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Mechanics Research Communications 36 (2009) 921–932

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Mechanics Research Communications

journal homepage: www.elsevier .com/ locate/mechrescom

Displacements and stresses due to nonuniform circular loadingsin an inhomogeneous cross-anisotropic material

C.D. Wang a,*, C.S. Tzeng b

a Department of Civil and Disaster Prevention Engineering, National United University, No. 1, Lien-Da, Kung-Ching-Li, Miao-Li, 360, Taiwan, ROCb Hu-Si Township, Peng-Hu County, No. 43-11, Hu-Si Village, 885, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 14 November 2006Received in revised form 3 August 2009Available online 8 August 2009

Keywords:DisplacementStressConicalParabolicCircular distribution of the vertical loadInhomogeneousCross-anisotropic material

0093-6413/$ - see front matter Crown Copyright �doi:10.1016/j.mechrescom.2009.08.001

* Corresponding author. Tel.: +886 37 381669; faE-mail addresses: [email protected], cdwang0

a b s t r a c t

This article presents the solutions for displacement and stress components along the cen-terline of nonuniform circular distribution of the vertical loads in a continuously inhomo-geneous cross-anisotropic material with Young’s and shear moduli varying exponentiallywith depth. The nonuniform loading types include a conical and a parabolic circular load.Planes of cross-anisotropy are assumed to be parallel to the horizontal ground surface. Theproposed solutions can be obtained by integrating the point load solutions in a cylindricalco-ordinate system for an inhomogeneous cross-anisotropic half-space, which werederived by Wang et al. [Wang, C.D., Tzeng, C.S., Pan, E., Liao, J.J., 2003. Displacements andstresses due to a vertical point load in an inhomogeneous transversely isotropic half-space.Int. J. Rock Mech. Min. Sci. 40(5), 667–685]. However, the resulting integrals of the nonuni-form circular solution for displacements and stresses cannot be given in closed form;hence, numerical integrations are required. Numerical results agree very well with the ana-lytical solutions of displacements and stresses subjected to both present loading types for ahomogeneous cross-anisotropic half-space, which are also yielded in Appendix A of thiswork. In addition, the proposed solutions are identical with Harr and Lovell’s [Harr, M.E.,Lovell, C.W. Jr., 1963. Vertical stresses under certain axisymmetrical loadings. High. Res.Board Rec. 39], and Geddes’s [Geddes, J.D., 1975. Vertical stress components produced byaxially symmetrical subsurface loadings. Can. Geotech. J. 12 (4), 482–497] solutions whenthe medium is isotropy. Two examples are illustrated to elucidate the effect of inhomoge-neity, and the type and degree of soil anisotropy on the vertical displacement and verticalnormal stress in the inhomogeneous isotropic/cross-anisotropic soils due to, respectively, aconical and a parabolic circular distribution of the vertical load acting on the ground sur-face. The generated solutions cannot only simulate the actual loading problem but also pro-vide the realistic stratum in many fields of engineering practice.

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1. Introduction

In most previous theoretical analysis of soil behavior, the properties of soil were assumed to be homogeneous andisotropic. However, many natural soils, such as flocculated clays, varved silts or sands, often deposited through a geologicprocess of sedimentation over a period of time. The effects of deposition, overburden, desiccation, etc., can lead geologicalmedia exhibit both the inhomogeneous and anisotropic deformability. The mechanical response of anisotropic materialswith spatial gradients in composition is called the anisotropic functionally graded materials (FGMs), and is of considerableinterest in soil/rock mechanics and foundation engineering. A very detailed survey work on this topic can be referred to

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922 C.D. Wang, C.S. Tzeng / Mechanics Research Communications 36 (2009) 921–932

Wang et al. (2003). However, theoretical understanding of inhomogeneous and anisotropic phenomena has not receivedmuch attention due to the mathematical difficulties involved in these media. Therefore, an elastic subsurface loading prob-lem for a continuously inhomogeneous cross-anisotropic half-space with Young’s and shear moduli varying exponentiallywith depth is relevant in this work.

In this article, the load with circular shape is chosen because the solutions produced are of practical importance in soil/rock mechanics and foundation engineering. Particularly, these solutions could have a direct application to problems asso-ciated with foundations under structures such as silos, chimneys, and tanks containing liquids (Gerrard and Wardle, 1973).Numerous existing analytical/numerical solutions for the inhomogeneous isotropic media owing to general types of circularload (i.e., uniform, parabolic, etc.) can be found in Wang et al. (2006). Nevertheless, in many engineering fields (Hooper,1976; Bauer et al., 1979; Hemsley, 1991; Wang and Liao, 2002a,b), applied loads are not uniformly distributed but more con-centrated towards the center of foundation. That means the loads might be realistically simulated as being distributed aslinearly varying or as parabola of revolution. Hence, in this paper, the displacement and stress components produced by non-uniform axisymmetric loadings, which include a conical and a parabolic circular distribution of the vertical load, acting in theinterior of an inhomogeneous cross-anisotropic half-space are concerned.

