Elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional buried parabolic rectangular loads

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 2002; 26: (DOI: /nag.253) Elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional buried parabolic rectangular loads Cheng-Der Wang 1,n,y and Jyh-Jong Liao 2 1 Department of Civil Engineering, Nanya Institute of Technology, Chung-Li 320, Taiwan, R.O.C. 2 Department of Civil Engineering, National Chiao-Tung University, Hsin-Chu 300, Taiwan, R.O.C. SUMMARY In many areas of engineering practice, applied loads are not uniformly distributed but often concentrated towards the centre of a foundation. Thus, loads are more realistically depicted as distributed as linearly varying or as parabola of revolution. Solutions for stresses in a transversely isotropic half-space caused by concave and convex parabolic loads that act on a rectangle have not been derived. This wor proposes analytical solutions for stresses in a transversely isotropic half-space, induced by three-dimensional, buried, linearly varying/uniform/parabolic rectangular loads. Load types include an upwardly and a downwardly linearly varying load, a uniform load, a concave and a convex parabolic load, all distributed over a rectangular area. These solutions are obtained by integrating the point load solutions in a Cartesian coordinate system for a transversely isotropic half-space. The buried depth, the dimensions of the loaded area, the type and degree of material anisotropy and the loading type for transversely isotropic half-spaces influence the proposed solutions. An illustrative example is presented to elucidate the effect of the dimensions of the loaded area, the type and degree of roc anisotropy, and the type of loading on the vertical stress in the isotropic/transversely isotropic rocs subjected to a linearly varying/uniform/parabolic rectangular load. Copyright # 2002 John Wiley & Sons, Ltd. KEY WORDS: analytical solutions; stresses; transversely isotropic half-space; three dimensional; buried; linearly varying, uniform, and parabolic rectangular loads 1. INTRODUCTION Anisotropic deformability is common in foliated metamorphic, stratified sedimentary and regularly jointed roc masses. Existing closed-form solutions, assuming linear and isotropic elasticity, for stress in such rocs or roc masses are normally not realistic. Better results can only be obtained by considering anisotropic deformability. Practically, an anisotropic roc can be modelled as either an orthotropic or a transversely isotropic material. This wor derives n Correspondence to: Cheng-Der Wang, Department of Civil Engineering, Nanya Institute of Technology, 414, Chung-Shan E. Road. Sec.3, Chung-Li 320 Taoyuan, Taiwan, R.O.C. y cdwang@nanya.edu.tw Contract/grant sponsor: National Science Council of the Republic of China; contract/grant number: NSC E Copyright # 2002 John Wiley & Sons, Ltd. Received 27 August 2001 Revised 19 June 2002

2 1450 C. D. WANG AND J.-J. LIAO elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional, buried, linearly varying/uniform/parabolic rectangular loads. Numerical methods, graphical methods, and closed-form solutions can be used to calculate the stresses induced by external loads in an anisotropic half-space. The numerical procedures can easily be automated with modern computers. However, most contributions have addressed the calculation of stresses/displacements in isotropic media. The authors have proposed graphical method of computing stress/displacement in transversely isotropic rocs subjected to three-dimensional, irregularly shaped surface loads [1,2]. The use of anisotropic influence charts to calculate the stresses/displacements is fast. However, the advantages of using the influence charts decline if the loading region is not uniform or stresses/displacements at multiple depths are simultaneously sought. Therefore, the closed-form solution method to estimate the stresses induced by non-uniform loads may be an alternative to the numerical or graphical method. A point load solution forms the basis of solutions to complex loading problems. Several researchers have presented solutions for stresses or displacements in response to a concentrated force applied to transversely isotropic half-spaces [3 5]. Wang and Liao [1,6] detailed various anisotropic loading conditions to obtain solutions in cases other than those that involved point loads. Stresses in a transversely isotropic half-space subjected to an arbitrary shape loaded area can be estimated by dividing the loaded area into several regularly shaped sub-areas, including triangles or rectangles. The authors recently derived closed-form solutions for stresses and displacements in a transversely isotropic half-space subjected to uniform/linearly varying triangular loads [7]. However, only uniform and some linearly varying rectangular loads acting on the transversely isotropic half-space were presented [6,8]. In many engineering fields [9,10], applied loads are not uniformly distributed but more concentrated towards the centre of the foundation. Hence, loads may be more realistically simulated as being distributed as linearly varying or as parabola of revolution. Teferra and Schultze [11] obtained solutions for stresses beneath the centre of a load in an isotropic half-space, for vertical concave and convex parabolic loads of infinite length. Nevertheless, existing closed-form solutions for anisotropic half-spaces are available only for axisymmetric problems. Gazetas [12,13] analytically investigated how soil s transverse isotropy affects stress distributions when it is subjected to axisymmetric parabolic vertical surface loading. To our nowledge, no closed-form solution for stresses in a transversely isotropic medium subjected to three-dimensional, buried, parabolic asymmetric loads has been proposed. Integrating the point load solutions in a Cartesian co-ordinate system [6] yields analytical solutions for stresses in the half-space caused by linearly varying/parabolic rectangular loads. The derived solutions are clear and concise. Also, according to our results, the buried depth, the dimensions of the loaded region, the type and degree of material anisotropy, and the loading type all affect stresses in a transversely isotropic half-space. An illustrative example is presented at the end of this paper to elucidate the effect of the dimensions of the loaded area, the type and degree of roc anisotropy, and the type of loading on vertical stress in isotropic/ transversely isotropic rocs subjected to a linearly varying/uniform/parabolic rectangular load. 2. SOLUTIONS FOR STRESSES INDUCED BY LINEARLY VARYING AND PARABOLIC RECTANGULAR LOADS In this wor, the solutions for stresses in a transversely isotropic half-space subjected to threedimensional, buried, linearly varying, uniform and parabolic rectangular loads are directly

3 ELASTIC SOLUTIONS FOR STRESSES 1451 integrated from the point load solutions in a Cartesian co-ordinate system [6]. Planes of transverse isotropy are assumed to be parallel to the horizontal surface. Appendix A provides closed-form solutions for stresses subjected to a point load ðp x ; P y ; P z Þ that acts at z ¼ h (measured from the surface) in the interior of a transversely isotropic half-space. In the case of point load solutions, p p is defined in Equations (A1) (A6) as the elementary functions for stresses. The solutions for stresses in a transversely isotropic half-space subjected to linearly varying and parabolic rectangular loads can be directly obtained by integrating the elementary functions of the point load solutions. The closed-form solutions for stresses induced by linearly varied loads distributed over a rectangular area are first given below Linearly varying rectangular loads A three-dimensional, upwardly linearly varying load, Pj linear ðj ¼ x; y; zþ (forces per unit area) distributed on a rectangle of length L and width W at a buried depth of h as shown in Figure 1 is Figure 1. The case of upwardly linearly varying rectangular loads with L * W area at the buried depth of hða > 0Þ:

