Effects of a cover layer on wave-induced pore pressure around a buried pipe in an anisotropic seabed

Ocean Engineering 27 (2000) 823–839 www.elsevier.com/locate/oceaneng

Effects of a cover layer on wave-induced pore pressure around a buried pipe in an anisotropic seabed X. Wanga, D.S. Jengb,*, Y.S. Linc a

Department of Civil and Environmental Engineering, The University of Adelaide, North Terrace, SA 5005, Australia b School of Engineering, Gold Coast Campus, Griffith University, QLD 9726, Australia c Department of Civil Engineering, National Chung Hsing University, Taichung, 402 Taiwan, ROC Received 29 November 1998; accepted 28 January 1999

Abstract In engineering practice, a cover layer of coarser material has been used to protect a buried marine pipeline from wave-induced seabed instability. However, most previous investigations of the wave–seabed–pipe interaction problem have been concerned only with such a problem either in an isotropic single layer or a rigid pipe. This paper proposes a two-dimensional finite element model by employing the principle of repeatability to investigate the wave-induced soil response around a buried pipeline. The elastic anisotropic soil bahavior and geometry of cover layer are included in the present model, while the pipe is considered to be an elastic medium. This study focuses on the effects of a cover layer (including thickness B and width W of the cover layer) on the wave-induced pore pressure in the vicinity of a buried pipeline.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Pore pressure; Cover layer; Anisotropic soil behavior; Pipeline

1. Introduction Design of marine pipelines regarding their stability is a rather complicated problem. One of the important factors that must be taken into account is the wave-induced * Corresponding author. E-mail: [email protected] 0029-8018/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 9 ) 0 0 0 1 2 - 8

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soil response around the pipeline. When gravity waves propagate over the ocean, they exert fluctuations of wave pressure on the sea floor. These fluctuations further induce excess pore pressures and effective stresses, which have been recognized as a dominant factor in analyzing the instability of a seabed. When the pore pressure becomes excessive with accompanying decrease in effective stress, a sedimentary bed may be moved in either a horizontal (shear failure) or vertical direction (liquefaction), leading to an instability of the seabed (Clukey et al., 1989; Barends and Spierenburg, 1991; Rahman, 1997; Zen et al., 1998). Thus, the evaluation of the wave-induced soil response, including pore pressure, effective stresses and soil displacements, is particularly important for marine geotechnical engineers involved in the design of foundations for offshore pipelines. Based on different assumptions of the rigidity of the soil skeleton and compressibility of pore fluid, numerous theories have been developed for the wave-induced soil response in an elastic medium since the 1940s. Among these, Yamamoto et al. (1978) proposed an analytical solution for the wave–seabed interaction problem in an isotropic uniform seabed of infinite thickness. Later, Mei and Foda (1981) proposed a rather simpler method, a boundary-layer approximation, for a similar problem. However, the boundary-layer approximation has been proved to be limited to a seabed of fine sand (Hsu and Jeng, 1994). Recently, Yamamoto’s model has been further extended to wave–seabed–wall interaction problem by including anisotropic soil behavior and variable permeability (Jeng, 1997a, b; Jeng and Seymour, 1997). Besides the development of analytical solutions, finite element models have been widely applied to examine the wave–seabed interaction problem in recent years. Among these, Thomas (1989) Thomas (1995) proposed a 1-D FEM model associated with a semi-analytical method. This framework has been further extended to discuss the effects of anisotropic soil behavior, as well as non-homogeneity (Jeng and Lin, 1996, 1997, 1998a). This 1-D model can only deal with the problem without any marine structure. From another aspect, a 2-D numerical FEM model was proposed by Gatmiri (1990, 1992). However, doubtful lateral boundary conditions were used in his model, as discussed in Jeng (1997b). Recently, a more rigorous 2-D model was proposed by the authors, employing the principle of repeatability (Jeng and Lin, 1999. The major advantage of this FEM model is that the whole wave–seabed– structure interaction problem can be solved by one program with different meshes. Similar to the studies of wave–seabed interaction without structures, the wave– seabed–pipe interaction problem was first studied using the assumptions of potential theory (MacPherson, 1978; MacDougal et al., 1988; Spierenburg, 1986). However, it has been submitted that the potential theory is somewhat far way from the realistic conditions of the soil and pore–fluid two-phase medium (Jeng, 1997b). Furthermore, the potential theory provides no information for the effective stresses and soil displacements in the seabed (Yamamoto et al., 1978). Recently, the wave-induced pore pressure around a buried pipeline has been studied through a boundary integral equation method (Cheng and Liu, 1986) and a finite element method (Magda, 1996, 1997). Among these, Cheng and Liu (1986) considered a buried pipe in a region surrounded by two impermeable walls. Magda (1996, 1997) considered similar cases with a wider range of degree of saturation.

