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A new method for formulation of dynamic responses of shallow foundations in simple general form
Soil Dynamics and Earthquake Engineering 25 (2005) 679–688 www.elsevier.com/locate/soildyn
A new method for formulation of dynamic responses of shallow foundations in simple general form T. Nogami*, A.Al Mahbub, S.H. Chen Department of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore, Singapore 117576 Accepted 1 November 2004
Abstract A single uniform rectangular area, either homogeneous or heterogeneous, is considered in a soil medium (fundamental cell). Two governing ordinary differential equations in special form are developed for the fundamental cell. The ground supporting a partially embedded foundation is divided into a number of coarse rectangular areas (secondary cells). Each secondary cell is treated as either a single fundamental cell for homogeneous ground or a stack of fundamental cells for inhomogeneous ground. Differential equations for the assembly of secondary cells are formed with those for the fundamental cells. These equations lead to the soil responses in each cell expressed in simple closed form. They also lead to the convenient treatment of soil with appropriate Winkler-type models along the foundation faces and concentrated forces acting at the foundation corners. With them, the foundation responses are finally expressed in simple closed form. The approach is demonstrated for various cases and confirmed to produce the results reasonable enough for civil engineering use. q 2005 Elsevier Ltd. All rights reserved. Keywords: Foundation; Inhomogeneous soil; Vibrations; Computational method
1. Introduction Simple expressions for the dynamic responses of foundations were presented previously. Novak and his colleagues [1,2] treated the soil at the side of foundation as a stack of mutually uncoupled thin layers. The stiffnesses of an individual layer at the foundation were formulated from vibrations of a horizontal, massless, rigid, circular slice of the foundation contained in a horizontal layer of unit thickness [2]. Those for foundations with rectangular base area were also formulated recently [3]. For the time-domain analysis, frequency-dependent soil stiffnesses at the side formulated in this manner were idealized as frequencyindependent spring–mass–dashpot systems for circular foundations [4,5]. Non-linear mechanism was introduced further in the side soil stiffnesses [6,7]. Nogami and his colleagues used another approach for shallow foundations [8–12] and deep foundations [13]. In * Corresponding author. Address: 13250 Kibbings Road, San Diego, CA 92130, USA. Tel.: C1 858 755 3952. E-mail address: [email protected] (T. Nogami).
0267-7261/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2004.11.013
this approach, the medium was idealized as a system of springs and closely spaced one-dimensional columns with distributed mass, in which the springs interconnect the two adjacent columns. Thus, these springs produce coupling among the columns, enabling the model to closely simulate the behaviour of a continuous medium. Without springs, the model is similar to Novak’s model. The above second simplified approach results in two governing ordinary differential equations in convenient form, to describe the dynamic behavior of a rectangular area in the plane-strain medium. Using these two equations as a base, a special method was used to develop simple closed form expressions for the foundation responses [14–16]. The method was called later the differential equation cell method (DECM). This paper refines DECM and demonstrates its application to shallow foundations in general soil profile in a systematic manner. The major emphasis of the present paper is placed on simplicity of the final formulations and generality to use them for any inhomogeneous soil profiles. Linear visco-elastic soils are assumed as commonly assumed in simple expressions proposed by many people.
