like suction effect and dissolved air in water have beenconsidered. The results have been drawn and verified by comparing with elastostatic corresponding solutions.
Introduction
Among the numerical methods for solving the governing differential equations of the soils phenomena, theboundary element method is of the great importance due to its advantages. The capability of modelinginfinite boundaries, no need for definition of interior mesh of element, and no unknowns associated withinterior points of the domain in a numerical implementation, less data preparation time and less required computer time and storage for the same level of accuracy have resulted in a unique interesting numericalmethod.
The development of the boundary element method has been restricted by the necessity of deriving the Green’s functions of the governing differential equations as a mathematical problem and in this regard the Green’s function is one of the interesting topics in the engineering mathematics. The Green’s functions havebeen presented for various sets of governing differential equations in exact and approximate forms. For the elastostatic equations the fundamental solutions have been derived by classical methods [1]. These Green’sfunctions have been all derived for fluid saturated soils [2,3]. Although the most of the difficulties arises inderiving the Green’s function for time dependent problems, they have not been presented for unsaturated case even for time independent problems. Some of the main difficulties between the formulation of saturated and unsaturated soils are one more equation and parameter (air pressure) , suction effects and dissolved air inwater.
In unsaturated soils the differential equations are different from those of saturated case due to thepresence of one more parameter (air pressure) and one more equation in one hand and the presence ofsuction and dissolved air in water effects in the other hand. This research is an attempt to derive such time-independent Green’s functions for a three-dimensional axisymmetric domain in polar coordinates as the two-dimensional case which have been presented in the previous paper.
Governing differential equations
The governing differential equations using the effective stress concept consist of [4]:Solid skeleton. Equilibrium and (linear-elastic) constitutive equations for soil’s solid skeleton including suction effects:
0ii,aj,aijij bpp (1)
wasaijij dpdpDdDpd (2)
considering the strain-deformation relations:
Advances in Boundary Element Techniques V 223
i,jj,iij uu2
1(3)
One can conclude:
01 ii,wsi,asjj,iij,j bpDpDuu (4)
where and are Lamé’s coefficients of soil elasticity and is the coefficient of deformations due to
suction effect. Also sD
, , , , and stand for stress, strain, soil’s dispacement in direction , air
and water pressure and body force in direction , respectively.iu ap wp ib i
i
Air
Water
Soil Particles
Fig. 1: Unsaturated soil scheme
Air phase. Time-independent continuity and transfer equations for the air phase, considering dissolved air inwater:
0waa Huudiv (5)
Zp
Kua
aaa (6)
au , , wu H , a and a are air and water velocity, Henry’s coefficient for dissolved air in water anddensity and specific weight of air, respectively. The air coefficient of permeability is defined as:
Er
a
aa SeDK 1 (7)
in which a , and are air dynamic viscosity, void ratio and degree of saturation, respectively and and are constants [5].
e rS D
EAs the two-dimensional case it seems reasonable to dispense of the variations of due to and
consequently of for the simplicity, since deriving the considered Green’s functions will become
too difficult, at least with common methods, which may be applied only to the linear differential equations. Therefore, the effects of have been considered in air coefficient of permeability by assuming as a
multilinear function of for each finite domain. However eq (5) may be simplified as:
aK rS
wa pp
rS aK
wa pp
022
ww
waa
a
aa pKH
pK
(8)
Water phase. Time-independent continuity and transfer equations for the water phase:
0wwudiv (9)
Zp
Kuw
www (10)
where the water coefficient of permeability is defined as:
224
53
0 1
.
ru
rurwzw S
SSKK (11)
0wzK is the intrinsic coefficient of permeability and is the residual degree of saturation. The same
discussion as said for shows that it is inevitable to dispense of the variations of in finite domains of
. Therefore:
ruS
aK wK
rS
02
ww
ww pK
(12)
In fact, we simplified the governing differential equations to a set of linear system to make possible toderive the acceptable Green’s functions, while the main features of the unsaturated soils like suction effectsand dissolved air in water have been kept in consideration. The governing differential equations (4,8,12) inthe matrix form may be written as:
fuCij (13)
where and: ijijij dcC
iuu apu4 wpu5 31,i (14)
ibf 04f 05f 31,i (15)
11c 12c sDc 113 sDc14
a
aaKc21
w
wa KHc22
w
wwKc31 (16)
ijd are the differential operators.
