Computing displacements in transversely isotropic rocks using influence charts

and Rock Engineering : Springer-Verlag 1999 Printed in Austria

Computing Displacements in Transversely Isotropic Rocks

Using In¯uence Charts

By

C. D. Wang and J. J. Liao

Department of Civil Engineering, National Chiao-Tung University, Hsinchu, Taiwan, Republic of China

Summary

This paper presents a simple graphical method for computing the displacement beneath/at the surface of a transversely isotropic half-space subjected to surface loads. The surface load can be distributed on an irregularly-shaped area. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the half-space. Based on the point load solutions presented by the authors, four in¯uence charts are constructed for calculating the three displacements at any point in the interior of the half-space. Then, by setting z ˆ 0 of the derived solutions, another four in¯uence charts for computing the surface displacements can also be proposed. These charts are composed of unit blocks. Each unit block is bounded by two adjacent radii and arcs, and contributes the same level of in¯uence to the displace-ment. Following, a theoretical study was performed and the results showed that the charts for interior displacements are only suitable for transversely isotropic rocks with real roots of the characteristic equation; however, the charts for surface displacements are suitable for all transversely isotropic rocks. Finally, to demonstrate the use of the new graphical method, an illustrative example of a layered rock subjected to a uniform, normal circular-shaped load is given. The results from the new graphical method agree with those of analytical solutions as well. The new in¯uence charts can be a practical alternative to the existing analytical or numerical solutions, and provide results with reasonable accuracy.

1. Introduction

In the design of foundations on rocks, the deformation response of the materials is an important factor. Conventionally, foundation materials are assumed to be linearly elastic and isotropic for calculating the stresses, strains and displacements. However, for most rocks, such as foliated metamorphic, strati®ed sedimentary, and regularly jointed rocks, their responses to deformation exhibit some degree of anisotropy. Hence, isotropic elasticity is not suitable for computing the stresses, strains, and displacements in these rocks.

To calculate the stress, strain, and displacement in an anisotropic half-space, one can use the closed-form solutions, numerical methods, or graphical methods.

Most of the existing closed-form solutions are limited to solving plane strain or axisymmetric problems with simple boundary conditions, such as loading types and shapes. The detailed review of the exact solutions for anisotropic media can be referred to Wang and Liao (1998a). In the past few decades numerical techniques have been developed for calculating the stresses, strains, and displacements un-derneath an irregular-shaped foundation (Wang and Liao, 1998b). Through the use of computer, these numerical methods can easily be automated and hence can be e½cient to use. However, most of them contributed to the calculation of stresses or displacements in isotropic media.

Several graphical methods for computing the displacements in an isotropic half-space have been used for decades. A graphical method using in¯uence charts was ®rst proposed by Newmark (1947). The in¯uence charts are e½cient to use in calculating displacement as compared to other complex mathematical or numeri-cal methods. However, the advantages of Newmark charts diminish if the loading area is not uniform, or displacements at multiple depths are sought simulta-neously. Uzan et al. (1980) constructed in¯uence charts for a special case of a two-layer system underlain by a rigid base. Nevertheless, the charts proposed by Uzan et al. (1980) were prepared for the materials with special elastic constants; hence, the applications are restricted. Recently, Huang (1995) extended the Newmark method to construct diagrams for computing the displacement in an isotropic solid subjected to an embedded distributed uniform vertical load. Poulos (1967) pro-posed a graphical method, called the sector method, for calculating the displace-ments in an elastic half-space. All of the existing graphical methods are limited to isotropic media. To the authors' knowledge, no graphical method of displacement calculation has been proposed for anisotropic media. The aim of this paper is to construct the new in¯uence charts for calculating the displacements in a trans-versely isotropic half-space subjected to three-dimensional loads with an irregular shape. By superposition of values corresponding to the in¯uence charts, the three displacements at any point in the half-space can be estimated. This paper describes the background of the new in¯uence charts and their application procedure. An illustrative example is presented at the end of the paper to demonstrate the cedure of computing induced interior and surface displacements using the pro-posed in¯uence charts in a layered rock mass. Veri®cations are also made by comparing the graphical solutions (by the in¯uence charts) with the analytical solutions.