For a homogeneous cross-anisotropic medium, Gazetas (1982a,b) analytically investigated how soil’s cross-anisotropy af-fects the surface displacement and stress distributions when it is caused by axisymmetric parabolic vertical surface loading.Hanson and Puja (1998a,b) estimated the stresses resulting from the combinations of uniform, linear, and quadratic loadings,applied over a circular area on the surface of a cross-anisotropic half-space. As for the inhomogeneous cross-anisotropicmaterials due to uniform or nonuniform axisymmetric circular loadings, the literatures are very limited since the mathemat-ical difficulties encountered in such analyses. Hooper (1975) utilized the finite element method to consider the surface dis-placement of an inhomogeneous cross-anisotropic half-space owing to a parabolic loading applied over a circular region with

O′ r

h pc

a

parrU , par

zU parzzσ , par

rzτ

z

Fig. 2. A parabolic circular distribution of the vertical load pc applied in the interior of an inhomogeneous cross-anisotropic half-space.

O′ r

h c

a

conrU , con

zU conzzσ , con

rzτ

z

p

Fig. 1. A conical circular distribution of the vertical load pc applied in the interior of an inhomogeneous cross-anisotropic half-space.

C.D. Wang, C.S. Tzeng / Mechanics Research Communications 36 (2009) 921–932 923

Young’s modulus following the linear law (E = E0 + kz). Pan (1989, 1997) used the vector functions and the propagator matrixmethod to solve the induced displacement and stress components of a cross-anisotropic and layered half-space under gen-eral surface loads. Wang et al. (2006) presented the solution for displacements and stresses along the centerline induced by auniform circular distribution of the vertical load in an inhomogeneous cross-anisotropic material with Young’s and shearmoduli varying exponentially with depth (Ee�kz, E

0e�kz, G

0e�kz). To our knowledge, no analytical/numerical solution for the

displacements and stresses in the cross-anisotropic half-space due to a conical and a parabolic circular load with Young’sand shear moduli varying exponentially with depth has been proposed. Integrating the point load solution of Wang et al.(2003), the displacements and stresses along the axisymmetric axis (z-axis), with conical and parabolic circular loads beingin the interior of an inhomogeneous cross-anisotropic half-space are generated. Then, the inhomogeneous solution can besimplified to the homogeneous solution by setting the inhomogeneity parameter to zero. Numerical results reveal thatthe induced displacement and stress by the proposed solutions agree very well with those by the analytical solutions. Inaddition, the present solutions also compare with Harr and Lovell’s (1963), and Geddes’s (1975) solutions when the mediumis isotropy. Two examples are illustrated to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropyon the induced vertical displacement and vertical normal stress along the axisymmetric axis, caused respectively by a conicaland a parabolic circular distribution of the vertical load, on the surface of the homogeneous/inhomogeneous isotropic/cross-anisotropic soils.

Table 1Differences between the homogeneous and inhomogeneous cross-anisotropic elastic constants.

Homogeneous (Liao and Wang, 1998) Inhomogeneous (Wang et al., 2003)

E Ee�kz

E0 E0e�kz

m mm0 m0

G0 G0e�kz

Table 2Soil types and their elastic properties (where E = 50 MPa, m = 0.3 are adopted in the numerical analysis).

Soil type E/E0 G0/E0 m/m0

Soil 1: Isotropy 1 0.385 1Soil 2: Cross-anisotropy 1 0.23 1Soil 3: Cross-anisotropy 1.35 0.385 1

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04

Normalized Vertical Displacement U zcon/pc (m/MPa)

Soil 1 with k=0 (numerical)

Soil 1 with k=0 (analytical)





Fig. 3. Comparison of numerical and analytical results for the normalized vertical displacement produced by a conical circular load along the z-axis for Soils1–3.