Green’s functions
Based on the method of Kupradze [6] which is a straightforward mathematical method, the Green’s functions of a set of differential equations with linear differential operators are the cofactors of :ijC
*ijij Cg (17)
in which is a potential function and satisfies the equation:
0xCdet ij (18)
in which is the Dirac delta function in three-dimensional space. By definition of the potential function x
, a set of fundamental solutions will be achieved. This leads to such equation:
010 xD 312112112
12 cccccD (19)
where is occurrence of the Laplacian operator. The solution of eq (19) in an axisymmetricthree-dimensional domain is:
nn 22 n
D
r
161280
7
(20)
and or the Green’s functions are: ijg
28
232
2
r
xxrg jiij
ij
Advances in Boundary Element Techniques V 225
aa
siai Kr
Dxg
28
13
wwa
awswasii KKr
KDHKDxg
28
14
aa
a
Krg
433ww
w
Krg
444
04334 gg 043 ii gg 31,j,i (21)
which while H and approach to zero, approach to elastostatic Green’s functions [1,7]:sD
28
232
2
r
xxrg jiij
ij
aa
iai Kr
xg
283 04ig 043 ii gg
aa
a
Krg
433ww
w
Krg
444
04334 gg 043 ii gg 31,j,i (22)
For instance, The derived Green's functions are shawn through Figs. 2 to 5 with the following initialvalues:
kPaE 4103 350. 020.H 2sD
28069 sm.g 1 13101 kPa
32931 mkg.a31000 mkgw smkg.a
510851
sm.aKw91021 5Kw 50.Sr 050.Sru
24101 mDKa 62.EKa 7500 .e m.z 001 (23)
105
05
10
X10
5
0
5
10
Y
5 10 91 10 8
1.5 10 82 10 8
05
05X
105
05
10
X10
5
0
510
Y
5 10 92.5 10 9
02.5 10 9
5 10 9
05
05X
Fig. 2: Green’s function Fig. 3: Green’s functionSolid skeleton displacement in direction one Solid skeleton displacement in direction one
11g 12g
due to Hevisaide point load in direction one. due to Hevisaide point load in direction two.
226
0.0 0.1 0.2 0.3 0.4 0.5
a0
12.0
16.0
20.0
24.0
28.0
32.0
Re(
Khh
)
0.0 0.1 0.2 0.3 0.4 0.5
a0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Im(K
hh)
Isotropic(a)(b)(c)(d)(e)(f)
0.0 0.1 0.2 0.3 0.4 0.5
a0
350
400
450
500
550
600
650
Re(
Km
m)
0.0 0.1 0.2 0.3 0.4 0.5
a0
0
50
100
150
200
Im(K
mm)
Isotropic(a)(b)(c)(d)(e)(f)
Figure 3: Horizontal and rocking impedance of a pile embedded in various soils (le/a = 50, ρp/ρ = 1.5,Ep/Eh = 1000)
• For the rocking impedance Kmm, the n3 index shows the largest influence, specially when n3 decreases.Again, the n1 index shows the smaller influence, although here this influence is larger than on the hor-izontal impedance. The n2 index show a surprising behavior. The Kmm value decreases both when n2
increases and when it decreases. But this behavior can be at least partially explained. The pile’s Young’smodule is not constant for all tests. Since the soil Eh depends on the elastic constants cij according tothe equation (24), Ep is also altered in order to maintain the Ep/Eh constant. When the n2 index varies,Eh reaches a maximum value at approximately n2 = 1.4. For both n2 = 2 and n2 = 0.5, Eh assumesmaller values than for n2 = 1. So, for the two anisotropic cases the pile-soil system stiffness is smallerthan for the isotropic case.
Acknowledgment
The work presented here was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo(FAPESP).
References
[1] Liu, W. and Novak, M. (1994). “Dynamic response of single piles embedded in transversely isotropiclayered media”. Earthquake Engineering and Structural Dynamics. 23, 1239–1257.
[2] Muki, R. (1960) “Asymmetric problems of the theory of elasticity for a semi-infinite solid and a thickplate.” Progress in Solid Mechanics. Vol. 1, North Holland Publishing Company, Amsterdam.
[3] Novak, M. (1977). “Dynamic stiffness and damping of piles.” Canadian Geotechnical Journal. 11(4), 574–598.
[4] Rajapakse, R.K.N.D. and Wang, Y. (1993), “Green’s functions for transversely isotropic elastic half space.”Journal of Engineering Mechanics. ASCE. 119, 1724–1746.
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