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Later, Jeng and Cheng (1999) proposed a finite difference model in a curvilinear coordinate system for the wave–seabed–pipe interaction problem. However, this FDM model has in a seabed of finer materials under a certain combination of wave and soil conditions. All aforementioned investigations have been only concerned with a buried pipe in a uniform single isotropic seabed. However, it is common in engineering practice to using a cover layer of coarser materials such as coarse sand, gravel and stones to protect a buried pipeline. It seems that theoretical investigations have been concerned with neither a cover layer nor with anisotropic soil behavior in the wave– seabed–pipe interaction problem. This paper will focus on the effects of a cover layer on wave-induced pore pressure in the vicinity of a buried pipeline, as well as the influences of anisotropic soil behavior. The influences of two parameters of geometry cover layer—width and thickness of cover layer—in the analysis of pore pressure have been detailed.

2. Anisotropic soil behavior In general, an isotropic material displays same elastic properties irrespective of the orientation of the samples. Such a material can be described by two elastic constants: Young’s modulus, E, and Poisson’s ratio, ␮. However, most marine sediments display some degree of anisotropy, possessing different properties in different directions. The elastic behavior of a cross-anisotropic material is described by five independent elastic parameters (Pickering, 1970). They are: (1) Ez: Young’s modulus in the vertical direction; (2) Ex: Young’s modulus in the horizontal direction; (3) ␮xx: Poisson’s ratio as the corresponding operators of lateral expansion in the horizontal direction due to horizontal direct stress normal to the former; (4) ␮zx: Poisson’s ratio as the corresponding operators of lateral expansion in the vertical direction due to horizontal direct stress; and (5) Gv: modulus of shear deformation in a vertical plane. For the possible ranges of the above anisotropic parameters, the reader can refer to the second author’s previous work (Jeng, 1997b). Two dependent elastic constants, ␮xz and Gx, can be interrelated by

␮xz Ex Ex ⫽ ⫽ n, and Gx ⫽ ␮zx Ez 2(1 ⫹ ␮xx)

(1)

For an isotropic soil, the non-dimensional parameter n is equal to 1. The vertical shear modulus in a vertical plane, Gz, can be expressed in terms of Young’s modulus, Ez, as (Gazetas, 1982) Gz ⫽ mEz

(2)

where m is the so-called anisotropic constant (Gazetas, 1982). It is equal to E/2(1 ⫹ ␮) for an isotropic soil. Under conditions of plane strains, the relationships between elastic incremental effective stresses and soil displacements are given by (Pickering, 1970)

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冤冥 冤 ␴x⬘

C11 C13 0

冥

␴z⬘ ⫽ Ez C13 C33 0 • ␶xz

0

0

m

∂u ∂x

冤 冥 ∂w ∂z

(3)

∂w ∂u ⫹ ∂x ∂z

where u and w are the soil displacements, while ␴x⬘ and ␴z⬘ are the effective normal stresses in the x- and z-directions, respectively. Shear stresses ␶xz denote the shear stress in the z-direction on a plane perpendicular to the x-direction. It is worth noting that the positive sign is used for tension stresses in the present paper. In Eq. (3), the coefficients Cij are given as C11 ⫽ (1 ⫺ ␮xz␮zx)/⌬, C13 ⫽ ␮xz(1 ⫹ ␮xx)/⌬,

(4)

C33 ⫽ (1 ⫺ ␮2xx)/⌬, ⌬ ⫽ (1 ⫹ ␮xx)(1 ⫺ ␮xx ⫺ 2␮xz␮zx)

3. Boundary value problem 3.1. Governing equations Considering a soil column in a porous seabed of finite thickness (h), a fully buried pipe (with a radius R) is located in the seabed (see Fig. 1). The wave crests are

Fig. 1.

Definition of wave–seabed–pipe interaction problem.