2. Displacement and stress components in an inhomogeneous cross-anisotropic material due to a vertical point load

In this work, the solutions for displacement and stress in an inhomogeneous cross-anisotropic material due to nonuni-form circular distribution of the vertical loads are directly integrated from Wang et al.’s point load solutions (2003). Theplanes of inhomogeneous cross-anisotropy are assumed to be parallel to the horizontal ground surface. The resulting solu-tions for the radial and vertical displacements, and vertical normal and shear stresses in Hankel domain (n, z) produced by astatic vertical point load, Pz, acting at z = h (h denotes the buried depth, as seen in Figs. 1 and 2) in the interior of an inho-mogeneous cross-anisotropic half-space are expressed as follows (Wang et al., 2006):

Fig. 4.(b) and

U�r ¼�Pz

4pC33C44A1e�u1n z�hj j þ A2eðk�u2nÞ z�hj j � A1

D1

De�u1nðzþhÞ � A3

D2

De�u1nzeðk�u2nÞh

�

� A4D3

Deðk�u2nÞze�u1nh � A2

D4

Deðk�u2nÞðzþhÞ

�ð1Þ

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04


Soil 1 with E/E'=1, G'/E'=0.385, ν/ν'=1k=0

k=-0.1

k=-0.3

k=-0.5

a

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04



k=-0.1

k=-0.3

k=-0.5

b

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04


Soil 3 with E/E'=1.35, G'/E'=0.385, ν/ν'=1k=0

k=-0.1

k=-0.3

k=-0.5

c

Effect of inhomogeneity parameter k on the normalized vertical displacement produced by a conical circular load along the z-axis for Soil 1 (a) Soil 2for Soil 3 (c).

Fig. 5.for Soil


U�z ¼Pz

4pC33C44B1e�u1n z�hj j þ B2eðk�u2nÞ z�hj j � B1

D1

De�u1nðzþhÞ � B3

D2


�

� B4D3

Deðk�u2nÞze�u1nh � B2

D4


�ð2Þ

r�zz ¼Pz

4pC33C44C1e�u1n z�hj j þ C2eðk�u2nÞ z�hj j � C1

D1

De�u1nðzþhÞ � C3

D2


�

� C4D3

Deðk�u2nÞze�u1nh � C2

D4


�ð3Þ

s�rz ¼�Pz

4pC33D1e�u1n z�hj j þ D2eðk�u2nÞ z�hj j þ D1

D1

De�u1nðzþhÞ þ D3

D2


�

þ D4D3

Deðk�u2nÞze�u1nh þ D2

D4


�ð4Þ

where A1 � A4, B1 � B4, C1 � C4, D1 � D4, D, D 1 � D4, Cij (i, j = 1–6), u1, and u2 can be referred to Wang et al. (2003, 2006).

� The differences between the homogeneous cross-anisotropic elastic constants (Liao and Wang, 1998) and the inhomoge-neous ones (Wang et al., 2003, 2006) adopted in this article are listed in Table 1. It is clear that, for the present inhomo-geneous cross-anisotropic medium, only three engineering elastic constants (E, E0, and G0) are assumed to be exponentiallydepending on the inhomogeneity parameter k (length�1); however, two Poisson’s ratios (m and m0) remain constants. Inaddition, according to the inhomogeneity parameter k, three different situations exist as follows:(1) k > 0, denotes a hardened surface, where E, E0, and G0 decrease with increasing depth.(2) k = 0, is referred to as the conventional homogeneous condition (Liao and Wang, 1998).(3) k < 0, denotes a soft surface, where E, E0, and G0 increase with increasing depth.The displacements Ur, Uz, and stresses

rzz, srz in the physical domain (r, z) for the inhomogeneous cross-anisotropic half-space can be obtained by taking theinversion of Hankel theorem for U�r (Eq. (1)), U�z (Eq. (2)), r�zz (Eq. (3)), s�rz (Eq. (4)) with respect to n of order 1, 0, 0, and1, in the following:

Ur

Uz

� �¼Z 1

0n

U�r J1ðnrÞU�zJ0ðnrÞ

� �dn ð5Þ

rzz

srz

� �¼Z 1

0n

r�zzJ0ðnrÞs�rzJ1ðnrÞ

� �dn ð6Þ

From Eqs. (1)–(4), it is noted that the integrands under the infinite integrals in Eqs. (5) and (6) involve products of Besselfunction of the first kind of order n (n = 0, 1), an exponential function, and a polynomial, which cannot be given in closed

5

4

3

2

1

0

z/a

0 0.2 0.4 0.6 0.8 1

Non-dimensional Vertical Normal Stress σzzcon/pc







Comparison of numerical and analytical results for the non-dimensional vertical normal stress produced by a conical circular load along the z-axiss 1–3.


form so that numerical integrations are required. The detailed numerical techniques to perform the integration can be foundin Wang et al. (2003, 2006).

3. Displacement and stress components in an inhomogeneous cross-anisotropic material due to nonuniform circulardistribution of the vertical loadings

In this article, considerations are given to the application of aforementioned vertical point load solutions to subsurfacecircular areas loaded in an axially symmetrical fashion in an inhomogeneous cross-anisotropic material. The subsurface cir-cular loadings include two types of nonuniform load are investigated as follows.