4 1452 C. D. WANG AND J.-J. LIAO considered. The loading type in Figure 1 can be expressed as the following form *P linear j ¼ Pj linear 1 þ a jxj L þ jyj jxyj W LW ð1þ where a is a constant. According to Equation (1), a specifies three different loading cases. Case 1: a > 0; the load is upwardly linearly varying as depicted in Figure 1; Case 2: a ¼ 0; the load is uniform, as shown in Figure 2(a); Figure 2. (a) The case of uniform rectangular loads with L * W area at the buried depth of hða ¼ 0Þ; (b) the case of a completely downwardly linearly varying rectangular loads with L * W area at the buried depth of hða ¼ 1Þ:

5 ELASTIC SOLUTIONS FOR STRESSES 1453 Case 3: a50; the load is downwardly linearly varying. Figure 2(b) shows zero contact stress applied to the region at the edges of Figure 1 in the case a ¼ 1: An elementary force *P linear j dz db; acting on an elementary surface dz dz; is extracted from the rectangle to calculate the stresses in the transversely isotropic half-spaces. Replacing the concentrated force P j by *P linear j dz dz; y by ðy ZÞ; and x by ðx zþ in Equations (A1) (A6) yields solutions for stresses under the action of the elementary force in the half-space. Integrating the solutions with respect to Z from 0 to W ; and with respect to z from 0 to L; we can derive the complete solutions: ½sŠ linear ¼ Z L Z W 0 0 ½sŠ p dz db where ½sŠ ¼½s xx ; s yy ; s zz ; t xy ; t yz ; t xz Š T (superscript, T, denotes the transpose matrix) and the superscripts, linear and p; refer to the stress components induced by a linearly varying rectangular load and a point load, respectively. The explicit solutions for stresses in a half-space can be regrouped in the forms of Equations (A1) (A6). The exact solutions in this case are therefore just Equations (A1) (A6), except that the stress elementary functions p ; p ;...; p are replaced by the stress integral functions N h1i N h4i ; N h1i N h4i ;...; N h1i N h4i ði ¼ 1; 2; 3; a; b; c; d; eþ for s linear xx ; s linear yy ; s linear zz ; t linear xy ; t linear yz ; t linear xz (Figure 1), respectively. For example, p is replaced by N h1i N h4i as follows. ½p Š)N h1i jxj þ a L þ jyj jxyj W LW * N h1i 1 jyj 1 L W * N h2i 1 jxj 1 W L * N h3i 1 LW *Nh4i ð3þ Similarly, the solutions under parabolic loading conditions can also be expressed as Equations (A1) (A6), except for the integral functions. Hence, only the stress integral functions are presented. N h1i ¼ S 1 S 2 ð4þ N h2i ¼ ys 3 þ y n S 4 z i S 5 ð5þ N h3i ¼ ðr i R xn i R yn i þ R xn y n iþ ð6þ N h4i ¼ 1 2 ½ðR i R yn iþx ðr xn i R xn y n iþx n þðy 2 þ z 2 i ÞS 3 ðy * 2 þ z 2 i ÞS 4Š ð7þ N h1i ¼ S 3 S 4 ð8þ ð2þ N h2i ¼ N h3i ð9þ N h3i ¼ xs 1 þ x n S 2 z i S 5 ð10þ N h4i ¼ 1 2 ½ðR i R xn iþy ðr yn i R xn y n iþy n þðx 2 þ z 2 i ÞS 1 ðx * 2 þ z 2 i ÞS 2Š ð11þ N h1i ¼ S 5 ð12þ

6 1454 C. D. WANG AND J.-J. LIAO N h2i ¼ z i N h1i ð13þ N h1i N h2i N h3i N h2i s4i N h3i s4i ¼ switch x; y in N h2i N h4i ¼ z i N h3i ð14þ ð15þ N h1i s4i ¼ S 6 S 7 ð16þ ¼ ys 8 þ y n S 9 z i N h1i ¼ switch x; y in N h2i s4i ð17þ ð18þ N h4i s4i ¼ 1 2 ðlw þ x2 S 10 x * 2 S 11 þ y 2 S 8 y * 2 S 9 z 2 i S 5Þ ð19þ N h4i N h2i N h1i ¼ S 5 þ S 8 S 9 ð20þ ¼ ys 12 y n S 13 þ z i N h1i ð21þ N h3i ¼ xs 6 þ x n S 7 ð22þ ¼ 1 2 ðz in h3i x 2 S 6 þ x * 2 S 7 þ y 2 S 12 y * 2 S 13 Þ ð23þ N h1i s6i N h2i s6i N h3i s6i N h4i s6i ¼ N h1i ¼ N h2i ¼ N h3i ¼ N h4i N h1i N h2i N h3i N h4i ¼ y þ y n R i þ z i R xn i þ z i R yn i þ z i R xn y n i þ z i x ¼ y xn x x þ y n n R i þ z i R xn i þ z i R yn i þ z i R xn y n i þ z i ys 3 þ y n S 4 þ z i ðs 8 S 9 Þ N h3i ¼ x x 1 1 * 2 þ z i ðs 6 S 7 Þ R i þ z i R yn i þ z i R xn i þ z i R xn y n i þ z i N h4i ¼ x 2 ðr i R yn iþþ xn 2 ðr x n i R xn y iþ n þ x x 1 1 * 3 R i þ z i R yn i þ z i R xn i þ z i R xn y n i þ z i ð24þ ð25þ ð26þ ð27þ ð28þ ð29þ ð30þ ðy2 þ z 2 i Þ 2 S 3 þ ðy * 2 þ z 2 i Þ S 4 2 ð31þ

7 ELASTIC SOLUTIONS FOR STRESSES 1455 where N h1i ¼ switch x; y in N h1i N h2i N h3i N h4i ¼ switch x; y in N h3i ¼ switch x; y in N h2i ¼ switch x; y in N h4i x n ¼ x L; y n ¼ y W ; R xn i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x * 2 þ y 2 þ z 2 i ; R yn i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y * 2 þ z 2 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R xn y n i ¼ x * 2 þ y * 2 þ z 2 i ði ¼ 1; 2; 3; a; b; c; d; eþ; S 1 ¼ ln R y n i þ y n R i þ y ; S 2 ¼ ln R x n y n i þ y n R xn i þ y S 3 ¼ R x n i þ x n R i þ x ; S 4 ¼ ln R x n y n i þ x n R yn i þ x ; 1 xy S 5 ¼ tan tan 1 z i R i S 6 ¼ ln R y n i þ z i R i þ z ; i xn y tan 1 xyn þ tan 1 x n y n z i R xn i z i R yn i z i R xn y n i S 7 ¼ ln R x n y n i þ z i R xn i þ z ; i S 8 ¼ tan 1 y2 þ z i ðr i þ z i Þ tan 1 y2 þ z i ðr xn i þ z i Þ xy x n y S 9 ¼ tan 1 y * 2 þ z i ðr yn i þ z i Þ tan 1 y 2 * þ z i ðr xn y n i þ z i Þ xy n x n y n S 10 ¼ tan 1 x2 þ z i ðr i þ z i Þ tan 1 x2 þ z i ðr yn i þ z i Þ xy xy n ð32þ ð33þ ð34þ ð35þ S 11 ¼ tan 1 x * 2 þ z i ðr xn i þ z i Þ tan 1 x * 2 þ z i ðr xn y n i þ z i Þ ; x n y x n y n S 12 ¼ ln R x n i þ z i R i þ z ; S 13 ¼ ln R x n y n i þ z i i R yn i þ z i Equations (A1) (A6), (3) and (4) (35) can easily be solved automatically to calculate the stresses in a transversely isotropic half-space subjected to three-dimensional, buried, linearly varying rectangular loads Parabolic rectangular loads A non-linear, three-dimensional, buried load distributed as a concave parabola on a rectangle (Figure 3) is considered to demonstrate the results for non-linearly distributed loads. Figure 3 depicts that the concave parabolic load applied over a rectangular region with sides L and W :