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assumed to propagate in the positive x-direction, while the z-direction is measured positive upward from the impermeable rigid bottom, as shown in Fig. 1. In this study, the consolidation equation (Biot, 1941), which has been generally accepted as the governing equation for flow of a compressible pore fluid in a compressible porous medium, is adopted to treat the wave–seabed interaction in a twodimensional domain as ∂2p ∂2p ␥wn⬘␤ ∂p ␥w ∂⑀ ⫹ 2⫺ ⫽ ∂x2 ∂z K ∂t K ∂t

(5)

where ␥w is the unit weight of pore water, n⬘ is soil porosity, p is pore pressure, and t is time. In Eq. (5), the compressibility of pore fluid (␤) and volumetric strain of the soil matrix (⑀) are defined by

␤⫽

1 1⫺S ∂u ∂w ⫹ and ⑀ ⫽ ⫹ Kw Pwo ∂x ∂z

(6)

where Kw is the true modulus of elasticity of water (taken as 2 ⫻ 109 N/m2 by Yamamoto et al., 1978), Pwo is the absolute water pressure, S is the degree of saturation, and u and w are the soil displacements in the x- and z-directions, respectively. Neglecting the effects of body forces and inertial terms, the equations governing the overall equilibrium of a porous medium can be expressed in terms of pore pressure and soil displacements as

冋 冋

Ez m

册 册

∂2 u ∂2 w ∂p ∂2 u ⫹ m ⫹ (C ⫹ m) ⫽ 13 2 2 ∂x ∂z ∂x∂z ∂x

(7)

∂2w ∂2 w ∂2 u ∂p ⫹ C33 2 ⫹ (C13 ⫹ m) ⫽ 2 ∂x ∂z ∂x∂z ∂z

(8)

Ez C11

in the x- and z-directions, respectively. 3.2. Boundary conditions For a porous seabed of finite thickness, as shown in Fig. 1, the evaluation of the wave-induced seabed response requires the solution of Eqs. (5), (7) and (8), together with the appropriate boundary conditions. For a buried pipe itself (with thickness of pipeline Ro), since the pipe is assumed to be an elastic medium here, the displacements and stresses of the pipe can be obtained by Eqs. (7) and (8). 3.2.1. Bottom boundary condition (BBC) Firstly, zero displacements and no vertical flow occur at the impermeable horizontal bottom, i.e. u⫽w⫽

∂p ⫽ 0, at z ⫽ 0 ∂z

(9)

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3.2.2. Boundary condition at the seabed surface (SBC) Secondly, we assume that the bottom frictional stress is small and negligible. The vertical effective normal stress and shear stress vanish and pore pressure is equal to the wave pressure at the surface of the seabed, i.e.

␴⬘z ⫽ ␶xz ⫽ 0, p⫽

␥wH cos(kx ⫺ ␻t) ⫽ poRe{ei(kx ⫺ ␻t)} at z ⫽ h, 2cos hkd

(10)

⫽ poRe{(cos kx ⫹ i sin kx)e−i␻t} where po denotes the amplitude of the wave pressure at the surface of the seabed, d is water depth, H is wave height, k is the wave number, and ␻ is the wave frequency. In Eq. (10), Re represents the real part of the function in brackets. 3.2.3. Lateral boundary condition (LBC) Since the existence of the pipeline only affects the wave-induced soil response near the pipeline, the “disturbed pressure” from the pipeline should vanish at points far away from the pipe. However, the porous seabed at these points is still under wave loading. Thus, the lateral boundary conditions at these points are given by the solution without pipeline, which will be described later. It is noted that the lateral boundary conditions used here are different from those of Magda (1996, 1997). He assumed that the soil displacements and the gradient of pore pressure vanished at points far away from the pipe. This implies that the soil response will vanish at these points under wave loading. However, it is clear that the wave-induced soil displacements and the gradient of pore pressure are not zero because the seabed is under wave loading. Thus, a more reasonable assumption may be that the effect of the pipe vanishes at points far away from the pipe, but the soil response owing to wave loading still exists. 3.2.4. Boundary condition of pipeline (PBC) It is noted that most previous studies of wave–seabed–pipe interaction have assumed the pipe to be rigid (Cheng and Liu, 1986 Magda, 1996, 1997; Jeng and Cheng, 1999). In this study, we assume the pipe to be an isotropic elastic medium. Thus, the boundary conditions along the pipe surface are kept free. Therefore, displacements of the pipe can also be evaluated in the present model.