3.1. Case A: A conical circular distribution of the vertical load with pressure pC

Considering a loading pressure to vary linearly with radial position from a maximum magnitude pc at the center to zeroat the periphery of a circular area of radius a, is depicted in Fig. 1. The displacement and stress components along theaxisymmetric axis (z-axis) are the most commonly quoted in axially symmetrical loading problems (Geddes, 1975). Theresponse can be obtained by double integration of the point load solutions between the correct limits (Barden, 1963). That

5

4

3

2

1

0

z/a

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

a

5

4

3

2

1

0

z/a

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

b

c

5

4

3

2

1

0

z/a

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

Fig. 6. Effect of inhomogeneity parameter k on the non-dimensional vertical normal stress produced by a conical circular load along the z-axis for Soil 1 (a)Soil 2 (b) and for Soil 3 (c).


means as long as integrating Eqs. (5) and (6) for the displacement and stress components with respect to radius q from 0 to a,and with respect to h from 0 to 2p, the desired displacements Ucon

r , Uconz , and stresses rcon

zz , sconrz can be obtained as:

Fig. 7.Soils 1–

Uconr

Uconz

( )¼Z 2p

0

Z a

0

Ur

Uz

� �q 1� q

a

� �dqdh ð7Þ

rconzz

sconrz

� �¼Z 2p

0

Z a

0

rzz

srz

� �qð1� q

aÞdqdh ð8Þ

It has to be stated that Uconr , Ucon

z , and rconzz , scon

rz are referred to the z-axis thus they are functions of z. The infinitesimalpoint load at the point (r = q, h, z = h) involved in the integration is pz ¼ pcq 1� q

a

� dqdh. The quantities Ur, Uz, rzz, srz at

the z-axis (r = 0) due to the above infinitesimal load are taken from Eqs. (5) and (6) who are referred in a cylindrical systemwith origin the point O0 (q, h, z = 0). The quantities at Ur, Uz, rzz, srz at r = 0 in O (r, h, z) (Fig. 1) are the same, due to the axi-symmetric problem (values independent of the angle h), with the values occurred from Eqs. (5) and (6) at the point withradius r = q in the cylindrical system with origin the point O0 (q, h, z = 0).

The values of displacements and stresses in Eqs. (7) and (8) should be calculated numerically by using the techniques pre-sented in Wang et al. (2003, 2006).

3.2. Case B: A parabolic circular distribution of the vertical load with pressure pC

The second loading case investigated herein is shown in Fig. 2. In this figure, a load distributed in the form of parabola ofrevolution, symmetrically about the z-axis is studied. The solution also can be obtained by double integration of the pointload solutions, i.e., integrating Eqs. (5) and (6) for the displacement and stress components with respect to q from 0 to a,and with respect to h from 0 to 2p, to obtain Upar

r , Uparz , rpar

zz , and sparrz as:

Uparr

Uparz

� �¼Z 2p

0

Z a

0

Ur

Uz

� �q 1� q2

a2

�dqdh ð9Þ

rparzz

sparrz

( )¼Z 2p

0

Z a

0

rzz

srz

� �q 1� q2

a2

�dqdh ð10Þ

In this article, both proposed solutions provide mathematical model for applications to the problems in solid mechanicswhere the materials are of inhomogeneity and cross-anisotropy. However, the most interesting results for soil/rock mechan-ics and foundation engineering are the vertical displacements (Ucon

z and Uparz ), and vertical normal stresses (rcon

zz and rparzz ).

Numerical results for them are presented in the following section.

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04

Normalized Vertical Displacement U zpar/pc (m/MPa)







Comparison of numerical and analytical results for the normalized vertical displacement produced by a parabolic circular load along the z-axis for3.

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04

Normalized Vertical Displacement Uzpar/pc

(m/MPa)


k=-0.1

k=-0.3

k=-0.5

b a

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04

Normalized Vertical Displacement Uzpar/pc (m/MPa)


k=-0.1

k=-0.3

k=-0.5

c

5

4

3

2

1

0

z/a

0 0.01 0.02 0.03 0.04

Normalized Vertical Displacement U zpar/pc (m/MPa)


k=-0.1

k=-0.3

k=-0.5

Fig. 8. Effect of inhomogeneity parameter k on the normalized vertical displacement produced by a parabolic circular load along the z-axis for Soil 1 (a) Soil2 (b) and for Soil 3 (c).