2. Deformability Anisotropy of Transversely Isotropic Rocks

Anisotropy is a general characteristic of foliated metamorphic rocks (e.g., argillite, slate, schist, phyllite, gneiss), strati®ed sedimentary rocks (e.g., shale, sandstone, coal, limestone), and regularly jointed rock masses. Deformability anisotropy implies that the deformability of a material is direction dependent. Depending on the planes of elastic symmetry, rock can be of general anisotropy, orthotropy, transverse isotropy, or complete isotropy. For a transverse isotropic rock, there is an axis of symmetry of rotation. The rock has isotropic properties in planes

normal to this axis. The deformability of a transversely isotropic material can be expressed in terms of ®ve elastic constants, i.e., C11, C13, C33, C44, C66 (Wang and

Liao, 1998b). These constants are directly related to the engineering elastic con-stants E, E0_{, u, u}0_{, and G}0_as:

C11 ˆ E 1 ÿ _EE₀u02 …1 ‡ u† 1 ÿ u ÿ 2E_E0u02 ; C13 ˆ Eu0 1 ÿ u ÿ2E_E0u0 2; C33 ˆ E 0_{1 ÿ u†} 1 ÿ u ÿ2E_E0u02; C44ˆ G 0_{; C} 66ˆ_{2…1 ‡ u†}E : …1†

Where E, E0 _{are the Young's moduli in the plane of transverse isotropy and its}

normal, respectively; u, u0_{are the Poisson's ratios characterizing the lateral strain}

in the plane of transverse isotropy to a normal stress acting parallel and normal to it, respectively; and G0 _{is the shear modulus in planes normal to the plane of}

transverse isotropy. These engineering elastic constants can be determined by static or dynamic experiments in the laboratory. Readers can refer to Wang and Liao (1998b) for details of these methods.

3. Construction of the Displacement In¯uence Charts for a Loaded Transversely Isotropic Half-Space

Similar to the Newmark charts (1947) for isotropic materials, the proposed charts for transversely isotropic media contain systematic unit blocks. Two radial lines and two adjacent arcs bound each block. The radii of the circles relate to the depth of the interested point for interior displacements, or the base length of loading area for surface displacements in the half-space. The in¯uence value of any unit block in displacement should be equal and independent of its location in the chart. To facilitate block counting, the plan of the surface load is drawn to a scale related to the depth of the interested point (for interior displacements) or the base length (for surface displacements). The unit blocks are made roughly square. The number of the blocks covered by the scaled loaded area is then counted.

Combining the solutions for displacements induced by di¨erent sectors with uniform loads (one of the sectors as shown in Fig. 1), one can obtain the dis-placements at point C with depth uiz due to the uniform load on a unit block.

Selecting proper values of coe½cients in the closed-form solutions for the dis-placements at point C, one can obtain the values of the radii and the central angle, which form the unit block, for the in¯uence displacements being a uniform amount, i.e. 0.004 or 0.01. In this section, the solutions for the displacements at a depth uiz under the vertex of a uniformly loaded sector of a circle in a transversely

isotropic half-space are presented ®rst. Then, four independent in¯uence charts for calculating the three interior displacements are proposed. Furthermore, by setting z ˆ 0 (at the surface) of the derived solutions, another four charts for calculating the surface displacements can also be proposed.

3.1 Interior Displacements under the Vertex of a Uniformly Loaded Sector The solutions of displacements in a transversely isotropic half-space subjected to a point load have been derived by several investigators (e.g., Liao and Wang, 1998). By integrating the point load solutions, one can obtain the displacements in the half-space, subjected to a uniform surface load of any irregularly-shaped area. Details of deriving displacements under the vertex of a uniformly loaded sector of a circle in a transversely isotropic half-space, based on the point load solutions of Liao and Wang (1998), are described as follows:

Figure 1 shows that a uniform load, Pj( j ˆ x; y; z, forces per unit of area) acts

on a sector bounded by two radial lines and a circle arc. In the ®gure, the depth of point C…0; 0; uiz† under the vertex is uiz, the radius is r, and the central angle is

b (positive counterclockwise with respect to X co-ordinate axis). Consider an ele-mentary area of rdrdb in the sector, the displacement at point C, ‰U ŠC, is derived by integrating the point load solutions (Liao and Wang, 1998) with dr from 0 to r and db from 0 to b (Gradshteynn and Ryzhik, 1994) as:

