Trans. Tianjin Univ. 2012, 18: 357-365 DOI 10.1007/s12209-012-1853-z
3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space* BA Zhenning(巴振宁)1,2，LIANG Jianwen(梁建文)1,2，REN Qiangqiang(任强强)1,2 (1. School of Civil Engineering, Tianjin University, Tianjin 300072, China; 2. Key Laboratory of Coast Civil Structure Safety of Ministry of Education, Tianjin University, Tianjin 300072, China) © Tianjin University and Springer-Verlag Berlin Heidelberg 2012
Abstract：The indirect boundary element method (IBEM) was established to solve the problem of 3-D seismic responses of 2-D topographies, by calculating the free-field responses with the direct-stiffness method and simulating the scattering wave fields with the dynamic Green’s functions of moving distributed loads. The proposed method yields accurate results, because the 3-D dynamic stiffness matrixes used are exact and the fictitious moving distributed loads can be acted directly on the interface between the alluvial valley and the layered half-space without singularity. The comparison with the published methods verifies the validity of the proposed method. And the numerical analyses are performed to give some beneficial conclusions. The study shows that 3-D scattering by an alluvial valley is essentially different from the 2-D case, and that the presence of soil layer affects not only the amplitude value of surface displacements but also the distribution of surface displacements. Keywords：3-D scattering; layered half-space; alluvial valley; plane P waves
Seismic responses of an alluvial valley can be 2-D (wave field with the incident wave perpendicular to the axis of the alluvial valley) or 3-D (wave field with an arbitrary incident angle). Since the pioneering work of Trifunac[1], there have been a large number of theoretical studies on seismic responses of an alluvial valley and most of the studies refer to the 2-D cases involving antiplane strain conditions or in-plane strain conditions. These methods include the wave function expansion method[2-4], the wave source method[5, 6], and the boundary element method[7-9]. Compared with the 2-D cases, studies on the 3-D responses of an alluvial valley are relatively few. Liang et al[10, 11] used wave function expansion method to obtain the 3-D responses of a uniform cylindrical valley embedded in a uniform half-space. Pedersen et al [12] also studied the same problem using indirect boundary element method (IBEM), but the free surface of the half-space must be discretized since the Green’s functions are fullspace ones. Barros and Luco[13] presented a boundary integral equation method to study the 3-D responses of an alluvial valley embedded in a layered half-space. The sources used in this method must be carefully located in
order to avoid singularities, and the location away from the interface between the valley and the layered halfspace leads to an approximate solution. Liao[14] studied the 3-D responses of a basin using the T-matrix method. In this study, IBEM was used to calculate the 3-D scattering of obliquely incident plane P waves by an alluvial valley embedded in a layered half-space. The proposed method used the dynamic Green’s functions of moving distributed loads acting on an inclined line in a layered half-space to simulate the scattering wave fields, and made it easy to treat an alluvial valley with arbitrary cross section in a layered half-space. The accuracy of the method was verified by comparison with related solutions, and a numerical analysis was carried out for an alluvial valley embedded in one soil layer over halfspace.
1
Model and method of solution
The model considered here corresponds to an infinite long, viscoelastic valley of arbitrary cross-section embedded in a horizontally layered viscoelastic halfspace. The valley is characterized by complex S-wave
Accepted date: 2012-06-14. *Supported by National Natural Science Foundation of China (No. 50978156 and 50908183) and Tianjin Research Program of Application Foundation and Advanced Technology(12JCQNJC04700). BA Zhenning, born in 1980, male, Dr, associate Prof. Correspondence to BA Zhenning, E-mail:
[email protected].