4. Illustrative examples

A parametric study is conducted to examine the effect of inhomogeneity, and the type and degree of material anisotropy onthe displacements and stresses. The displacements and stresses frequently of major interest are the vertical components (Ged-des, 1975); hence, the distributions of vertical displacements (Ucon

z and Uparz ) and vertical normal stresses (rcon

zz and rparzz ) along

the centerline of a conical and a parabolic loaded circle acting on the surface (h = 0) of the inhomogeneous isotropic/cross-anisotropic materials are presented. The effect of the inhomogeneity parameter k, and the type and degree of material anisot-ropy, specified by the ratios E/E0, G0/E0, and m/m0 (E and E0 are Young’s moduli in the plane of cross-anisotropy and in a directionnormal to it, respectively; m and m0 are Poisson’s ratios characterizing the lateral strain response in the plane of cross-anisotropyto a stress acting parallel or normal to it, respectively; G0 is the shear modulus in planes normal to the plane of cross-anisot-ropy) on the displacements and stresses is studied. For typical ranges of cross-anisotropic parameters, Gazetas (1982a) sug-gested that the ratio E/E0 for clays ranging from 0.6 to 4, and was as low as 0.2 for sands. However, for the heavilyoverconsolidated London clay, the ratio for E/E0 is in the range of 1.35–2.37, and for G0/E0 is in 0.23–0.44 (Gibson, 1974; Leeand Rowe, 1989; Tarn and Lu, 1991). The elastic properties for the three types of isotropic and cross-anisotropic soils are listedin Table 2. The values adopted in Table 2 for E and m are 50 MPa (Chou and Bobet, 2002) and 0.3 (Karakus and Fowell, 2003),respectively. The variation of proposed solutions for k varies between 0 (homogeneous) to �0.5 (k < 0 denotes a soft surface).

5

4

3

2

1

0

z/a

0 0.2 0.4 0.6 0.8 1

Non-dimensional Vertical Normal Stress σzzpar/pc







Fig. 9. Comparison of numerical and analytical results for the non-dimensional vertical normal stress produced by a parabolic circular load along the z-axisfor Soils 1–3.


The reason why the situation with k > 0 (i.e., the hardened surface) is not chosen since it corresponds to an underground withdecreasing elastic moduli, which might not be the usual case for an earth material. Based on Eqs. ((1)–(4), (7)–(10)), a FORTRANprogram is written to calculate the displacements and stresses. The computed results of displacement and stress by aforemen-tioned numerical techniques for a conical and a parabolic distributed pressure are respectively demonstrated in Figs. 3–10.

Firstly, the influence of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement, producedby a conical circular distribution of the vertical load is analyzed. Fig. 3 shows the variations of the normalized verticaldisplacement (Ucon

z =pc) vs. the non-dimensional factor z/a (a is the radius of a loaded circle) for Soils 1–3 with k = 0 (thehomogeneous case). It is observed that the numerical results are nearly the same as those obtained from the analytical solu-tions, which are derived in Appendix A of this article. Fig. 4a–c plot the effect of k (from 0 to -0.5) on Ucon

z =pc for the isotropicSoil 1, and cross-anisotropic Soil 2 and Soil 3, respectively. The ranges of Ucon

z =pc in Fig. 4a–c are within 0–0.04. These figuresreveal that as the degree of inhomogeneity of a soil increases (from k = 0 to �0.5), the magnitude of Ucon

z =pc along the z-axisalmost decreases for all soils. In addition, it is found that the decrease of G0/E0 (from 0.385 (Fig. 4a) ? 0.23 (Fig. 4b)), andincrease of E/E0 (from 1 (Fig. 4a) ? 1.35 (Fig. 4c)), the magnitudes of induced vertical displacement, Ucon

z =pc, are quite differ-ent. That means in Fig. 4, the inhomogeneity parameter k, and the type and degree of soil anisotropy do have a remarkablyinfluence on the displacement, owing to a conical circular distribution of the vertical load. Following, Fig. 5 depicts the non-dimensional vertical normal stress rcon

zz =pc along the z-axis for Soils 1–3 vs. z/a when k = 0. The numerical results for thecross-anisotropic Soil 2 and Soil 3 are in good agreement with the analytical solutions, which are provided in Appendix A.Identically, the numerical result for the isotropic Soil 1 is well matching the analytical solutions given by Harr and Lovell(1963) and Geddes (1975). Fig. 6a–c illustrate the effect of k on rcon

zz =pc for Soils 1–3, respectively. The values of rconzz =pc in

Fig. 6a–c vary from �0.4 to 1.0. It is noted that with increasing k from 0 to -0.5, the magnitude of rconzz =pc nearly decreases

in all soils. Notably, the non-dimensional vertical normal stress, rconzz =pc, might be transferred to tension in Soil 1 (Fig. 6a)

when k = � 0.1, �0.3, �0.5; in Soil 2 (Fig. 6b) when k = � 0.5; and in Soil 3 (Fig. 6c) when k = � 0.3, �0.5.Secondly, the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement caused by a

parabolic circular distribution of the vertical load is explored. The normalized vertical displacement (Uparz =pc) vs. z/a for Soils