‰U ŠCˆ …_b 0 …_r 0‰U Š p_{r dr db; …2†}

where ‰U Š ˆ ‰ux; uy; uzŠT (superscript T denotes the transpose of matrix); the

superscript C denotes the point C at which the induced displacements are eval-uated; the superscript p indicates a point load acting at point O. Upon integration,

‰U ŠChas the following components: uC x ˆ Pxz  ÿ ku1 cF1‡kmu_u 2 1  cF2‡ 1 C44 cF3 ‡ ku1 eG1ÿkmu_u 2 1  eG2‡ 1 C44 eG3  ‡ Pyz  ku1 dG1ÿkmu_u 2 1  dG2‡ 1 C44 dG3  ‡ Pzku2z…u1 aH1ÿ m  aH2†; …3† uC y ˆ Pxz ku1 dG1ÿkmu_u 2 1  dG2‡ 1 C44 dG3   ‡ Pyz  ÿ ku1 cF1‡kmu_u 2 1  cF2‡ 1 C44 cF3 ÿ ku1 eG1‡kmu_u 2 1  eG2ÿ 1 C44 eG3  ‡ Pzku2z…u1 bH1ÿ m  bH2†; …4† uC z ˆ Pxkz…mu1 aH1ÿ u22 aH2† ‡ Pykz…mu1 bH1ÿ u22 bH2† ‡ 2Pzku1u2z…m  cF1ÿ u2 cF2†: …5† where:

.

u1u₂2…mÿu1†…C11C33ÿC₁₃2†; a ˆ sin b, b ˆ 1 ÿ cos b, c ˆ

b 2p, d ˆ 1ÿcos 2b

2 , e ˆsin 2b2 ; Fiˆ12 cos a1 iÿ 1

 

, Giˆ_4p1 ÿ1 ‡_{cos a}1 _iÿ 2 ln1‡cos a_{2 cos ai}i

 

, Hiˆ 1

2p tan aiÿ ln1‡sin acos aii   

 

, and tan aiˆ r=uiz…i ˆ 1; 2; 3†;

.

u4_{ÿ su}2_{‡ q ˆ 0; …6†}

where s ˆC11C33ÿC13…C13‡2C44†

C33C44 , q ˆ

C11

C33. If the strain energy is assumed to be pos-itive de®nite in the medium (Amadei et al., 1987), the roots of Eq. (6), u1 and

u2are restricted to the following three cases:

case 1. When s2_{ÿ 4q > 0, u} 1; 2ˆ G  f1 2‰s G  …s2_{ÿ 4q†} p Šg q

are two real distinct roots; case 2. When s2_{ÿ 4q ˆ 0, u} 1; 2ˆ G  s=2 p

; Gps=2are real double roots (i.e., complete isotropy);

case 3. When s2_{ÿ 4q < 0, u}

1ˆ1₂p…s ‡ 2 qp †ÿ i1₂p…ÿs ‡ 2 qp †ˆ g ÿ id,

u2ˆ g ‡ id are two conjugate complex roots [where g cannot be equal

to zero (Liao and Wang, 1998)].

root type of Eq. (6). G G0  2 …1 ‡ u† ÿ _EE₀   1 ÿ u ‡ _GE₀   u0_{ÿ 2} E E0   u02   > 0; for case 1 ˆ 0; for case 2 < 0; for case 3 : …7† Gerrard (1975) and Amadei et al. (1987) demonstrated that, for most trans-versely isotropic rocks, the values of E=E0 _{and G=G}0 _{are between 1 and 3, the}

Poisson's ratios u and u0 _{are between 0.15 and 0.35, and the value of u}0_E=E0 _is

between 0.1 and 0.7. Based on these data, Liao and Wang (1998) presented that approximately two thirds of transversely isotropic rocks belong to the case 1 (i.e., two real distinct roots).