Transactions of Tianjin University
Vol.18 No.5 2012
V
velocity C s CsV (1 2i V )1/ 2 , mass density V , and the valley interface. Poisson’s ratio V . The term CsV represents the real part For an infinite long alluvial valley, although the dyV of S-wave velocity and represents the hysteretic ratio. namic responses of the alluvial valley are 3-D, the wave The soil layers are characterized by complex S-wave ve- fields at two different cross sections are identical but shift L locities C sj CsLj (1 2i jL )1/ 2 , mass density Lj , Poisson’s with time. This allows only one cross section of the valratios Lj , and the thicknesses H Lj of different layers ley used for calculation to obtain the 3-D dynamic re( j 1, , N ). And the bedrock is characterized by com- sponses anywhere, and the computational cost can be reduced greatly. The solving steps are as follows: first, R plex S-wave velocity C s CsR (1 2i R )1/ 2 , mass denthe free-field responses for the obliquely incident plane P sity R , and Poisson’s ratio R . waves are obtained by the direct-stiffness method, and The excitation is represented by obliquely incident then moving distributed loads are applied to the interface plane P waves from the bedrock. The normal to the wave- between the alluvial valley and the layered half-space to front forms an angle v with the vertical axis y on plane simulate the scattering wave fields. The fictitious loads y - z , as shown in Fig.1(b), and forms an angle h with densities are determined by boundary conditions. Finally, the horizontal axis y on plane x - y , as shown in the total responses are obtained by adding the responses Fig.1(a), and Fig.1(c) gives the profile of the alluvial of the free field and the fictitious loads. valley cross section. c is the apparent velocity and s is
(a) Horizontal plane, definition of angle h
(b) Vertical plane, definition of angle v
(c) Cross section of the valley
Fig.1 Model
The free-field responses for obliquely incident plane P waves can be calculated by the direct-stiffness method. The displacements at the upper and bottom interfaces of each soil layer can be determined as (1) Px 0 u0 P SR v P-SV-SH 0 y0 iPz 0 iw0 SP SV SHU Q
(2)
where u0 , v0 and w0 are the three bedrock outcrop input R motions; SP-SV-SH and SP-SV-SH are the 3-D half-space dynamic stiffness matrix of the rock and the total 3-D dynamic stiffness matrix of the site, respectively. More details about the free-field responses of the layered halfspace can be found in Ref. [15]. The amplitude coefficients of up-going and downgoing waves are obtained. Finally, the dynamic responses such as displacements and stresses at any point are evaluated following the relationship between the dynamic re—358—
sponses and the amplitude coefficients of up-going and down-going waves. The scattering wave-field responses can be simulated by the dynamic Green’s functions of moving distributed loads acting on the interface between the alluvial valley and the layered half-space discretized by inclined lines. And the Green’s functions can be calculated in the following way. Firstly, the 3-D dynamic Green’s functions of moving distributed loads acting on inclined lines in a layered half-space are deduced, and then by integrating the Green’s functions along the moving direction, the dynamic Green’s functions of moving loads acting on an inclined line are obtained. More details of the Green’s functions can be found in Ref.[16]. If gu ( x) and gt ( x) are the displacements and stresses in Green’s functions of moving loads, the displacements and stresses (in the x , y and z coordinate directions) at any point in the layered half-space can be expressed as
BA Zhenning et al: 3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space
u p , v p , wp
gu ( x, y, z ) px , p y , pz
T
T
T
p p p t x , t y , t z gt ( x, y, z ) px , p y , pz
T
(3)
Tf W T ( s )tf ( s )ds
(4)
Vp L W T ( s )guL ( s )ds
s
s
Vp V W T ( s )guV ( s )ds
where px , p y and pz are the densities of moving loads acting on the inclined lines in the three coordinate directions. The superscript “p” in the displacement terms
u
p T
T
and stress terms t , t , t indicates the results induced by the moving distributed loads. The continuity conditions of displacements and stresses along the interface between the valley and the layered half-space can be written as p
p
,v , w
p x
p y
p z
t xL ( s ) t xf ( s) L f T s W (s) t yL ( s) t yf ( s) ds t ( s ) t ( s ) z z
t xV ( s ) V T s W (s) t yV (s) ds t z ( s )
(5)
s
Vf W T ( s )vf ( s )ds s