1–3 with k = 0 is demonstrated in Fig. 7. Apparently, the numerical results are equivalent to those by the analytical solutions(in Appendix A). The normalized vertical displacement (Upar

z =pc) for Soils 1–3 with k from 0 to �0.5 is respectively shown inFig. 8a–c. In these figures, the values of Upar

z =pc are also within 0-0.04 (the same with those of Uconz =pc). As it can be seen in

Fig. 8 that, as the degree of inhomogeneity of a soil increases (from k = 0 to �0.5), the magnitude of Uparz =pc along the z-axis

still approximately decreases for all soils. Finally, the non-dimensional vertical normal stress (rparzz =pc) vs. z/a for Soils 1–3

with k = 0 is plotted in Fig. 9. Once more, the numerical results of rparzz =pc for the isotropic Soil 1, and the cross-anisotropic

Soil 2 and Soil 3 are respectively in excellent agreement with the isotropic solutions of Harr and Lovell (1963) and Geddes(1975), and the cross-anisotropic solutions proposed in Appendix A. Fig. 10a–c depict the influence of k on rpar

zz =pc for Soils1–3, respectively. The variations of rpar

zz =pc in Fig. 10a–c are within -0.6-1.0, which are little distinct from those of rconzz =pc in

Fig. 6a–c. However, the similar phenomenon can be found in Soil 1 (Fig. 10a), Soil 2 (Fig. 10b), and Soil 3 (Fig. 10c) that withthe increase of the degree of inhomogeneity (k = 0 to �0.5), the magnitude of rpar

zz =pc decreases. Additionally, the inducedtensile stress by this loading type would be appeared in all soils, especially for the isotropic Soil 1 (Fig. 10a) withk = � 0.5, an extra tensile stress (rpar

zz =pc ¼ �0:507) is found at z/a = 1.5.

5

4

3

2

1

0

z/a

- 0.6 -0.4 - 0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

a

5

4

3

2

1

0

z/a

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

b

5

4

3

2

1

0

z/a

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1



k=-0.1

k=-0.3

k=-0.5

c

Fig. 10. Effect of inhomogeneity parameter k on the non-dimensional vertical normal stress produced by a parabolic circular load along the z-axis for Soil 1(a) Soil 2 (b) and for Soil 3 (c).


From Figs. 4, 6, 8, 10, the effect of inhomogeneity on the displacements and stresses owing to a conical and a paraboliccircular distribution of the vertical load can be evidently observed. The most interesting feature perhaps is that for k = � 01,�0.3, �0.5 in Soil 1, for k = � 0.5 in Soil 2, and for k = � 0.3, �0.5 in Soil 3, tensile stresses could be induced by both proposednonuniform loads. These figures are not only theoretically but also practically in the fields of engineering to determine thevertical displacements and vertical normal stresses on the loading axis, resulting from the nonuniform circular distributionof vertical loads in the inhomogeneous cross-anisotropic materials.

5. Conclusions

In this article, solutions are generated for the displacement and stress components along the axisymmetric axis due tononuniform circular distribution of the vertical loadings in a continuously inhomogeneous cross-anisotropic material withYoung’s and shear moduli varying exponentially with depth. The nonuniform loading types contain a conical and a paraboliccircular load. The planes of cross-isotropy are assumed to be parallel to the boundary surface. These solutions can be ob-tained by integrating the point load solutions, which were developed by Wang et al. (2003). The numerical techniques uti-lized herein include the integration over each of the first twenty half-cycles of Bessel functions, the Gauss quadrature


formula, and the well-known Simpson’s Rule for solving the present loading problem. Numerical results agree very well withthe analytical solution of displacements and stresses in a homogeneous cross-anisotropic half-space (derived in Appendix Aof this work) when the inhomogeneity parameter k is zero. In addition, the calculated results are identical with Harr andLovell’s (1963), and Geddes’s solutions (1975) when the material is isotropy. A parametric study by two illustrative examplesfor London clay has yielded that the inhomogeneity parameter k, and the type and degree of soil anisotropy do have a pro-foundly impact on the vertical displacement and vertical normal stress caused by proposed loading types.