8 1456 C. D. WANG AND J.-J. LIAO Figure 3. The case of concave parabolic rectangular loads with L*W area at the buried depth of hðb > 0Þ: The type of loading shown in Figure 3 has the following form [14]: *P par j ¼ P par j 1 þ b x2 L 2 þ y2 W 2 x2 y 2 L 2 W 2 ð36þ where b is a constant, and that specifies the following three cases: Case 1: b > 0; the load is concave parabolic, as shown in Figure 3; Case 2: b ¼ 0; the load is uniform; this case is identical to that of a ¼ 0 (Figure 2(a)); Case 3: b50; the load is convex parabolic. Figure 4 depicts zero contact stress at the region shown at the edges of Figure 3, for the case of b ¼ 1: The elementary force P j ; acting on a small rectangle, can also be expressed as *P par j dz dz ðj ¼ x; y; zþ (forces per unit area). Similarly, as for the linearly varying rectangular load, the solutions for stresses can be obtained by direct integration as follows: ½sŠ par ¼ Z L Z W 0 0 ½sŠ p dz db ð37þ

9 ELASTIC SOLUTIONS FOR STRESSES 1457 Figure 4. The case of completely convex parabolic rectangular loads with L * W area at the buried depth of hðb ¼ 1Þ: where the superscripts, par and p; refer to the stress components that are induced by a parabolic load and a point load, respectively. The explicit solutions for stresses in a half-space can be regrouped in the forms of Equations (A1) (A6). The exact solutions in this case are the same as Equations (A1) (A6), except that the stress elementary functions p ; p ;...; p are replaced by the stress integral functions, N h1i N h9i ; N h1i N h9i ;...; N h1i N h9i ði ¼ 1; 2; 3; a; b; c; d; eþ for s par xx ; spar yy ; spar zz ; tpar xy ; tpar yz ; tpar xz (Figure 3), respectively. In this loading case, for instance, p should be replaced by N h1i N h9i as follows: ½p Š)N h1i þ b x 2 L 2 þ y2 W 2 *S 14 *N h1i 2x L 2 * S 15 * N h2i 2y W 2 * S 14 * N h3i 4xy L 2 W 2 * N h4i þ S 15 L 2 * N h5i þ S 14 W 2 * N h6i þ 1 L 2 W 2 ð2x * N h7i þ 2y * N h8i N h9i Þ where S 14 ¼ 1 x2 ; S L 2 15 ¼ 1 y2 : Equations (4) (35) are the stress integral functions for N h1i W 2 N h4i ; N h1i N h4i ;...; N h1i N h4i : Hence, only the integral functions for N h5i N h9i ; N h5i N h9i ;...; N h5i N h9i are given as follows: ð38þ N h5i ¼ðR i R xn iþy ðr yn i R xn y n iþy n z 2 i N h1i ð39þ N h6i ¼ N h4i ð40þ

10 1458 C. D. WANG AND J.-J. LIAO N h8i N h7i ¼ 1 3 ½ðyR i y n R yn iþx ðyr xn i y n R xn y iþx n n þ x 3 S 1 x * 3 S 2 þ y 3 S 3 y * 3 S 4 z 3 i S 5Š ¼ ðx2 2y 2 2z 2 i Þ 3 þ ðx2 2y * 2 2z 2 i Þ 3 R i þ ðx * 2 2y 2 2z 2 i Þ R xn i 3 R yn i ðx * 2 2y * 2 2z 2 i Þ R xn y 3 n i ð41þ ð42þ N h9i ¼ 1 4 f½ðx2 2y 2 z 2 i ÞR i ðx * 2 2y 2 z 2 i ÞR x n išy ½ðx 2 2y * 2 z 2 i ÞR y n i ðx * 2 2y * 2 z 2 i ÞR x n y n išy n þðx 4 z 4 i ÞS 1 ðx * 4 z 4 i ÞS 2g N h6i N h5i ¼ N h4i ¼ðR i R yn iþx ðr xn i R xn y n iþx n z 2 i N h1i N h7i N h9i N h6i N h8i ¼ switch x; y in N h8i N h8i ¼ N h7i ¼ switch x; y in N h9i N h5i ¼ z i N h2i ¼ switch x; y in N h5i N h7i ¼ z i N h6i ¼ switch x; y in N h7i N h9i ¼ z i N h7i ð43þ ð44þ ð45þ ð46þ ð47þ ð48þ ð49þ ð50þ ð51þ ð52þ ð53þ N h5i s4i ¼ z i N h3i y 2 S 12 þ y * 2 S 13 ð54þ N h6i s4i ¼ switch x; y in N h5i s4i ð55þ N h7i s4i ¼ LW ðy þ yn Þ þ z i 3 3 ½ðR i R yn iþx ðr xn i R xn y iþx n Š n 2 3 ðx3 S 6 x * 3 S 7 Þ 1 3 ðy3 S 8 y * 3 S 9 Þþ z3 i 3 ðs 3 S 4 Þ ð56þ N h8i s4i ¼ switch x; y in N h7i s4i ð57þ