where p 1 and p 2 are the densities of moving loads acting on the interface between the alluvial valley and the layered half-space, respectively. The subscripts “p” and “ f ” represent the moving distributed loads and the free field. By solving Eqs. (7) and (8), the load densities p1 and p2 are obtained, and from Eq. (3), the displacements inside and outside the alluvial valley can be expressed as f u ( x, y , z ) u ( x, y , z ) p1x f L v( x, y, z ) v ( x, y, z ) gu ( x, y, z ) p1 y f w( x, y, z ) w ( x, y, z ) p1z
u ( x, y , z ) p2 x V v( x, y, z ) gu ( x, y, z ) p2 y w( x, y, z ) p2 z
u L (s) u f (s) L f T s W (s) v L( s) v f (s) ds w (s) w (s) u V (s) V s W (s) v V( s) ds w (s) T
(6)
where W ( s) is the weighting function, which can be taken as a unit matrix, and the integral can be evaluated over each element separately. The superscripts “L” and “V” represent the displacements and stresses in the layered half-space and in the valley caused by the moving distributed loads, and the superscript “ f ” represent the displacements and stresses of the free filed. Substituting Eqs.(3) and (4) into Eqs.(5) and (6) yields p 1x p 2x V Tp p 1 y Tf Tp p 2 y p 1z p 2z
(7)
p 1x p 2x V Vp p 1 y Vf Vp p 2 y p 1z p 2z
(8)
L
L
where Tp L W T ( s )gtL ( s )ds
2
(9)
(10)
Validation of the method
The validity of the presented method was verified by comparison with the results of Ref. [13] for a semicircular valley embedded in a uniform half-space and in a layered medium consisting of a layer overlying a halfspace. For the uniform half-space, the valley ( v ) and the half-space (1) are characterized by damping ratios 1 ＝ v ＝0.005, Poisson’s ratios 1 ＝ v ＝1/3, density ratio v 1 ＝2/3, and S-wave velocity ratio v 1 ＝0.5. For the layered half-space, the valley ( v ), the soil layer (1) and the half-space (2) are characterized by 1 v 2 0.005, 1 ＝ v ＝ 2 ＝ 1/3, v 1 ＝ 2/3, 2 1 ＝ 4/3, v 1 ＝ 0.5, and 2 1 ＝ 2.0. The horizontal incident angle h ＝0°, and the vertical incident angle v ＝30°. The dimensionless incident frequency a / 1 ＝0.5, in which is the incident frequency and a is the radius of the valley. AP is the incident amplitude of P waves. Fig.2 shows that the results agree very well with those of Ref. [13].
s
Tp V W T ( s )gtV ( s )ds s
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(a) Amplitude u / AP
(b) Amplitude v / AP
Fig.2
(c) Amplitude w / AP
Comparison with the results of Ref.[13]
alluvial valley and discuss whether the obliquely incident plane P waves can be decomposed into the plane of val3.1 Comparisons between 2-D scattering and 3-D ley cross section, the surface displacement amplitudes of 2-D case and 3-D case for a semi-circular cross-section scattering In order to study the differences between 2-D scat- alluvial valley embedded in a uniform half-space are iltering and 3-D scattering of incident plane P waves by an lustrated in Fig. 3. The valley and the half-space(1)are
3
Numerical results and discussion
(a)v＝30º, ＝0.5
(b)v＝60º, ＝0.5
(c)v＝90º, ＝0.5
(d)v＝30º, ＝0.5
(e)v＝60º, ＝0.5
(f)v＝90º, ＝0.5
(g)v＝30º, ＝1.0
(h)v＝60º, ＝1.0
(i)v＝90º, ＝1.0
(k)v＝60º, ＝1.0
(l)v＝90º, ＝1.0
(j)v＝30º, ＝1.0
Fig.3
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Comparisons between the 3-D results and 2-D results
BA Zhenning et al: 3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space
characterized by 1 ＝ v ＝0.001, 1 ＝ v ＝1/3, v 1 2/3, v 1 ＝ 0.5, ＝ 0.5 and 1.0. For 3-D scattering, h 45°, v ＝30°, 60°and 90°. And for 2-D scattering, ＝30°, 60°and 90°. The results in Fig. 3 include amplitudes u / AP and w / AP for 2-D case and u / AP sin h and w / AP sin h for 3-D case. As shown in Fig. 3，for non-perpendicular incident plane P waves ( v ≠,90°), the 3-D responses are obviously different from the 2-D ones, and the obliquely incident plane P waves cannot be simply decomposed into plane of valley cross section as those in the 2-D case. But for vertically incident plane P waves ( v ＝ 90°), the method presented in this paper has the same results as the
2-D case. The results in Fig. 3 also indicate that the “screen effect” is more obvious for the 2-D case than for the 3-D case. 3.2 Effects of the horizontally incident angles The effects of the horizontally incident angles on 3-D responses of an alluvial valley of semi-circular cross section embedded in a uniform half-space are illustrated in Fig.4. The parameters for calculation and the definition of the dimensionless frequency are the same as those in Fig.3. The results are presented for obliquely incident plane P waves with v ＝45°, h ＝0°, 30°, 60° , 90°, and ＝0.5, 1.0 , 2.0.