4. Finite element model 4.1. Finite element formulations Since the wave-induced oscillatory soil response is periodically fluctuating in the temporal domain, the wave-induced pore pressure, effective stresses and soil displacements can be expressed as

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冦 冧 冤冦 冧 冦 冧冥 p(x,z;t)

Pr(x,z)

Pc(x,z)

u(x,z;t)

Ur(x,z)

Uc(x,z)

w(x,z;t)

Wr(x,z)

⫽

␴x⬘(x,z;t)

Wc(x,z)

⫹i

Sxr(x,z)

e−i␻t

Sxc(x,z)

␴z⬘(x,z;t)

Szr(x,z)

Szc(x,z)

␶xz(x,z;t)

Txzr(x,z)

Txzc(x,z)

829

(11)

where subscripts r and c represent the real and imaginary parts of the soil response, respectively. Substituting Eq. (11) into Eqs. (5), (7) and (8) then directly applying the Galerkin method (Zienkiewicz and Taylor, 1989) to these equations, the finite element analytical formulations can be expressed in matrix form as

冕

NI[Qe] dS ⫽

S

冕 冕 冕

冕

BT1 D1B1 dV[P] ⫹

V

⫹

BT2 D2B2 dV[P]

V

BT2 D3B3 dV[U]

(12)

V

冕

Ni[Fe] dS ⫽

S

B4TD4B4 dV[U] ⫹

V

[Qe] ⫽

冤 冤

(qnr)1

[Fe] ⫽

(qnr)ne

0

… (qnc)1

(fxr)1

B5TB1 dV[P]

0

0

(qnc)ne

(fzr)1

0

冥

(fxr)ne

(14)

0

(fzr)ne

qnr

qnc

⫽

i

fxr

0

(fxc)1

⫽

fzr

fzc

K ␥w

∂Pr ∂x

∂Pc ∂x

Sxr

fxc

i

0

… 0

(fzc)1

0

(fxc)ne

冢 冦 冧 冦 冧冣 冢 冣 冢冦 冧 冦 冧 冣 冉 冊

(13)

V

0

0

冕

Sxc

Txzr

Txzc

nx ⫹

∂Pr ∂z

K ␥w

∂Pc ∂z

nz

0

(fzc)ne

冥

(15)

(16)

i

Txzr

nx ⫹

Txzc Szr

Szc

nz

i

(17)

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冤 冤

(Pr)1

[P] ⫽

[U] ⫽

0

(Pr)ne

0

… 0

(Pc)1

0

(Pc)ne

(Ur)1

0

(Wr)1

0

冥

(18)

(Pr)ne

0

(Wr)ne

… 0

(Pc)1

0

(Wc)1

0

(Pc)ne

0

0

冥

(19)

(Wc)ne

where ne is the number of nodes per element, Ni is the shape function of the ith node, and coefficient matrices Bi and Di are given in Appendix A. 4.2. Numerical scheme The first step in solving the wave–soil–pipe interaction with variable permeability and shear modulus is to obtain the lateral boundary conditions. To do so, the wave– seabed interaction can be solved by employing the principle of repeatability (Zienkiewicz and Scott, 1972). This concept is particularly convenient for periodical loading such as the present problem (Jeng and Lin, 1999). Once the later boundary conditions are obtained, the whole wave–seabed interaction problem with the pipeline can be solved. Because the concentration of stresses is to be expected, the local refinement of the finite-element (FE) mesh always has to be taken into account in the region near a structure. To improve the accuracy of the solution in this region, we use two different mesh systems in the present model. As seen in Fig. 2, a four-node iso-parametric element is used in the region near the pipeline. Outside this region, a four-nodal rectangular element is used. This kind of

Fig. 2.

Finite-element mesh in the vicinity of the pipeline.

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mesh has been used for treating the problem around a pipe-like structure (Magda, 1996, 1997). 4.3. Verification Since investigations for the wave-induced seabed response around a buried pipeline in an anisotropic non-homogeneous seabed have not been available until now, a possible verification of the present model is by comparison with previous experimental data in a uniform isotropic seabed. The experimental study conducted in the J.H. DeFree Hydraulic Laboratory at Cornell University (Turcotte et al., 1984) is used for the verification of the present model. The input data of the experiment is listed in Fig. 3. In the figure, the solid line represents the present results in a uniform isotropic seabed, and the symbols denote the laboratory data. As seen in the figure, the present solution overall agrees with the experimental data.

5. Numerical results and discussion Based on the proposed finite element model, the wave-induced pore pressure, effective stresses and soil displacements can be calculated. The effects of wave and soil characteristics on the wave-induced seabed response without pipeline have been discussed in general terms in the second author’s previous work (Jeng, 1997b). Thus, this paper is aimed at investigating the effects of (1) a cover layer and (2) anisotropic

Fig. 3. Distribution of the amplitude of the pore pressure on the pipeline surface versus polar angle ␪. The solid line denotes the present results and symbols denote the laboratory data from Turcotte et al. (1984).