3.2 In¯uence Charts for the Interior Displacements

Presented displacement in¯uence charts include an index length representing the depth of the desired point, and numbers of concentric circles and radial lines. A unit block, except for those adjacent to the point C, is formed by two radial lines and two concentric circle arcs. ‰U ŠCdepends on the geometry of the loaded sector as described in Eqs. (3)±(5). The geometry is de®ned by a set of coe½cients a; b; c; d; e, Fi, Giand Hi. The values of a; b; c; d and e depend on the central angle

b. The coe½cients Fi, Gi and Hi relate to the ratio of r=uiz. The value of c is

positive regardless of the value of b. The others (i.e., a; b; d and e) can be either positive or negative. For a given depth uiz, the values of Fi, Gi, Hionly depend on

the radius r, and F1ˆ F2ˆ F3, G1ˆ G2ˆ G3, H1ˆ H2. Hence, only the charts

for aHi, bHi, cFi, dGi, and eGi are required for estimating ‰U ŠC in a half-space

graphically. The data of r are calculated using the Newton-Raphson method. Combining the numerical value to be 0.4% of a unit load intensity, a series of b and r can be obtained for aHi, bHi, dGi, and eGi, and 1% of a unit load intensity

for cFi, respectively. Considering the symmetric properties of triangular functions,

the charts for aHi and bHi are identical, except that the X- and Y-axes are

exchanged. Consequently, only four independent charts (i.e., aHi, cFi, dGi, eGi)

are needed for computing ‰U ŠC. Figures 2±5 depict the in¯uence charts of aHi,

cFi, dGi and eGi, respectively. The index length of depth uiz in these ®gures is set

to 0.8 cm. The calculated r is symmetrical with respect to the original point O, therefore, only one quarter of the chart is drawn. The sign ``ÿ'' in the ®gures indicates that the values of a; b; d and e are negative. The in¯uence value is nega-tive for the blocks with a ``ÿ'' sign. The details of the preparation of the in¯uence charts can be referred to Wang and Liao (1998b).

For the medium with conjugate complex roots of the characteristic equation [Eq. (6)], the value of uiz is a complex variable. Hence, the presented four in¯uence

charts are not suitable for computing the displacements in a transversely isotropic half-space categorized into case 3. For case 3 material, the preparation of in¯uence charts requires elastic constants as a prior and uiz being replaced by z. This means

that the charts prepared for case 3 material are valid only for a particular medium. The Appendix illustrates the method for constructing the in¯uence chart and

Fig. 2. In¯uence chart for aHi(in¯uence value per block is G0:004, negative in¯uences are indicated by a minus, …ÿ†, sign)

Fig. 4. In¯uence chart for dGi(in¯uence value per block is G0:004, negative in¯uences are indicated by a minus, …ÿ†, sign)

Fig. 5. In¯uence chart for eGi(in¯uence value per block is G0:004, negative in¯uences are indicated by a minus, …ÿ†, sign)

calculating the vertical displacement in a half-space subjected to a uniform normal load for case 3 material.

3.3 Closed-Form Solutions and the In¯uence Charts for Surface Displacements Equations (3)±(5) are limited to solving the displacements in the interior of a transversely isotropic half-space subjected to three-dimensional uniform loads. Also, the proposed charts (Figs. 2±5) are only suitable for computing the displacements in the half-space subjected to irregularly-shaped surface loads. Practically, displacements at the surface of a half-space induced by surface loads are important. In order to prepare the in¯uence charts for computing the surface displacements of a transversely isotropic half-space, the closed-form solutions for the surface displacements (at point O) have to be derived. The exact solutions for surface displacements can be derived from Eqs. (3)±(5) by setting z ˆ 0. Then, the closed-form solutions for the surface displacements at point O due to uniform loads can be expressed as:

u0 xˆP₂xL ÿ k ÿkm_u 1 ÿ 1 u3C44    cI ‡_2p1 k ÿkm_u 1 ‡ 1 u3C44    eI   ‡P_4pyL k ÿkm_u 1 ‡ 1 u3C44    dI ‡P_2pzLk…u2ÿ m†  aI; …8† u0 y ˆP_4pxL k ÿkm_u 1 ‡ 1 u3C44    dI ‡P₂yL ÿ k ÿkm_u 1 ÿ 1 u3C44    cI ÿ_2p1 k ÿkm_u 1 ‡ 1 u3C44    eI   ‡P_2pzLk…u2ÿ m†  bI; …9† u0 z ˆP_2pxLk…m ÿ u2†  aI ‡P_2pyLk…m ÿ u2†  bI ‡ PzLku2…m ÿ u1†  cI; …10†