The yielded solutions are never mentioned in literature, and they could realistically imitate the actual stratum of loadingsituations in many areas of engineering practice. Furthermore, the present solutions provide a new mathematical model forthe loading problems in solid mechanics where the materials are inhomogeneous and cross-anisotropy.

Acknowledgement

We acknowledge the funds support by NSC of ROC under Grant No. 96-2628-E-239-022-MY3.

Appendix A. The solutions of displacements (Upr , Up

z ) and stresses (rpzz, sp

rz) produced by a vertical point load Pz acting in theinterior of a homogeneous cross-anisotropic half-space (z = h) are quoted from Liao and Wang (1998), and are re-expressed as:

Upr ¼ �

Pz

4p½gðpdll � pdl2Þ þm1ðT1pdla � T3pdlcÞ �m2ðT2pdlb � T4pdldÞ� ðA:1Þ

Upz ¼ �

Pz

4p½m1ðgpd21 þ T1m1pd2a � T2m2pd2bÞ �m2ðgpd22 þ T3m1pd2c � T4m2pd2dÞ� ðA:2Þ

rpzz ¼

Pz

4p½ðC13 � u1m1C33Þðgps11 þ T1m1ps1a � T2m2ps1bÞ � ðC13 � u2m2C33Þðgps12 þ T3m1ps1c � T4m2ps1dÞ� ðA:3Þ

sprz ¼

Pz

4pC44½ðu1 þm1Þðgps21 þ T1m1ps2a � T2m2ps2bÞ � ðu2 þm2Þðgps22 þ T3m1ps2c � T4m2ps2dÞ� ðA:4Þ

where

� Cij (i, j = 1–6), mj (j = 1, 2), u1 and u2 of this case can be found in Liao and Wang (1998).� g ¼ ðC13þC44Þ

C33C44ðu21�u2

2Þ, T1 ¼ g

m1

u1þu2u2�u1

, T2 ¼ gm2

2u1ðu2þm2Þðu2�u1Þðu1þm1Þ

, T3 ¼ gm1

2u2ðu1þm1Þðu2�u1Þðu2þm2Þ

, T4 ¼ gm2

u1þu2u2�u1

.

� Defining pd1i ¼R�irRi

, pd2i ¼ 1Ri

(i = 1, 2, a, b, c, d) in Eqs. (A.1) and (A.2), and ps1i ¼ zi

R3i, ps2i ¼ r

R3i

(i = 1, 2, a, b, c, d) in Eqs. (A.3) and

(A.4) are respectively the elementary functions for the displacements and stresses.

� Ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2

i

q, R�i ¼ Ri � zi (i = 1, 2, a, b, c, d); z1 = u1(z � h), z2 = u2(z � h), za = u1(z + h), zb = u1z + u2h, zc = u1h + u2z,

zd = u2(z + h).

Hence, the solutions for displacements and stresses in a homogeneous cross-anisotropic medium due to a conical and aparabolic circular distribution of the vertical load can be directly integrated from aforementioned elementary functions(pd1i � pd2i and ps1i � ps2i). For example, the analytical solutions of displacement and stress along the axisymmetric axis(r = 0), resulting from a conical circular load can be regrouped as the forms of Eqs. ((A.1)–(A.4)) except for the elementary func-tions (pd1i � pd2i and ps1i � ps2i) are replaced by the displacement integral functions cd1i � cd2i, and stress integral functionscs1i � cs2i. The same is hold for the subjected parabolic one (i.e., the displacement integral functions are fd1i � fd2i, and the stressones are fs1i � fs2i). Therefore, only the displacement and stress integral functions for each loading type are presented below.

A.1. Displacement and stress integral functions for a conical circular distribution of the vertical load applied in the interior of ahomogeneous cross-anisotropic half-space 0 1

cd1i ¼ 2pa12� n2

i þ ni

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2

i

qþ ni ln

ni

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2

i

q

B@ CA ðA:5Þ

cd2i ¼ paffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2

i

q� 2ni � n2

i lnni


i

q

0B@

1CA ðA:6Þ

cs1i ¼ 2p 1þ ni lnni


i

q

0B@

1CA ðA:7Þ

cs2i ¼ 2p 2ni � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2

i

q� ln

ni


i

q

0B@

1CA ðA:8Þ

where ni = zi/a (i = 1, 2, a, b, c, d), and a is the radius of a circular loading area.