11 ELASTIC SOLUTIONS FOR STRESSES 1459 N h9i s4i ¼ LW ðx þ xn Þðy þ y n Þ ðx4 S 6 x * 4 S 7 þ y 4 S 12 y * 4 S 13 Þ þ z i 6 ½ðx2 þ y 2 2z 2 i ÞR i ðx * 2 þ y 2 2z 2 i ÞR x n i ðx 2 þ y * 2 2z 2 i ÞR y n i ðx * 2 þ y * 2 2z 2 i ÞR x n y n iš ð58þ N h7i N h8i N h5i ¼ LW y 2 S 8 þ y * 2 S 9 2z i ðys 3 y n S 4 Þ z 2 i S 5 ð59þ N h6i ¼ LW þ x 2 S 10 x * 2 S 11 þ z i ðxs 1 x n S 2 Þ ð60þ ¼ 2LW ðx þ xn Þ þ ðx3 S 10 x * 3 S 11 Þþ 1 3 ðy3 S 12 y * 3 S 13 Þ z i 6 ½ðR i R xn iþy ðr yn i R xn y iþy n ð3x 2 z 2 n i ÞS 1 þð3x * 2 z 2 i ÞS 2Š ð61þ ¼ 2LW ðx þ xn Þ ðx3 S 6 x * 3 S 7 Þ 2 3 ðy3 S 8 y * 3 S 9 Þ z i 3 ½ðR i R yn iþx ðr xn i R xn y iþx n þð3y 2 þ z 2 n i ÞS 3 ð3y * 2 þ z 2 i ÞS 4Š ð62þ N h9i ¼ LW ðx2 þ xx n 2 þ x * y 2 yy n y * 2 Þ þ ðx4 S 10 x * 4 S 11 y 4 S 8 þ y * 4 S 9 Þ z i 6 ½ðyR i y n R yn iþx ðyr xn i y n R xn y iþx n Š n þ z i 3 x3 S 1 x * 3 S 2 2y 3 S 3 þ 2y * 3 S 4 þ z3 i 2 S 5 N h5i s6i N h6i s6i N h7i s6i N h8i s6i N h9i s6i ¼ N h5i ¼ N h6i ¼ N h7i ¼ N h8i ¼ N h9i N h5i N h6i N h7i N h8i N h9i N h5i ¼ y R i R xn i þ y R i þ z i R xn i þ z i y n R yn i R xn y n i þ y 1 1 * 2 R yn i þ z i R xn y n i þ z i þ 2z i ðys 12 y n S 13 Þ ð63þ ð64þ ð65þ ð66þ ð67þ ð68þ ð69þ

12 1460 C. D. WANG AND J.-J. LIAO N h6i ¼ y!! x 2 x 2 * x 2 x y n * 2 R i þ z i R xn i þ z i R yn i þ z i R xn y n i þ z i þðx 2 z 2 i ÞS 1 ðx * 2 z 2 i ÞS 2 2z i ðxs 10 x n S 11 Þ ð70þ N h7i ¼ y 3 ðxr i x n R xn iþþ yn 3 ðxr y n i x n R xn y iþ n!! x 3 þ y x 3 * x 3 x y n * 3 R i þ z i R xn i þ z i R yn i þ z i R xn y n i þ z i þ 1 3 ð2x3 S 1 2x * 3 S 2 y 3 S 3 þ y * 3 S 4 þ z 3 i S 5Þ z i ðx 2 S 10 x * 2 S 11 Þ N h8i ¼ 1 3 ðr3 i R 3 x n i R3 y n i þ 1 1 R3 x n y n i Þþy4 R i þ z i R xn i þ z i y 1 1 * 4 þ z i ðy 2 S 12 y * 2 S 13 Þ R yn i þ z i R xn y n i þ z i ð71þ ð72þ N h9i ¼ 2LWz iðx þ x n Þ þ ½yðx2 R i x * 2 R xn iþ y n ðx 2 R yn i x * 2 R xn y iþš n þ 1 6 ½ð3x4 þ z 4 i ÞS 1 ð3x * 4 þ z 4 i ÞS 2Š þ y y 1 1 * 5 R i þ z i R xn i þ z i R yn i þ z i R xn y n i þ z i 2z i 3 ðx3 S 10 x * 3 S 11 y 3 S 12 þ y * 3 S 13 Þ þ z2 i 6 ½yðR i R xn iþ y n ðr yn i R xn y n iþš ð73þ N h5i N h6i N h7i N h8i ¼ switch x; y in N h6i ¼ switch x; y in N h5i ¼ switch x; y in N h8i ¼ switch x; y in N h7i ð74þ ð75þ ð76þ ð77þ N h9i ¼ switch x; y in N h9i Equations (A1) (A6), (4) (35), (38), and (39) (78) can be applied to compute the stresses in a transversely isotropic half-space subjected to three-dimensional, buried and parabolic rectangular loads. Also, the presented formulae for stresses are consistent with those presented by Teferra and Schultze [11] as the medium is isotropic and is in a state of plane strain. Moreover, stresses in the media in response to non-uniform, irregularly shaped loads can be estimated by superposing values that correspond to the rectangular sub-areas. Figure 5 shows a ð78þ

13 ELASTIC SOLUTIONS FOR STRESSES 1461 Figure 5. Flow chart to compute the stresses in a transversely isotropic half-space subjected to presented loading types. flow chart of the computation for stresses induced by linearly varying, uniform, and parabolic rectangular loads in a transversely isotropic half-space. 3. ILLUSTRATIVE EXAMPLE This section presents a parametric study to confirm the derived solutions and elucidate the effect of the type and degree of material anisotropy, the dimensions of the loaded area, and the types of loading on the stresses. An example illustrates the solution of the vertical stress, as depicted in Figures 6 11, for the action of vertical, linearly varying, uniform and parabolic loads on a rectangle. Several types of isotropic and transversely isotropic rocs are considered as foundation materials. Table I lists their elastic properties, with E=E 0 and G=G 0 ranging between 1 and 3 and n=n 0 varying between 0.75 and 1.5. The values of E and n adopted in Table I are 50 GPa and 0.25, respectively. The selected domains of anisotropic variation follow the

14 1462 C. D. WANG AND J.-J. LIAO Figure 6. Effect of the type and degree of roc anisotropy on vertical stress induced by the loading case of a ¼ 1: (a) for Rocs 1, 2, 3 with E=E 0 ¼ 1; 2; 3; and n=n 0 ¼ G=G 0 ¼ 1; respectively; (b) for Rocs 1, 4, 5 with n=n 0 ¼ 1; 0:75; 1:5; and E=E 0 ¼ G=G 0 ¼ 1; respectively; (c) for Rocs 1, 6, 7 with G=G 0 ¼ 1; 2; 3; and E=E 0 ¼ n=n 0 ¼ 1; respectively.

15 ELASTIC SOLUTIONS FOR STRESSES 1463 Figure 7. Effect of the type and degree of roc anisotropy on vertical stress induced by the loading case of b ¼ 1: (a) for Rocs 1, 2, 3 with E=E 0 ¼ 1; 2; 3; and n=n 0 ¼ G=G 0 ¼ 1; respectively; (b) for Rocs 1, 4, 5 with n=n 0 ¼ 1; 0:75; 1:5; and E=E 0 ¼ G=G 0 ¼ 1; respectively; (c) for Rocs 1, 6, 7 with G=G 0 ¼ 1; 2; 3; and E=E 0 ¼ n=n 0 ¼ 1; respectively.