where I ˆ r=L, and L is the base length for the surface displacements. The loaded area can be drawn to any scale whatever and the base length L determined for the particular scale used. Equations (8)±(10) indicate that the displacements can be computed from knowing the loads, the base length, the material constants, and the geometry of sector (a; b; c; d; e; I). Hence, one can draw the charts for aI, bI, cI, dI and eI for computing the three surface displacements, using graphical methods. Similarly to preparing the charts for interior displacements, except that the radii of the circles relate to the base length (L) of surface displacements, the charts for aI, bI, cI, dI, and eI can be constructed. The chart for bI is the same as for aI if the X- and Y-axes are exchanged. Consequently, only four independent charts, aI (Fig. 6), cI (Fig. 7), dI (Fig. 8), and eI (Fig. 9) are needed for computing the surface displacements. The in¯uence value of each unit block on surface displace-ments is 0.04 of load intensity for aI, dI and eI, and of 0.01 of load intensity for cI, respectively. Since the base length L in the right corner of Figs. 6±9 is always

Fig. 6. In¯uence chart for aI (in¯uence value per block is G0:04, negative in¯uences are indicated by a minus, …ÿ†, sign)

Fig. 8. In¯uence chart for dI (in¯uence value per block is G0:04, negative in¯uences are indicated by a minus, …ÿ†, sign)

Fig. 9. In¯uence chart for eI (in¯uence value per block is G0:04, negative in¯uences are indicated by a minus, …ÿ†, sign)

a real number (i.e., 0.8 cm), the values of L are una¨ected by the media with conjugate complex roots (case 3) of the characteristic equation. Hence, the charts for aI, cI, dI and eI can be adopted to compute the surface displacements for all transversely isotropic rocks.

4. Use of the In¯uence Charts

The three displacement components at a point in the interior or at the surface of the half-space subjected to three-dimensional loads on an arbitrary shape can be estimated from the new proposed in¯uence charts. For this purpose, one must know (1) the elastic constants of half-space materials, (2) the types and magnitudes of surface loads, (3) the types of loading shapes, and (4) the depth of desired point …uiz† or the base length (L) of the surface displacements. Detailed procedure to use

the in¯uence charts and their applications are described as follows:

(1) Identify the type of rock (i.e., isotropic, transversely isotropic, orthotropic or generally anisotropic). If the rock is isotropic, the desired displacements can be computed using the Newmark charts (1947). However, the charts can only be used to compute the vertical displacement beneath/at the surface of an elastic mass induced by a uniform normal load. If the rock is orthotropic or generally anisotropic, there are no in¯uence charts available.

(2) Verify if the planes of isotropy are parallel to the surface. The in¯uence charts presented herein are applicable only if the planes of isotropy are parallel to the surface.

(3) Determine the root type of characteristic equation [i.e., case 1, 2 or 3, in Eq. (7)] for the half-space. Continue to step (4) through step (9) if the root type is case 1 or case 2. If the root type is case 3, the in¯uence charts will have to be prepared individually and the following steps do not apply.

(4) Calculate the characteristic root ui (i ˆ 1; 2; 3), functions m and k from Eqs.

(5) and (6).

(5) Adopt a scale that should be equal to the depth uiz …i ˆ 1; 2; 3† as shown in

Figs. 2±5 for the interior displacements, or equal to the base length L in Figs. 6±9 for the surface displacements.

(6) Redraw the plan of the loaded area, using the scale obtained in step (5). Transparent paper is recommended.

(7) Place the plan of the loaded area plotted in step (6) on the in¯uence charts. The point at which the displacements are desired should be placed over the center of the circles on these charts.

(8) Count the number of blocks on the in¯uence charts covered by the plan of the loaded area.

(9) Compute the interior displacements from Eqs. (3)±(5), or the surface dis-placements from Eqs. (8)±(10), based on functions m, k [from step (4)] and the number of blocks covered by plan of the loaded area [from step (8)].