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
2. Differential equations for assembly of cells 2.1. Fundamental differential equations for a single cell A soil medium is assumed to be a horizontally layered visco-elastic medium, in which an individual layer is assumed to be either homogeneous or heterogeneous. Its material properties are defined by complex Lame`’s constants (l* and G*, in which G* is the complex shear modulus) and unit mass (r). With the Cartesian coordinates x and z taken, respectively, in the horizontal and vertical directions, the complex moduli dependent on z are defined herein as ðG ðzÞ; l ðzÞÞ Z ð1 C 2DiÞðGðzÞ; lðzÞÞ (1) pffiffiffiffiffiffi where iZ K1; and DZfrequency-independent damping factor of the medium. Linear variations of the moduli with z are stated as ðGðzÞ; lðzÞÞ Z ð1 C Cðz K z0 ÞÞðG0 ; l0 Þ
(2)
where z0Zreference location in z0 and l0ZG(z) and l(z) at z0, respectively; CZconstant (0 for homogeneous material); and denoting n as Poisson’s ratio lðzÞ l 2n Z 0 Z GðzÞ G0 1 K 2n
(3)
It is noted that Eq. (5) is also the equation of motion of a column in a system of closely spaced columns that are interconnected by lateral springs distributed along the side [9,17]: where kc and ks correspond, respectively, to the complex stiffnesses of the column and spring per length of the column; and mc corresponds to the mass per length of the column. The non-dimensional parameters, kc and ks , are dependent rather uniquely on n and very little on soil profile [9,17]. A rectangular cell is considered in a continuous medium, in which the sides of the cell are parallel to the x–z coordinates (Fig. 1). The displacement of the medium in the cell is assumed to be expressed in the form of uðx; zÞ Z XðxÞZðzÞ
(7)
Substituting Eq. (7) into Eq. (5) and denoting f(x) as a weight function, the Galerkin method for weighted residual over x in the cell yields ð xb 2 d XðxÞ ks ðzÞ fðxÞdx ZðzÞ 2 xa dx ð xb d dZðzÞ kc ðzÞ C XðxÞfðxÞdx dz dz xa ðx b C mc XðxÞfðxÞdx ZðzÞ Z 0
(8)
Equations of motion of the medium in the frequency xa domain are then stated for the plane-strain conditions as 2 3 2 v v2 v v 9 v v ( ) 8 G C l < Kru2 ux ðx; zÞ = 6 ðl C 2G Þ vx2 C vz G vz 7 ux ðx; zÞ vz vx vxvz 6 7 Z 6 7 2 : Kru2 u ðx; zÞ ; 4 v2 5 u ðx; zÞ v v v v v z z l ðl C C C 2G Þ G G vz vz vxvz vz vx2 vz where ux and uzZfrequency-domain displacements in the horizontal and vertical directions, respectively; and uZ circular frequency. Off-diagonal terms couple the lateral and vertical soil motions. It is known that ignorance in the minor component of these motions affects the computed foundation responses very little. Ignoring the off-diagonal terms in Eq. (4) and slightly modifying, the following equation is obtained v vuðx; zÞ v vuðx; zÞ k ðzÞ k ðzÞ C Cmc u2 uðx; zÞ Z 0 vx s vx vz c vz (5) where uZeither ux or uz; and according to Refs. [9,17] ðks ðzÞ; kc ðzÞ; mc Þ Z ðAs ðzÞks ; Ac ðzÞ; kcr; m c Þ
(6a)
with ks , kc and m c Znon-dimensional factors [9], and ( * ðl ðzÞ C 2G* ðzÞ; G* ðzÞÞ for ux ðAs ðzÞ; Ac ðzÞÞ Z (6b) ðG* ðzÞ; l* ðzÞ C 2G* ðzÞÞ for uz
(4)
pa a, a
pa( )
b, a
pb
ux a, b
h
b, b
p
b
G( ), ( ),
D
pa a, a
b, a
uz
pa( ) a, b
pb
h
b, b
pb G( ), ( ),
D
homogeneous or heterogeneous cell Fig. 1. Fundamental cell subjected to tractions.
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
Integrating the first term by parts, Eq. (8) results in d dZðzÞ nðzÞ K C ðkðzÞ K u2 mÞZðzÞ dz dz
FZ ðn; k; mÞZðzÞ Z fðxa Þpa ðzÞ K fðxb Þpb ðzÞ
u1, (n1, k1, m1), (N1, K1, M1) u2, (n2, k2, m2), (N2, K2, M2)
1 2
hj uj, (nj, kj, mj), (Nj, Kj, Mj)
j
Z fðxa Þpa ðzÞ K fðxb Þpb ðzÞ or in compact form
J
(9)
where xa and xbZleft and right ends of the cell in the x coordinate, respectively; pa,b(z)Zthe traction acting along z at (xa,b,z), expressed as dXðxÞ (10) pa;b ðzÞ Z K ks ðzÞZðzÞ dx xa;b
681
uJ, (nJ, kJ, mJ), (NJ, KJ, MJ)
(a) Assembly of vertically connected cells hj 1
u1, (n1, k1, m1), (N1, K1, M1)
2
L
u2, (n2, k2, m2), (N2, K2, M2)
j
L
J
uj, (nj, kj, mj), (Nj, Kj, Mj)
(b) Assembly of horizontally connected cells
and Fig. 2. Assembly of vertically or horizontally connected cells.