16 1464 C. D. WANG AND J.-J. LIAO Figure 8. Effect of the type and degree of roc anisotropy on vertical stress induced by the loading case of a ¼ b ¼ 0: (a) for Rocs 1, 2, 3 with E=E 0 ¼ 1; 2; 3; and n=n 0 ¼ G=G 0 ¼ 1; respectively; (b) for Rocs 1, 4, 5 with n=n 0 ¼ 1; 0:75; 1:5; and E=E 0 ¼ G=G 0 ¼ 1; respectively; (c) for Rocs 1, 6, 7 with G=G 0 ¼ 1; 2; 3; and E=E 0 ¼ n=n 0 ¼ 1; respectively

17 ELASTIC SOLUTIONS FOR STRESSES 1465 Figure 9. Effect of the type and degree of roc anisotropy on vertical stress induced by the loading case of a ¼ 1: (a) for Rocs 1, 2, 3 with E=E 0 ¼ 1; 2; 3; and n=n 0 ¼ G=G 0 ¼ 1; respectively; (b) for Rocs 1, 4, 5 with n=n 0 ¼ 1; 0:75; 1:5; and E=E 0 ¼ G=G 0 ¼ 1; respectively; (c) for Rocs 1, 6, 7 with G=G 0 ¼ 1; 2; 3; and E=E 0 ¼ n=n 0 ¼ 1; respectively.

18 1466 C. D. WANG AND J.-J. LIAO Figure 10. Effect of the type and degree of roc anisotropy on vertical stress induced by the loading case of b ¼ 1: (a) for Rocs 1, 2, 3 with E=E 0 ¼ 1; 2; 3; and n=n 0 ¼ G=G 0 ¼ 1; respectively; (b) for Rocs 1, 4, 5 with n=n 0 ¼ 1; 0:75; 1:5; and E=E 0 ¼ G=G 0 ¼ 1; respectively; (c) for Rocs 1, 6, 7 with G=G 0 ¼ 1; 2; 3; and E=E 0 ¼ n=n 0 ¼ 1; respectively.

19 ELASTIC SOLUTIONS FOR STRESSES 1467 Figure 11. Effect of the presented loading types on vertical stress: (a) for Roc 1 with E=E 0 ¼ n=n 0 ¼ G=G 0 ¼ 1; (b) for Roc 2 with E=E 0 ¼ 2; n=n 0 ¼ G=G 0 ¼ 1; (c) for Roc 3 with E=E 0 ¼ 3; n=n 0 ¼ G=G 0 ¼ 1; (d) for Roc 4 with n=n 0 ¼ 0:75; E=E 0 ¼ G=G 0 ¼ 1; (e) for Roc 5 with n=n 0 ¼ 1:5; E=E 0 ¼ G=G 0 ¼ 1; (f) for Roc 6 with G=G 0 ¼ 2; E=E 0 ¼ n=n 0 ¼ 1; (g) for Roc 7 with G=G 0 ¼ 3; E=E 0 ¼ n=n 0 ¼ 1: suggestions of Gerrard [15] and Amadei et al. [16]. The loads act on the horizontal surface ðh ¼ 0Þ of isotropic/transversely isotropic rocs in this example. The effect of the degree of anisotropy, specified by the ratios E=E 0 ; u=u 0 ; and G=G 0 on the stresses is considered.

20 1468 C. D. WANG AND J.-J. LIAO Figure 11. (continued ) Using Equations (A1) (A6), (3), (4) (35), (38), and (39) (78), a FORTRAN program was written to calculate the six stress components under linearly varying, uniform and parabolic loading. In this study, only the vertical stress at the right corner, C (at depth z from the surface) of the loaded area was calculated. Figures 6 10 present the normalized vertical stress ðs linear P linear z or s par zz =P par z zz = Þ at point C; induced by a completely downwardly linearly varying load

21 ELASTIC SOLUTIONS FOR STRESSES 1469 Table I. Elastic properties and root types for different rocs. Roc type E=E 0 n=n 0 G=G 0 Root type Roc 1. Isotropic Equal Roc 2. Transversely isotropic Complex Roc 3. Transversely isotropic Complex Roc 4. Transversely isotropic Complex Roc 5. Transversely isotropic Distinct Roc 6. Transversely isotropic Distinct Roc 7. Transversely isotropic Distinct ða ¼ 1Þ, a completely convex parabolic load ðb ¼ 1Þ; a uniform load ða ¼ b ¼ 0Þ; an upwardly linearly varying load ða ¼ 1Þ and a concave parabolic load ðb ¼ 1Þ; respectively, over a rectangular area for various roc types (Rocs 1 7, Table I) and for various dimensions of the loaded area (m ¼ L=z or n ¼ W =z). The figures plot the relationship between two nondimensional factors, m and s linear zz =Pz linear (or s par par zz =Pz Þ for n ¼ 0:1; 0:5; 1:0; 1: In these figures, the non-dimensional factor, n; is adopted to elucidate the effect of the dimensions of the loaded region on the vertical stress. The stress beneath a strip is that beneath a rectangle as either L or W approaches infinity ð1þ [17]. However, practically, a rectangle for which W =L > 4 to 5 can be regarded as a strip [18]. In this example, n ¼ 40 is selected in the simulation of an infinite load. Hence, with nowledge of the type and magnitude of the loading, the dimensions of the loaded area and the type of roc the vertical stress at point C can be estimated from these figures. Figures 11(a) 11(g) summarize the results of applying the same stress as induced by the loads mentioned above (a ¼ 1; b ¼ 1; a ¼ b ¼ 0; a ¼ 1; b ¼ 1) to Rocs 1 7, respectively, with n ¼ 1:0: With reference to Figures 6 11, the effects of the type and degree of roc anisotropy, loading types and the dimensions of the loaded region on the stress induced by surface loading are investigated below. Figures 6(a) 6(c), 7(a) 7(c), 8(a) 8(c), 9(a) 9(c), and 10(a) 10(c) plot the vertical stress induced by loading with a ¼ 1; b ¼ 1; a ¼ b ¼ 0; a ¼ 1; and b ¼ 1 for Rocs 1 ðe=e 0 ¼ n=n 0 ¼ G=G 0 ¼ 1Þ; 2 ðe=e 0 ¼ 2; n=n 0 ¼ G=G 0 ¼ 1Þ; 3 ðe=e 0 ¼ 3; n=n 0 ¼ G=G 0 ¼ 1Þ; Rocs 1, 4 ðn=n 0 ¼ 0:75; E=E 0 ¼ G=G 0 ¼ 1Þ; 5 ðn=n 0 ¼ 1:5; E=E 0 ¼ G=G 0 ¼ 1Þ; Rocs 1, 6 ðg=g 0 ¼ 2; E=E 0 ¼ n=n 0 ¼ 1Þ; 7 ðg=g 0 ¼ 3; E=E 0 ¼ n=n 0 ¼ 1Þ with variable non-dimensional factors ðm; nþ: From Figure 6(a), the stress induced in Rocs 2 and 3 is less than that in Roc 1 within the smaller loaded region (horizontal scale, m); however, the calculated result changes as m increases. For a given n ð¼ 0:1; 0:5; 1:0; 1Þ; the non-dimensional factor, m; significantly affects the induced vertical stress at point C. The trend of vertical stress in Figure 6(c) for Rocs 6 and 7 is the opposite of that in Figure 6(a) for Rocs 2 and 3. Figures 6(a) and 6(c) preliminarily imply that the induced stress depends on the type and degree of anisotropy for a given loading ða ¼ 1Þ: Figure 7(a) indicates that the induced stress with b ¼ 1 for Rocs 1 3 increases as E=E 0 increases, whereas for Rocs 6 and 7 (Figure 7(c)) the results are totally different. Notably, a little of the stress in Figure 7(c) might be transferred by tension in Roc 7, when n ¼ 0:1 and 0.5, within a very small loaded area ðm40:5þ: Figures 8(a), 9(a) and 10(a) show that for a given loading type (a ¼ b ¼ 0; a ¼ 1; and b ¼ 1), depth ðzþ; and loaded region (L or W ), the magnitude of the vertical stress decreases as E=E 0 increases (Rocs 2, 3). Comparing Figures 8(c), 9(c) and 10(c) with Figures 8(a), 9(a) and 10(a) reveals that the non-dimensional vertical stress increases with increases in G=G 0 (Rocs 6, 7). Figures 6(b), 7(b), 8(b), 9(b) and 10(b),