Figure 10 presents a ¯ow chart that illustrates the use of the in¯uence charts. Although the charts are proposed for uniform loads, the displacement induced by

Fig. 10. Flow chart for computing the displacements induced by irregular loading shapes using in¯uence charts

non-uniform loads can be estimated by dividing the entire loading area into several sub-areas, each with an approximately uniform load.

5. Illustrative Example

To demonstrate and verify the usage of the proposed displacement in¯uence charts, an example is illustrated in this section. The interior and surface displace-ments of a half-space constituted by the layered rocks subjected to a uniform circular-shaped load are computed (Fig. 11a). The load with circular shape is chosen because there are several analytical solutions (i.e., Hanson and Puja, 1996) for verifying the results obtained from the presented graphical methods.

Fig. 11a shows that the vertical displacement at point C …uC

z† with a depth of 5

meters below the surface point O is desired. The layered rocks can equivalently be transversely isotropic rock, and the planes of transverse isotropy are parallel to the horizontal surface. The equivalent transversely isotropic properties of the layered rocks can be obtained from Wardle and Gerrard (1972). The mechanical properties of the hypothetical layered rock are given in Table 1. This ten-layer hypothetical rock satis®es the assumption of Salamon's model (1968) that the representative sample must contain a large number of layers. The deformability properties (Ei, ui)

of the hypothetical layers are adopted from Kulkawy (1975) for various sedimen-tary rocks obtained from uniaxial compression tests. The adopted Ei increases

with the increase of depth. Then, the ®ve equivalent elastic constants of the layered rock are E ˆ 42:5 GPa, E0_{ˆ 30:5 GPa, u ˆ 0:24, u}0_{ˆ 0:14, G}0_{ˆ 13:3 GPa. The}

calculated ®ve elastic constants satisfy the chosen domains of Gerrard (1975) and Amadei et al. (1987) for most transversely isotropic rocks. From Eq. (7), the me-dium belongs to case 1 with two real distinct roots. The half-space is subjected to a uniform normal load (Pz) on the horizontal surface with a loading area shown in

Fig. 11a. Equation (5) then is rewritten as: uC

z=Pzˆ 2ku1u2z  …m  cF1ÿ u2 cF2†: …11†

Equation (11) indicates that, knowing the elastic constants, ui, m, and k, one

independent in¯uence chart cFi (Fig. 3) is enough for computing the normalized

vertical displacement uC

z=Pz in this example. For illustration, the procedures of

calculating uC

z=Pzare described as follows:

(1) Calculate the characteristic roots and functions m and k: u1ˆ 0:801, u2ˆ

1:495, m ˆ 2:224 and k ˆ 0:029.

(2) Set the unit length as: u1z ˆ 4:005, u2z ˆ 7:475 for cF1and cF2, respectively.

(3) Redraw the plan of the loaded area, using the scales obtained in step (2) on transparent papers (for cF1and cF2).

(4) Place the transparent papers prepared in step (3) on the in¯uence chart …cFi†.

Point C should be placed over the center of the chart. Figs. 11b and 11c demonstrate the procedure for overlapping planes of the loaded area on the chart for cF1 and cF2, respectively.

Fig. 11a, b. a Plan of loaded area acting on the surface, b the blocks covered by the plan of the loaded area for cF1

Fig. 11c, d. c The blocks covered by the plan of the loaded area for cF2, d the blocks covered by the plan of the loaded area for cI

the loaded area. The numbers of blocks are about 41 in Fig. 11b, and 20 in Fig. 11c.

(6) From Eq. (11), the normalized vertical displacement …uC

z=Pz† at point C is

computed as: uC

z=Pzˆ 2  0:029  0:801  1:495  5  …2:224  41 ÿ 1:495  20†  0:01

ˆ 0:2128…m=GPa†:

Comparing the result with analytic solutions of Hanson and Puja (1996), the vertical displacement computed using the in¯uence chart agrees with the analytic result within 2%.