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soil behavior on the wave-induced pore pressures along the pipe surface. To gain a basic understanding of the whole problem, the input data of the case study are tabulated in Table 1. 5.1. Effect of width of cover layer To examine the influences of width of a cover layer (W), values of W are taken as 2.0 m, 2.5 m, 3.0 m and 4.0 m, with fixed thickness of cover layer B ⫽ 2.5 m as an example. Contours of wave-induced pore pressure in the vicinity of a buried pipeline in an anisotropic seabed with various W are illustrated in Fig. 4. In the figure, dotted lines represent the region of a cover layer. As shown in Fig. 4, the width of a cover layer directly affects the wave-induced pore pressures. For example, the region covered by the contour level p/po ⫽ 0.95 extends in the horizontal direction as the width of cover layer W increases. This implies that the wave-induced pore pressure increases as W increases in the vicinity of a buried pipeline. This phenomenon is more significant in the cover layer. Table 1 Input data for case study Wave characteristics Wave period T Water depth d Wave length L Wave height H Soil characteristics Porosity n⬘ Poisson’s ratio ␮xx Poisson’s ratio ␮zx Anisotropic parameter n Anisotropic parameter m Seabed thickness h Young’s modulus Ez Permeability K

10.0 s 20 m 121.1 m 5.0 m

Degree of saturation S

0.45 0.45 0.45 0.8 0.6 50.0 m 7.0 ⫻ 107 N/m2 10−4 m/s (soil A: fine sand) 10−2 m/s (soil B: coarse sand) 0.98

Geometry of the pipe Radius of the pipe R Thickness of the pipe Ro Young’s modulus of the pipe Ep

0.5 m 0.02 m 10 ⫻ Ez

Geometry of cover layer Width of cover layer W Thickness of cover layer B

2.0–4.0 m 2.0–4.5 m

Note: For an isotropic seabed: ␮ ⫽ ␮xx ⫽ 0.45, E ⫽ Ez.

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Fig. 4. Contours of wave-induced pore pressure in the vicinity of a buried pipeline in anisotropic seabeds with various values of width of a cover layer: (a) W ⫽ 2.0 m, (b) W ⫽ 2.5 m, (c) W ⫽ 3.0 m and (d) W ⫽ 4.0 m. (B ⫽ 2.5 m).

Fig. 5 illustrates the distributions of pore pressure along the pipeline surface versus radial angle (␪, as defined in Fig. 1) for various values of W. The figure indicates that the pore pressure p/po increases as W increases. It is noted that the maximum p/po occurs at ␪ ⫽ 90°, while the minimum occurs at ␪ ⫽ 270°. This implies that the maximum pore pressure occurs at the top of the pipeline, and the minimum occurs at the bottom of the pipe. 5.2. Effect of thickness of cover layer Besides the width of a cover layer (W), thickness of a cover layer (B) is another important factor in the design of a cover layer. To gain a basic understanding of the

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Fig. 5. Distribution of wave-induced pore pressure along a pipeline surface for various widths of cover layer. Solid lines denote the anisotropic solution and dashed lines denote the isotropic solution (B ⫽ 2.5 m).

effects of thickness of a cover layer, the thickness of a cover layer B ⫽ 2.0 m, 2.5 m, 3.5 m and 4.5 m is considered with fixed W ⫽ 2.0 m as an example. Contours of wave-induced non-dimensional pore pressure (p/po) for various values of B with fixed W ⫽ 2.0 m are illustrated in Fig. 6. The figure clearly shows that the thickness of a cover layer has a great influence on the wave-induced pore pressure near a buried pipeline. A thicker cover layer increases the affected region of a buried pipe in the vertical direction. For example, in Fig. 6(a) for B ⫽ 2.0, the cover layer is surrounded by the contour level p/po ⫽ 0.9, and the concentration of pore pressure near the bottom of a cover layer is observed. However, in Fig. 6(b), (c) and (d), the contour levels around the cover layer are 0.85, 0.85 and 0.8, respectively. Furthermore, the phenomenon of pore pressure concentration reduces as B increases. Fig. 7 illustrates the distribution of wave-induced pore pressure along pipeline surface for various B in an anisotropic seabed. The figure clearly shows that the pore pressure p/po on the pipeline surface increases as the thickness of a cover layer (B) decreases. This implies that a thinner cover layer may enhance the potential of waveinduced seabed instability around the buried pipeline. 5.3. Effect of anisotropic soil behavior It is of interest to investigate the differences of pore pressure along the pipeline surface between isotropic and anisotropic soil behavior. In Figs. 5 and 7, the waveinduced pore pressures on a pipeline surface in an isotropic seabed are also included (dashed lines). As shown in Fig. 5, the relative differences of pore pressure p/po between isotropic