ðnðzÞ; kðzÞ; mÞ ð xb dXðxÞ dfðxÞ Z ; mc XðxÞfðxÞ dx kc ðzÞXðxÞfðxÞ; ks ðzÞ dx dx xa (11) Similarly, substituting Eq. (7) into Eq. (5) and using j(z) as a weight function, the Galerkin method for weighted residual over z in the cell yields d2 XðxÞ KNðzÞ C ðKðzÞ K u2 MÞXðxÞ dx2
Xj ðxÞ Z djK1=j XjK1 ðxÞ
djK1=j Z
ZjK1 ðzb Þ Zj ðza Þ
(15a)
(15b)
where djK1 / jZfactor; d0 / 1Zreference factor (assumed to be d0 / 1Z1); and X0(x) and Z0(z)Zreference functions. In addition, the force equilibrium condition at this interface leads to
Z jðza Þpa ðxÞ K jðzb Þpb ðxÞ or in compact form FX ðN; K; MÞXðxÞ Z jðza Þpa ðxÞ K jðzb Þpb ðxÞ
the bottom, as shown in Fig. 2a. Between the jK1th and jth fundamental cells, the compatibility conditions along x lead to
(12)
where za and zbZupper and lower ends of the cell in the z coordinate, respectively; pa,b(x)Ztraction acting along x at (x,za,b), expressed as dZðzÞ pa;b ðxÞ Z K kc ðzÞXðxÞ (13) dz za;b
paj ðxÞ K pbjK1 ðxÞ Z 0
Successive applications of Eqs. (15a) from jZ1 through j in order result in Xj ðxÞ Z d0=j X0 ðxÞ
ð zb dZðzÞ djðzÞ Z ;mc ZðzÞjðzÞ dz ks ðzÞZðzÞjðzÞ;kc ðzÞ dz dz za (14) Eqs. (9) and (12) are the fundamental differential equations for a single cell (fundamental cell). The weight functions are selected as (f(x),j(z))Z(X(x),Z*(z)), in which Z*(z) is the conjugator of Z(z) [12]. For the soil profiles considered herein, integrations yield closed form expressions for (n, k, m) and (N, K, M). 2.2. Differential equations for assembly of cells J cells of equal length in x are assumed to be connected vertically in series and are numbered in order from the top to
(17a)
where
and ðN;K; MÞ
(16)
d0=j Z
j Y
djK1=j
(18)
jZ1
Multiplying Eq. (12) by d0/j and replacing Xj(x) with d0/jX0(x), Eq. (12) for the jth cell can be rewritten as FX ðd20=j Nj ; d20=j Kj ; d20=j Mj ÞX0 ðxÞ Z d0=j jj ðza Þpaj ðxÞ K d0=j jj ðzb Þpbj ðxÞ
(19)
Since d0=jK1 ZjK1 ðzb ÞZ d0=j Zj ðza Þ, summing up Eq. (19) from jZ1 through J and denoting X0(x)ZX(x) result in a single differential equation for X(x). With this and Eq. (9), the differential equations for the assembly are FX ðN 0 ; K 0 ; M 0 ÞXðxÞ Z d0=1 j1 ðza Þpa1 ðxÞ K d0=J jJ ðzb ÞpbJ ðxÞ
(20a)
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
FZ ðnj ; kj ; mj ÞZj ðzÞ Z fðxa Þpaj ðzÞ K fðxb Þpbj ðzÞ (20b) j Z 1 wJ where
J X N 0; K 0; M 0 Z ðd0=j Þ2 ðNj ; Kj ; Mj Þ
(21)
jZ1
Next, J rectangular cells of equal length in z are assumed to be connected horizontally in series and are numbered from the left to the right in order as shown in Fig. 2b. Similarly, the following relations exist in this connection: Zj ðzÞ Z djK1=j ZjK1 ðzÞ Z d0=j Z0 ðzÞ djK1=j Z
XjK1 ðxb Þ Xj ðxa Þ
paj ðzÞ K pbjK1 ðzÞ Z 0
(22a) (22b) (22c)
Then, the differential equations for the assembly are expressed as
Fig. 3. Foundation partially embedded in soil and soil medium divided into a number of secondary cells.