Also, the vertical surface displacement at point O …u0

z† is computed. For a

uniform normal load acting on the horizontal surface shown in Fig. 11a, Eq. (10) can be rewritten as:

u0

z=Pzˆ Lku2…m ÿ u1†  cI: …12†

Equation (12) indicates that the normalized vertical surface displacement u0

z=Pz can be calculated from a single in¯uence chart cI (Fig. 7). Figure 11d

demonstrates the procedure for overlapping plan of the loaded area on the chart for cI. The base length (L) is set to be 5 meters. Then, approximately 71 blocks are located in the loaded area of Fig. 11d. Hence, the value of u0

z=Pz[Eq. (12)] is equal

to 0.2190. Comparing with the analytical result (0.2174), the di¨erence between them is less than 1%. Similarly, one can easily compute the horizontal displace-ments (uC

x; ux0; uyC; uy0) following the above procedures.

6. Conclusions

Based on the derived closed-form solutions for displacements under the vertex of a uniformly loaded sector in a transversely isotropic half-space, four independent in¯uence charts are proposed for calculating the three displacement components at

Table 1. Thickness and deformability properties of sedimentary rocks for the illustrative layered rock

Rock layer Rock type Thickness Ei ui

i ti(m) (GPa) 1 conglomerate 2.0 13.0 0.15 2 subgraywacke 4.0 14.6 0.07 3 sandstone 4.0 18.4 0.21 4 marlstone 10.0 19.3 0.04 5 graywacke 10.0 20.1 0.08 6 siltstone 10.0 24.0 0.18 7 shale 10.0 26.0 0.09 8 limestone 10.0 47.5 0.23 9 dolomite 20.0 59.0 0.30 10 anhydrite 20.0 75.8 0.27

any point in the interior of a half-space subjected to three-dimensional surface loads on an irregularly-shaped area. Then, by setting z ˆ 0 in the derived solutions, another four in¯uence charts for computing the surface displacements are also proposed. The desired displacements are computed from the charts by counting the number of elements covered by a plan of the loaded area, drawn to a proper scale and laid upon the charts. The in¯uence values from the four in¯uence charts are then summed up. Since the in¯uence charts for computing the interior displacements are prepared on the basis of the index length …uiz†, the proposed

charts for computing the interior displacements are only suitable for a transversely isotropic half-space with real roots of the characteristic equation. However, the charts for calculating the surface displacements can be adopted for all transversely isotropic media because of the base length (L) always being a real number. The new in¯uence charts are easy to use, and the computed results are reasonably ac-curate. These charts o¨er a practical alternative to the analytical and numerical solutions.

Acknowledgments

The authors wish to thank the National Science Council of the R.O.C. for ®nancially supporting this research under contract No. NSC 86-2621-E009-011.

Appendix: Illustration for Constructing the In¯uence Chart for Case 3 To demonstrate the construction and usage of in¯uence charts for case 3, an ex-ample for computing the vertical displacement …uC

z† subjected to a uniform normal

load …Pz† is illustrated. From Eq. (6), u1ˆ g ÿ id, u2ˆ g ‡ id. Then, the

normal-ized vertical displacement …uC

z=Pz† can be expressed in terms of the central angle b

and a depth ratio r/z as follows: uC z=Pzˆ 2kmu1u2z  cF1ÿ 2ku1u22z  cF2 ˆ z  cF0_{; …13†} where c ˆ_2pb, F0_{ˆ ku} 2 ÿu1…m ÿ u2† ‡ m  r z ÿ
₂ ‡u2 1 q ÿ u1  …r z†2‡ u22 q   .

Similar to the method for drawing the charts for aHi, cFi, dGi, and eGi, the

chart for cF0_{can also be constructed, except that the elastic constants of the}

me-dium are involved in this chart. Assuming that the elastic constants are E ˆ 50 GPa, E0_{ˆ 25 GPa …E=E}0_{ˆ 2†, G=G}0_{ˆ 1, and u ˆ u}0_{ˆ 0:25, and solving Eq. (6),}

the characteristic roots are complex and the values of g and d are 1.0082 and 0.5914, respectively. Figure 12 is the in¯uence chart for cF0_{. For a uniform load as}

shown in Fig. 11a and using z as the scale (right corner of Fig. 12), one can redraw the plan of the loaded area. The number of blocks covered by the loaded area is approximately 37. Using Eq. (13), the normalized vertical displacement uC

z=Pzis

equal to 0.185 (ˆ5  37  0:001, m/GPa). The value is very close to the exact solu-tions (0.188) of Hanson and Puja (1996).