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Fig. 6. Contours of wave-induced pore pressure in the vicinity of a buried pipeline in anisotropic seabeds with various values of thickness of a cover layer: (a) B ⫽ 2.0 m, (b) B ⫽ 2.5 m, (c) B ⫽ 3.5 m and (d) B ⫽ 4.5 m. (W ⫽ 2.0 m).

and anisotropic solutions become more significant as the width of a cover layer (W) decreases. Similar trends are also observed in Fig. 7 for various B. It is noted that the effect of anisotropic soil behavior near the bottom of the pipe (180° ⱕ ␪ ⱕ 360°) is more significant than that near the top of the pipe (0° ⱕ ␪ ⱕ 180°). 6. Conclusions In this paper, a two-dimensional finite element model is proposed to examine the wave-induced pore pressure around a buried pipeline in an anisotropic seabed. The numerical results of the present model overall agree with the experimental data for

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Fig. 7. Distribution of wave-induced pore pressure along a pipeline surface for various thickness of cover layer. Solid lines denote the anisotropic solution and dashed lines denote the isotropic solution. (W ⫽ 2.0 m).

an isotropic seabed (Turcotte et al., 1984). The effects of a cover layer on the waveinduced pore pressure have been discussed in this paper. Based on the numerical results, some conclusions can be stated: 1. A cover layer significantly affects the wave-induced pore pressure near a buried pipeline. The region surrounded by a certain contour level (for example, p/po ⫽ 0.95) extends in the horizontal direction as the width of a cover layer W increases. The wave-induced pore pressure p/po on the surface of a buried pipe increases as W increases. 2. Numerical results indicate that a larger thickness of cover layer enhances the affected region in the vertical direction. The wave-induced pore pressure p/po on the pipeline surface increases as the thickness of a cover layer B decreases. 3. Besides the geometry of a cover layer, anisotropic soil behavior also affects the wave-induced pore pressure on the pipeline surface. Effects of anisotropic soil behavior increase as W and B increase. It is noted that effects of anisotropic soil behavior are more important near the bottom of the pipe than the top of the pipe.

Acknowledgements The author is grateful to Professor Philip L.-F. Liu from the School of Civil and Environmental Engineering, Cornell University, for his generosity in providing his experimental data. The work described here is part of the activities of the Special Research Centre for Offshore Foundation Systems at the University of Western Aus-

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tralia, established and supported under the Australian Research Council’s Research Centres Program. Financial support by the National Science Council of Taiwan under project NSC88-2611-E-005-006 is also appreciated.

Appendix A List of coefficient matrices The coefficient matrices Bi(i ⫽ 1–5) in Eqs. (12) and (13) are listed as follows. Bi ⫽ [bi1 bi2$bine]

b1i ⫽

b2i ⫽

∂Ni ∂x

0

0

∂Ni ∂x

(20)

冤 冥 ∂Ni ∂z

0

0

∂Ni ∂z

(21)

冋 册 Ni 0

(22)

0 Ni

b3i ⫽ bT1i

(23)

 ∂Ni ∂x

 

0

0

0

0

∂Ni ∂x

0

0

0

0

∂Ni ∂z

0

bi4 ⫽ 



0

0

0

∂Ni ∂z

∂Ni ∂z

0

∂Ni ∂x

0

0

∂Ni ∂z

0



  

∂Ni ∂x 

(24)

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冤

Ni 0 0 0

b5i ⫽

0 Ni 0 0 0 0 Ni 0 0 0 0 Ni

冥

(25)

The coefficient matrices Di in Eqs. (12) and (13) are given:

冤 冥 1 0 0 0

D1 ⫽

␥w 0 1 0 0 K 0 0 1 0

(26)

0 0 0 1

D2 ⫽ n⬘␤␻

冋

D3 ⫽ ␻

冋

0 ⫺1 1

0

0 ⫺1 1

冤

0

册

C11 0

D4 ⫽ E z

0 C11 C13

0

册

(27) (28)

C13

0 0 0

0 C13 0 0 C33 0

0 0

0 C13

0 C33 0 0

0

0

0

0

0 m 0

0

0

0

0

0 0 m

冥

(29)

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