A.2. Displacement and stress integral functions for a parabolic circular distribution of the vertical load applied in the interior of ahomogeneous cross-anisotropic half-space 2 3

fd1i ¼ pa43þ ni


i

qþ nið2þ n2

i Þ lnni


i

q

64 75 ðA:9Þ

fd2i ¼2pa

32ð1þ n2

i Þ32 � nið3þ 2n2

i Þh i

ðA:10Þ

fs1i ¼ 2pð1þ 2n2i � 2ni


i

qÞ ðA:11Þ

fs2i ¼ �p 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2

i

qþ ð2þ 3n2

i Þ lnni


i

q

264

375 ðA:12Þ

The derived formulations for vertical stress are identical with the analytical solutions given by Harr and Lovell (1963), andGeddes (1975) when the medium is isotropy.

References

Barden, L., 1963. Stresses and displacements in a cross-anisotropic soil. Géotechnique 13 (3), 198–210.Bauer, G.E., Shields, D.H., Scott, J.D., Nwabuokei, S.O., 1979. Normal and shear measurements on a strip footing. Can. Geotech. J. 16 (1), 177–189.Chou, W.I., Bobet, A., 2002. Predictions of ground deformations in shallow tunnels in clay. Tunnel. Underground Space Technol. 17 (1), 3–19.Gazetas, G., 1982a. Stresses and displacements in cross-anisotropic soils. J. Geotech. Eng. Div., ASCE 108 (GT4), 532–553.Gazetas, G., 1982b. Axisymmetric parabolic loading of anisotropic halfspace. J. Geotech. Eng. Div., ASCE 108 (GT4), 654–660.Geddes, J.D., 1975. Vertical stress components produced by axially symmetrical subsurface loadings. Can. Geotech. J. 12 (4), 482–497.Gerrard, C.M., Wardle, L.J., 1973. Solutions for Point Loads and Generalized Circular Loads Applied to a Cross-anisotropic Half-space. CSIRO Technical Paper

13.Gibson, R.E., 1974. The analytical method in soil mechanics. Géotechnique 24 (2), 115–140.Hanson, M.T., Puja, I.W., 1998a. Elastic subsurface stress analysis for circular foundations I. J. Eng. Mech., ASCE 124 (5), 537–546.Hanson, M.T., Puja, I.W., 1998b. Elastic subsurface stress analysis for circular foundations II. J. Eng. Mech., ASCE 124 (5), 547–555.Harr, M.E., Lovell Jr., C.W., 1963. Vertical stresses under certain axisymmetrical loadings. High. Res. Board Rec., 39.Hemsley, J.A., 1991. Elastostatic deformation of a half-space under conical loading. Int. J. Numer. Anal. Method Geomech. 15 (11), 759–783.Hooper, J.A., 1975. Elastic settlement of a circular raft in adhesive contact with a transversely isotropic medium. Géotechnique 25 (4), 691–711.Hooper, J.A., 1976. Parabolic adhesive loading of a flexible raft foundation. Géotechnique 26 (3), 511–525.Karakus, M., Fowell, R.J., 2003. Effects of different tunnel face advance excavation on the settlement by FEM. Tunnel. Underground Space Technol. 18 (5),

513–523.Lee, K.M., Rowe, R.K., 1989. Deformation caused by surface loading and tunneling: the role of elastic anisotropy. Géotechnique 39 (1), 125–140.Liao, J.J., Wang, C.D., 1998. Elastic solutions for a transversely isotropic half-space subjected to a point load. Int. J. Numer. Anal. Method Geomech. 22 (6),

425–447.Pan, E., 1989. Static response of a transversely isotropic and layered half-space to general surface loads. Phy. Earth Planet. Inter. 54 (3–4), 353–363.Pan, E., 1997. Static Green’s functions in multilayered half-spaces. Appl. Math. Model. 21 (8), 509–521.Tarn, J.Q., Lu, C.C., 1991. Analysis of subsidence due to a point sink in an anisotropic porous elastic half space. Int. J. Numer. Anal. Method Geomech. 15 (8),

573–592.Wang, C.D., Liao, J.J., 2002a. Elastic solutions of displacements for a transversely isotropic half-space subjected to three-dimensional buried parabolic

rectangular loads. Int. J. Solids Struct. 39 (18), 4805–4824.Wang, C.D., Liao, J.J., 2002b. Elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional parabolic rectangular loads.

Int. J. Numer. Analyt. Method Geomech. 26 (14), 1449–1476.Wang, C.D., Tzeng, C.S., Pan, E., Liao, J.J., 2003. Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space.

Int. J. Rock Mech. Min. Sci. 40 (5), 667–685.Wang, C.D., Pan, E., Tzeng, C.S., Han, F., Liao, J.J., 2006. Displacements and stresses due to a uniform vertical circular load in an inhomogeneous cross-

anisotropic half-space. Int. J. Geomech., ASCE 6 (1), 1–10.

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