22 1470 C. D. WANG AND J.-J. LIAO respectively, plot the effect of n=n 0 (Rocs 1, 4, 5) with a ¼ 1; b ¼ 1; a ¼ b ¼ 0; a ¼ 1; and b ¼ 1 on the vertical stress. According to these figures, the induced stress is slightly influenced by n=n 0 : Figures 6 10 show that the vertical stress increases with the non-dimensional factor, n; for all rocs, implying that the stress calculated from plane strain exceeds the three-dimensional solution. The figures also indicate that the stress induced by the considered loading types strongly depends on the dimensions of the loaded area, and the type and degree of roc anisotropy. Figures 11(a) 11(g) clarify the effect of the loading (a ¼ 1; b ¼ 1; a ¼ b ¼ 0; a ¼ 1; b ¼ 1) on the non-dimensional vertical stress (s linear zz =Pz linear or s par par zz =Pz ) for Rocs 1 7, respectively, with n ¼ 1:0: Figures 11(a) 11(e) show that the order induced stress follows a ¼ 1 > b ¼ 1 > a ¼ b ¼ 0 > b ¼ 1 > a ¼ 1: The magnitude of the calculated stress can be reasonably judged from the geometry of loading in Figures 1 4. However, the trend in the induced stress in Figures 11(f) and 11(g) (for Rocs 6 and 7) differs a little from that in Figures 11(a) 11(e), within the smaller loaded region ðm42þ: Figures 11(a) 11(e) show the effect of the loading type on the stress is explicit. The average induced vertical stress by a ¼ 1 and b ¼ 1 is approximately double that induced by a ¼ b ¼ 0 for all rocs; however, the average stress by a ¼ 1and b ¼ 1 is disproportional to that by a ¼ b ¼ 0: The example was presented to elucidate the solutions and clarify how the dimensions of the loaded area, the type and degree of roc anisotropy, and the type of loading affect the nondimensional vertical stress. The analysis results indicate that the induced stress in isotopic/ transversely isotropic rocs under various types of loading is easily calculated. Hence, the anisotropic deformability must be considered when estimating the stresses in a transversely isotropic half-space subjected to linearly varying/uniform/parabolic rectangular loads. 4. CONCLUSIONS Integrating the elementary functions of a point load yields the elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional, buried, linearly varying, uniform and parabolic rectangular loads. The solutions are limited to planes of transverse isotropy that are parallel to the horizontal surface of the half-space. The loading types include an upwardly linearly varying load, a downwardly linearly varying load, a uniform load, a concave parabolic load and a convex parabolic load, all acting on a rectangular area. The proposed closed-form solutions for stresses are affected by the buried depth ðhþ; the dimensions of the loaded area ðl; W Þ; the type and degree of roc anisotropy (E=E 0 ; n=n 0 ; G=G 0 ), and the type of loading ( a > 0; a ¼ b ¼ 0; a50; b > 0; b50) in transversely isotropic half-spaces. A parametric study of an illustrative example has yielded the following conclusions: 1. The ratios E=E 0 (u=u 0 ¼ G=G 0 ¼ 1) and G=G 0 ðe=e 0 ¼ n=n 0 ¼ 1Þ strongly influence the non-dimensional vertical stress in transversely isotropic rocs subjected to a downwardly linearly varying, and a convex parabolic rectangular load. However, u=u 0 ðe=e 0 ¼ G=G 0 ¼ 1Þ has little effect on this stress. 2. In a very small loaded area of m and n; a little stress may be transferred by tension in the medium.

23 ELASTIC SOLUTIONS FOR STRESSES The stress induced in transversely isotropic rocs by a uniform, an upwardly linearly varying and a concave parabolic rectangular load decreases as E=E 0 increases, and increases as G=G 0 increases, but is only slightly influenced by u=u 0 : 4. The plane strain solution overestimates the induced stress, as compared to the presented three-dimensional solution. 5. The induced stress heavily relies on the dimensions of the loaded area, the type and degree of material anisotropy, and the type of loading types. 6. The formulated stresses correlate well with those in the isotropic medium under plane strain. The calculation of induced stresses by various types of loading, distributed over a rectangular area in an isotropic/transversely isotropic half-space is fast and correct, and the presentation of the derived solutions is clear and concise. These solutions can more realistically simulate actual loading circumstances in many engineering practices. Furthermore, the presented solutions also offer an alternative to the numerical or graphical methods, and provide reasonable results for practical purposes. Similarly, the elastic solutions for displacements in a transversely isotropic half-space subjected to the types of loading considered here can also be derived. The solutions will be presented in forthcoming papers [19]. APPENDIX A The point load solutions for stresses in a transversely isotropic half-space in Cartesian forms can be expressed as follows [6]: s p xx ¼ P x 4p ða 11 u 1 m 1 A 13 Þ p s11 T 1 p s1a þ T 2 p s1b m 1 ða 11 u 2 m 2 A 13 Þ p s12 T 3 p s1c þ T 4 p s1d m 2 2A 66 p s71 p s72 T 1 p s7a þ T 2 p s7b þ T 3 p s7c T 4 p s7d þ 2u 3 ðp s73 þ p s7e Þ m 1 m 2 þ P y 4p ða 11 u 1 m 1 A 13 2A 66 Þ ða 11 u 2 m 2 A 13 2A 66 Þ þ 2A 66 m 1 p s21 T 1 p s2a þ T 2 p s2b m 2 p s22 T 3 p s2c þ T 4 p s2d m 1 p s81 m 2 p s82 T 1 p s8a þ T 2 p s8b þ T 3 p s8c T 4 p s8d 2u 3 ðp s83 þ p s8e Þ þ P z 4p fða 11 u 1 m 1 A 13 2A 66 Þðp s31 þ T 1 m 1 p s3a T 2 m 2 p s3b Þ ða 11 u 2 m 2 A 13 2A 66 Þðp s32 þ T 3 m 1 p s3c T 4 m 2 p s3d Þ þ 2A 66 ½ðp s51 p s52 Þþm 1 ðt 1 p s5a T 3 p s5c Þ m 2 ðt 2 p s5b T 4 p s5d ÞŠg ða1þ