(a) Amplitude u / AP , =0.5
(b) Amplitude v / AP , =0.5
(c) Amplitude w / AP , =0.5
(d) Amplitude u / AP , =1.0
(e) Amplitude v / AP , =1.0
(f) Amplitude w / AP , =1.0
(g) Amplitude u / AP , =2.0
(h) Amplitude v / AP , =2.0
(i) Amplitude w / AP , =2.0
Fig.4 Surface displacements around a valley in uniform half-space with different horizontally incident angles
It is shown that the horizontally incident angles have great effects on the amplitudes of surface displacements around the valley. As h increases, the horizontal displacements in the x direction increase gradually, while the horizontal displacements in the y direction decrease gradually. And the “screen effect” is more significant for larger horizontally incident angle. Both the horizontal and vertical displacements become symmetric about the
axis of the valley when h ＝90. It should be noted that when the incident waves are perpendicular to the axis of the valley ( h ＝90), there are only the in-plane displacements (displacements in the x and z directions) and the out-plane displacements (in y direction) vanishing. 3.3 Responses of a valley in a layered half-space In order to describe the effects of the presence of soil layers, 3-D responses of obliquely incident plane P —361—
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waves by a semi-circular valley embedded in one soil layer over bedrock are shown in Figs. 5—7 for different stiffness ratios of bedrock to soil layers. The bedrock is characterized by S-wave velocity CsR , mass density R , Poisson’s ratio R and damping ratio R ; the soil layer is characterized by S-wave velocity CsL , mass density L , damping ratio L , Poisson’s ratio L and depth H ; and the valley is characterized by S-wave velocity CsV , mass density V , damping ratio V , Poisson’s ratio V and radius a . The parameters for calculation are H / a ＝2.0, R / L ＝ 1.0, L R ＝ 1/3, R ＝ 0.02, L ＝ 0.05, V ＝ 0.05, CsR / CsL ＝ 2.0, 5.0 and , h ＝ 45, v ＝ 5, 30, 60 and 90, a / CsL ＝0.25, 0.5, and 1.0. The displacements shown in Figs. 5—7 have been normalized by AP , which is the amplitude of incident displacement at the bedrock. As shown in Figs. 5—7, the presence of soil layer has a significant effect on the 3-D responses of the alluvial valley. From Figs.5—7, it can be indicated that the presence of soil layer causes marked changes that cannot be attributed to the simple amplification of free-field ground motion by the soil layer. There exist some inter-
actions between the valley and the soil layer. The results in Figs.5—7 indicate that the spatial distributions of displacement amplitudes are similar for different stiffness ratios of bedrock to soil layer, although the values of displacement amplitudes get higher with the increase of the stiffness ratio of bedrock to soil layer. For example, the maximum values of the vertical displacement amplitudes are 5.29, 9.72, 27.34 for the S-wave velocity ratios of bedrock to soil layer CsR / CsL ＝ 2.0, CsR / CsL ＝5.0 and CsR / CsL ＝ , respectively. The reason for this phenomenon is that although the stiffness ratio of bedrock to soil layer changes, the dynamic characteristics keep unchanged. In order to study the effects of soil layer depth on 3D responses, Fig.8 and Fig.9 illustrate the surface displacement amplitudes around the alluvial for different soil layer thicknesses H / a ＝1.0 and H / a ＝4.0, respectively. The S-wave velocity ratio CsR / CsL ＝5.0, ＝0.25, 0.5, 1.0, and other parameters are the same as those in Fig.5. The displacements in Figs. 8 and 9 are also normalized by the amplitudes of incident displacement at the bedrock.