List of Symbols

a; b; c; d; e functions of central angle, b Cij …i; j ˆ 1 @ 6† elastic constants

E, E0_{; u; u}0_{; G}0 _{engineering elastic constants of a transversely isotropic rock}

Ei; ui deformability properties of the i-th layer of the layered rock

F0 _{functions of the complex roots g, d, and the depth ratio, r/z}

Fi; Gi; Hi functions of the depth ratio, r=uiz …i ˆ 1; 2; 3†

L the base length

Pj… j ˆ x; y; z† uniform loads (forces per unit of area)

r radius of a circle

r=uiz the depth ratio

ti thickness of the i-th layer of the layered rock

u1; u2; u3 roots of the characteristic equation

uC

x; uyC; uzC interior displacements induced by irregularly-shaped loads

u0

x; uy0; uz0 surface displacements induced by irregularly-shaped loads

X, Y, Z Cartesian co-ordinate system

b central angle

g; d real and imaginary part of the complex roots, respectively

References

Amadei, B., Savage W. Z., Swolfs, H. S. (1987): Gravitational stresses in anisotropic rock masses. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 24, 5±14.

Fig. 12. Plan of loaded area on the in¯uence chart for cF0_{(E ˆ 50 GPa, E}0_{ˆ 25 GPa, G=G}0_{ˆ 1, u ˆ} u0_{ˆ 0:25, in¯uence value per block is 0.001)}

Gerrard, C. M. (1975): Background to mathematical modeling in geomechanics: the roles of fabric and stress history. In: Proc., Int. Symp. on Numerical Methods, Karlsruhe, 33± 120.

Gradshteynn, I. S., Ryzhik, I. M. (1994): Tables of integrals. Series and products, Academic Press, San Diego, California.

Hanson, M. T., Puja, I. W. (1996): Love's circular patch problem revisited: closed form solutions for transverse isotropy and shear loading. Q. Appl. Math. 54, 359±384. Huang, S. Y. (1995): Graphical solution for vertical stress distributions and settlement in

soil due to uniformly applied vertical loads acting on a buried area. M.Sc. Thesis, The Cooper Union for the advancement of science and art.

Kulhawy, F. H. (1975): Stress deformation properties of rock and rock discontinuities. Engng. Geol. 9, 327±350.

Liao, J. J., Wang, C. D. (1998): Elastic solutions for a transversely isotropic half-space subjected to a point load. Int. J. Numer. Anal. Methods Geomech. 22, 425±447. Newmark, N. M. (1947): In¯uence charts for computation of vertical displacements in

elastic foundations. Bull. No. 367, Eng. Expt. Station, University of Illinois.

Poulos, H. G. (1967): The use of the sector method for calculating stresses and displace-ments in an elastic mass. In: Proc., 5th Aust.-New Zealand Conf. Soil Mech. Fndn. Engng., Auckland, 198±204.

Salamon, M. D. G. (1968): Elastic moduli of a strati®ed rock mass. Int. J. Rock Mech. Min. Sci. 5, 519±527.

Uzan, J., Ishai, I., Ho¨man, M. S. (1980): Surface de¯exion in a two-layer elastic medium underlain by a rough rigid base. Geotechnique 30, 39±47.

Wang, C. D., Liao, J. J. (1998a): Elastic solutions for a transversely isotropic half-space subjected to buried asymmetric-loads. Int. J. Numer. Anal. Methods Geomech (in press).

Wang, C. D., Liao, J. J. (1998b): Stress in¯uence charts of transversely isotropic rocks. Int. J. Rock Mech. Min Sci. 35, 771±785.

Wardle, L. J., Gerrard, C. M. (1972): The equivalent anisotropic properties of layered rock and soil masses. Rock Mech. 4, 155±175.

Authors' address: Prof. J. J. Liao, Department of Civil Engineering, National Chiao-Tung University, Hsinchu 30050, Taiwan, Republic of China.