24 1472 C. D. WANG AND J.-J. LIAO s p yy ¼ P x 4p ða 11 u 1 m 1 A 13 2A 66 Þ p s11 T 1 p s1a þ T 2 p s1b m 1 ða 11 u 2 m 2 A 13 2A 66 Þ p s12 T 3 p s1c þ T 4 p s1d m 2 þ 2A 66 p s71 p s72 T 1 p s7a þ T 2 p s7b þ T 3 p s7c T 4 p s7d m 1 m 2 þ P y 4p ða 11 u 1 m 1 A 13 Þ p s21 T 1 p s2a þ T 2 p s2b m 1 ða 11 u 2 m 2 A 13 Þ 2A 66 m 2 p s22 T 3 p s2c þ T 4 p s2d m 1 p s81 m 2 p s82 T 1 p s8a þt 2 p s8b þt 3 p s8c T 4 p s8d 2u 3 ðp s73 þ p s7e Þ þ 2u 3 ðp s83 þ p s8e Þ þ P z 4p fða 11 u 1 m 1 A 13 2A 66 Þðp s31 þ T 1 m 1 p s3a T 2 m 2 p s3b Þ ða 11 u 2 m 2 A 13 2A 66 Þðp s32 þ T 3 m 1 p s3c T 4 m 2 p s3d Þ þ 2A 66 ½ðp s61 p s62 Þþm 1 ðt 1 p s6a T 3 p s6c Þ m 2 ðt 2 p s6b T 4 p s6d ÞŠg s p zz ¼ P x 4p ða 13 u 1 m 1 A 33 Þ p s11 T 1 p s1a þ T 2 p s1b m 1 ða 13 u 2 m 2 A 33 Þ p s12 T 3 p s1c þ T 4 p s1d m 2 þ P y 4p ½ðA 13 u 1 m 1 A 33 Þ p s21 T 1 p s2a þ T 2 p s2b m 1 ða 13 u 2 m 2 A 33 Þ p s22 T 3 p s2c þ T 4 p s2d m 2 ða2þ t p xy ¼ P x 4p 2A 66 þ P z 4p ½ðA 13 u 1 m 1 A 33 Þðp s31 þ T 1 m 1 p s3a T 2 m 2 p s3b Þ ða 13 u 2 m 2 A 33 Þðp s32 þ T 3 m 1 p s3c T 4 m 2 p s3d ÞŠ m 1 p s81 m 2 p s82 T 1 p s8a þ T 2 p s8b þ T 3 p s8c T 4 p s8d u 3 ð2p s83 þ 2p s8e p s23 p s2e ÞŠ þ P y 4p 2A 66 p s71 p s72 T 1 p s7a þ T 2 p s7b þ T 3 p s7c T 4 p s7d m 1 m 2 u 3 ð2p s73 þ 2p s7e p s13 p s1e ÞŠ P z 2p A 66½ðp s41 p s42 Þþm 1 ðt 1 p s4a T 3 p s4c Þ m 2 ðt 2 p s4b T 4 p s4d ÞŠ ða3þ ða4þ

25 ELASTIC SOLUTIONS FOR STRESSES 1473 t p yz ¼ P x 4p ðu 1 þ m 1 ÞA 44 p s41 T 1 p s4a þ T 2 p s4b m 1 ðu 2 þ m 2 ÞA 44 p s42 T 3 p s4c þ T 4 p s4d ðp s43 þ p s4e Þ m 2 þ P y 4p ðu 1 þ m 1 ÞA 44 ðu 2 þ m 2 ÞA 44 m 1 p s61 T 1 p s6a þ T 2 p s6b m 2 p s62 T 3 p s6c þ T 4 p s6d þðp s53 þ p s5e Þ P z 4p A 44½ðu 1 þ m 1 Þðp s21 þ T 1 m 1 p s2a T 2 m 2 p s2b Þ ðu 2 þ m 2 Þðp s22 þ T 3 m 1 p s2c T 4 m 2 p s2d ÞŠ t p xz ¼ P x 4p ðu 1 þ m 1 ÞA 44 p s51 T 1 p s5a þ T 2 p s5b m 1 ðu 2 þ m 2 ÞA 44 p s52 T 3 p s5c þ T 4 p s5d m 2 P y 4p ðu 1 þ m 1 ÞA 44 p s41 T 1 p s4a þ T 2 p s4b m 1 ðu 2 þ m 2 ÞA 44 p s42 T 3 p s4c þ T 4 p s4d m 2 P z 4p A 44½ðu 1 þ m 1 Þðp s11 þ T 1 m 1 p s1a T 2 m 2 p s1b Þ ðu 2 þ m 2 Þðp s12 þ T 3 m 1 p s1c T 4 m 2 p s1d ÞŠ þðp s63 þ p s6e Þ ðp s43 þ p s4e Þ ða5þ ða6þ where * A ij (i; j ¼ 1 6) are the elastic moduli or elasticity constants of the medium, and can be expressed in terms of five independent elastic constants for a transversely isotropic half-space as A 11 ¼ Eð1 ðe=e 0 Þu 02 Þ ð1 þ uþð1 u ð2e=e 0 Þu 02 Þ ; A 13 ¼ Eu 0 1 u ð2e=e 0 Þu 02; E 0 ð1 uþ A 33 ¼ 1 u ð2e=e 0 Þu 02; A 44 ¼ G 0 E ; A 66 ¼ ða7þ 2ð1 þ uþ where E and E 0 are Young s moduli in the plane of transverse isotropy and in a direction normal to it, respectively; n and n 0 are Poisson s ratios characterizing the lateral strain response in the plane of transverse isotropy to a stress acting parallel and normal to it, respectively; p G 0 is the shear modulus in planes normal to the plane of transverse isotropy. * u 3 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 66 =A 44 ; u 1 and u 2 are the roots of the following characteristic equation: u 4 su 2 þ q ¼ 0 ða8þ