(a)Amplitude u / AP , CsR / CsL 2.0
(b) Amplitude v / AP , CsR / CsL 2.0
(c) Amplitude w / AP , CsR / CsL 2.0
(d)Amplitude u / AP , CsR / CsL 5.0
(e) Amplitude v / AP , CsR / CsL 5.0
(f) Amplitude w / AP , CsR / CsL 5.0
(g)Amplitude u / AP , CsR / CsL
(h) Amplitude v / AP , CsR / CsL
(i) Amplitude w / AP , CsR / CsL
Fig.5 Surface displacement amplitudes( H / a 2.0 , 0.25 )
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BA Zhenning et al: 3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space
(a)Amplitude u / AP , CsR / CsL 2.0
(b) Amplitude v / AP , CsR / CsL 2.0
(c) Amplitude w / AP , CsR / CsL 2.0
(d)Amplitude u / AP , CsR / CsL 5.0
(e) Amplitude v / AP , CsR / CsL 5.0
(f) Amplitude w / AP , CsR / CsL 5.0
(g)Amplitude u / AP , CsR / CsL
(h) Amplitude v / AP , CsR / CsL
(i) Amplitude w / AP , CsR / CsL
Fig.6 Surface displacement amplitudes( H / a 2.0 , 0.5 )
(a)Amplitude u / AP , CsR / CsL 2.0
(b) Amplitude v / AP , CsR / CsL 2.0
(c) Amplitude w / AP , CsR / CsL 2.0
(d)Amplitude u / AP , CsR / CsL 5.0
(e) Amplitude v / AP , CsR / CsL 5.0
(f) Amplitude w / AP , csR / csL 5.0
(g)Amplitude u / AP , CsR / CsL
(h) Amplitude v / AP , CsR / CsL
(i) Amplitude w / AP , CsR / CsL
Fig.7 Surface displacement amplitudes( H / a 2.0 , 1.0 )
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(a)Amplitude u / AP , 0.25
(b) Amplitude v / AP , 0.25
(c) Amplitude w / AP , 0.25
(d)Amplitude u / AP , 0.5
(e) Amplitude v / AP , 0.5
(f) Amplitude w / AP , 0.5
(g)Amplitude u / AP , 1.0
(h) Amplitude v / AP , 1.0
(i) Amplitude w / AP , 1.0
Fig.8 Surface displacement amplitudes( H / a 1.0 , C / C 5.0 ) R s
L s
(a)Amplitude u / AP , 0.25
(b) Amplitude v / AP , 0.25
(c) Amplitude w / AP , 0.25
(d)Amplitude u / AP , 0.5
(e) Amplitude v / AP , 0.5
(f) Amplitude w / AP , 0.5
(g)Amplitude u / AP , 1.0
(h) Amplitude v / AP , 1.0
(i) Amplitude w / AP , 1.0
Fig.9 Surface displacement amplitudes( H / a 4.0 , C / C 5.0 ) R s
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L s
BA Zhenning et al: 3-D Scattering of Obliquely Incident Plane P Waves by Alluvial Valley Embedded in Layered Half-Space
The comparison of Figs.8 and 9 with Figs.5 to 7 shows that the spatial distributions of displacements are completely different for different soil layer thicknesses because the variation of soil layer depth directly leads to the changes of the dynamic characteristic of soil layer. For example, the first vertical resonance frequency is ＝1.0 for H / a ＝1.0, while the first vertical resonance frequency is ＝ 0.25 for H / a ＝ 4.0. The surface displacement amplitudes become more complex for deeper soil layer. The surface displacements decrease greatly for higher incident frequency and deeper soil layer since the material damping of the valley and the soil layer are considered in the presented method.
4
Conclusions
(1) 3-D scattering by an alluvial valley is essentially different from 2-D case, and the obliquely incident plane P waves cannot be simply decomposed into the valley cross-section plane as those in a 2-D case. (2) The horizontally incident angles have a significant effect on the variation and values of displacement amplitudes. With the increase of horizontally incident angles, the horizontal displacement amplitudes in the x direction increase gradually, while those in the y direction decrease gradually. (3) The presence of soil layer causes marked changes compared with uniform half-space case. The variation of the stiffness ratio of bedrock to soil layer leads to the change in the values of displacement amplitudes，and the change of soil layer depth affects both the values and the distribution of displacement amplitudes.
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