26 1474 C. D. WANG AND J.-J. LIAO whereas s ¼ A 11A 33 A 13 ða 13 þ 2A 44 Þ ; q ¼ A 11 : A 33 A 44 A 33 Since the strain energy is assumed to be positive definite in the medium, the values of elastic constants are restricted. Hence, there are three categories of the characteristic roots, u 1 and u 2 as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Case 1: u 1;2 ¼ f 1 2 ½s p ðs 2 4qÞŠg are two real distinct roots when s 2 4q > 0; p Case 2: u 1;2 ¼ ffiffiffiffiffiffiffi pffiffiffiffiffiffiffi s=2 ; s=2 are double equal real roots when s 2 4q ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Case 3: u 1 ¼ 1 p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðs þ 2 q Þ i 1 p 2 ð s þ 2 q Þ ¼ g id; u 2 ¼ g þ id are two complex conjugate roots (where g cannot be equal to zero) when s 2 4q50: * m j ¼ ða 13 þ A 44 Þu j A 33 u 2 j A 44 ¼ A 11 A 44 u 2 j ðj ¼ 1; 2Þ; ¼ ða 13 þ A 44 Þ ða 13 þ A 44 Þu j A 33 A 44 ðu 2 1 u2 2 Þ; T 1 ¼ u 1 þ u 2 m 1 u 2 u 1 T 2 ¼ 2u 1 ðu 2 þ m 2 Þ m 2 ðu 2 u 1 Þðu 1 þ m 1 Þ ; T 3 ¼ 2u 2 ðu 1 þ m 1 Þ m 1 ðu 2 u 1 Þðu 2 þ m 2 Þ ; T 4 ¼ u 1 þ u 2 ; m 2 u 2 u 1 p ¼ x R 3 i p ¼ y R 3 i ; p ¼ z i R 3 ; p s4i ¼ xyð2r i þ z i Þ i R 3 i ðr i þ z i Þ 2; p 1 ¼ R i ðr i þ z i Þ x2 ð2r i þ z i Þ R 3 i ðr i þ z i Þ 2 p s6i ¼ 1 R i ðr i þ z i Þ y2 ð2r i þ z i Þ R 3 i ðr i þ z i Þ 2; p ¼ x R 3 i 3x R i ðr i þ z i Þ 2 þ x3 ð3r i þ z i Þ R 3 i ðr i þ z i Þ 3 p ¼ y R 3 i 3y R i ðr i þ z i Þ 2 þ y3 ð3r i þ z i Þ R 3 i ðr i þ z i Þ 3 p * R i ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 þ z 2 i ði ¼ 1; 2; 3; a; b; c; d; eþ; z 1 ¼ u 1 ðz hþ; z 2 ¼ u 2 ðz hþ; z 3 ¼ u 3 ðz hþ; z a ¼ u 1 ðz þ hþ; z b ¼ u 1 z þ u 2 h; z c ¼ u 1 h þ u 2 z; z d ¼ u 2 ðz þ hþ; z e ¼ u 3 ðz þ hþ: The load is applied at the surface when the buried depth h ¼ 0: APPENDIX B: NOMENCLATURE A ij ði; j ¼ 126Þ elastic moduli or elastic constants in Equation (A7) dz; dz infinitesimal element along Y -orx-axis, respectively E; E 0 ; u; u 0 ; G 0 engineering elastic constants of transversely isotropic materials h the buried depth, as pseen in Figures 1 4 i complex number ð¼ ffiffiffiffiffiffi 1 Þ ; m 1 ; m 2 ; T 1 ; T 2 ; T 3 ; T 4 coefficients in Equations (A1) (A6) L; W length along X -axis and width along Y -axis, respectively N h1i N h9i ;...; N h1i N h9i integral functions for stresses induced by linearly varying, uniform, and parabolic rectangular loads p p elementary functions for stresses induced by a point load P j ðj ¼ x; y; zþ a point load (force)

27 ELASTIC SOLUTIONS FOR STRESSES 1475 P linear P par j ðj ¼ x; y; zþ linearly varying rectangular loads (forces per unit of area) j ðj ¼ x; y; zþ parabolic rectangular loads (forces per unit of area) q; s coefficients in Equation (A8) u 1 ; u 2 ; u 3 roots of the characteristic equation (Equation (A8)) X ; Y ; Z Cartesian co-ordinate system Gree letters a; b the constant controlling the linearly varying and parabolic loads, respectively g; d real and imaginary part of the complex roots, respectively s s linear xx ; s linear yy ; s linear zz s p xx ; sp yy ; sp zz s par xx ; spar yy ; spar zz t linear xy ; t linear yz t p xy ; tp yz ; tp xz t par xy ; tpar yz ; tpar xz ; t linear xz stress components normal stresses induced by linearly varying rectangular loads normal stresses induced by a point load normal stresses induced by parabolic rectangular loads shear stresses induced by linearly varying rectangular loads shear stresses induced by a point load shear stresses induced by parabolic rectangular loads Superscripts linear p par T stresses induced by linearly varying rectangular loads stresses induced by a point load stresses induced by parabolic rectangular loads transpose matrix ACKNOWLEDGEMENT The authors would lie to than the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC E REFERENCES 1. Wang CD, Liao JJ. Stress influence charts for transversely isotropic rocs. International Journal of Roc Mechanics and Mining Sciences 1998; 35: Wang CD, Liao JJ. Computing displacements in transversely isotropic rocs using influence charts. Roc Mechanics and Roc Engineering 1999; 32: Lehnitsii SG. Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day Press: San Francisco, Pan YC, Chou TW. Green s function solutions for semi-infinite transversely isotropic materials. International Journal of Engineering Science 1979; 17: Liao JJ, Wang CD. Elastic solutions for a transversely isotropic half-space subjected to a point load. International Journal for Numerical and Analytical Methods in Geomechanics 1998; 22: Wang CD, Liao JJ. Elastic solutions for a transversely isotropic half-space subjected to buried asymmetric-loads. International Journal for Numerical and Analytical Methods in Geomechanics 1999; 23: Wang CD, Liao JJ. Elastic solutions for a transversely isotropic half-space subjected to arbitrarily-shaped loads using triangulating technique. International Journal of Geomechanics 2001; 1: Lin W, Kuo CH, Keer LM. Analysis of a transversely isotropic half space under normal and tangential loadings. Journal of Tribology ASME 1991; 113: Hooper JA. Parabolic adhesive loading of a flexible raft foundation. Geotechnique 1976; 26: Bauer GE, Shields DH, Scott JD, Nwabuoei SO. Normal and shear measurements on a strip footing. Canadian Geotechnical Journal 1979; 16:

28 1476 C. D. WANG AND J.-J. LIAO 11. Teferra A, Schultze E. Formulae, Charts and Tables in the Area of Soil Mechanics and Foundation Engineering, Stresses in Soils. A. A. Balema Publishers: Broofield, Gazetas G. Stresses and displacements in cross-anisotropic soils. Journal of Geotechnical Engineering Division ASCE 1982; 108: Gazetas G. Axisymmetric parabolic loading of anisotropic halfspace. Journal of Geotechnical Engineering Division ASCE 1982; 108: Davis RO, Selvadurai APS. Elasticity and Geomechanics. Cambridge University Press: New Yor, Gerrard CM. Bacground to mathematical modeling in geomechanics: the roles of fabric and stress history. Proceedings of International Symposium on Numerical Methods, Karlsruhe, 1975; Amadei B, Savage WZ, Swolfs HS. Gravitational stresses in anisotropic roc masses. International Journal of Roc Mechanics and Mining Sciences & Geomechanics Abstract 1987; 24: Feda J. Stress in Subsoil and Methods of Final Settlement Calculation. Elsevier Scientific Publishing Company: Amsterdam, Giroud JP. Tables Pour Le Calcul Des Fondations. Tome.2. Tassement: Dunod, Wang CD, Liao JJ. Elastic solutions of displacements for a transversely isotropic half-space subjected to threedimensional buried parabolic rectangular loads. International Journal of Solids and Structures 2002; in press.