FX ðN 0I ; K 0I ; M 0I ÞX I ðxÞ Z jIa1 pIa1 ðxÞ K jIbJ pIbJ ðxÞ
(25a)
FX ðNj ; Kj ; Mj ÞXj ðxÞ Z jj ðza Þpaj ðxÞ K jj ðzb Þpbj ðxÞ (23a) j Z 1 wJ 0
0
I I I;II II II I;II Z dI;II 0=I fa paj ðzÞ K d0=II fb pbj ðzÞ j Z 1 wJ
0
FZ ðn ; k ; m ÞZðzÞ Z d0=1 f1 ðxa Þpa1 ðzÞ K d0=J fJ ðxb ÞpbJ ðzÞ
(23b)
J X
ðd0=j Þ2 ðnj ; kj ; mj Þ
(24)
(25b)
FX ðN 00II;III ; K 00II;III ; M 00II;III ÞX II;III ðxÞ III III II;III II II Z dII;III 0=III ja1 pa1 ðxÞ K d0=II jbJ pbJ ðxÞ
where d0/j is expressed by Eq. (18) and ðn 0 ; k 0 ; m 0 Þ Z
FZ ðnj00I;II ; kj00I;II ; mj00I;II ÞZjI;II ðzÞ
(25c)
III III III III III III III FZ ðnIII j ; kj ; mj ÞZj ðzÞ Z fa paj ðzÞ K fb pbj ðzÞ j
jZ1
Z 1 wJ III
3. Application to shallow foundation problems 3.1. Differential equations for assembly of secondary cells A partially embedded rigid foundation is considered in a layered inhomogeneous soil medium underlain by bedrock as shown in Fig. 3a. Axisymmetry is assumed with respect to the vertical line at the center of the foundation and thus only the shaded portion is considered for formulation. The soil domain is divided into three rectangular areas (secondary cells) as shown in Fig. 3b, in which each layer within the secondary cell forms a fundamental cell. Fundamental cells are connected vertically in secondary cells. In each secondary cell as an assembly of fundamental cells, it is set that Zj(za)ZZjK1(zb) and d0/1Z1. Cells I and II are mutually connected horizontally and Cells II and III are vertically. Thus, from Eqs. (20a) and (20b) and Eqs. (23a) and (23b), the differential equations for the assembly of secondary cells can be written as
(25d)
where the quantities with superscripts I,II or II,III indicate the quantities common in Cells I and II or in Cells II and III; jZlayer number; JI,II and JIIIZJ for Cells I and II and for Cell III, respectively, ðfa;b ; ja;b ÞZ ðfðxa;b Þ; jðza;b ÞÞ and ðnj00I;II ; kj00I;II ; mj00I;II Þ 2 I I I I;II 2 II II II Z ðdI;II 0=I Þ ðnj ; kj ; mj Þ C ðd0=II Þ ðnj ; kj ; mj Þ
ðN 0I ; K 0I ; M 0I Þ Z
JI X
ðNjI ; KjI ; MjI Þ
(26a)
(26b)
jZ1
ðN 00II;III ; K 00II;III ; M 00II;III Þ 2 0III 0III 0III II;III 2 Z ðdII;III 0=III Þ ðN ; K ; M Þ C ðd0=II Þ
!ðN 0II ; K 0II ; M 0II Þ I;II ðZjI ; ZjII Þ Z ZjI;II ðdI;II 0=I ; d0=II Þ
(26c) (26d)
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688 II;III ðX II ; X III Þ Z X II;III ðdII;III 0=II ; d0=III Þ
(26e)
with X I ðxb Þ I;II ; d Þ Z 1; ðdI;II 0=I 0=II X II ðxa Þ II;III ðdII;III 0=II ; d0=III Þ
Z
ZJIII ðzb Þ Z1II ðza Þ
;1
(27a) (27b)
3.2. Implementation of boundary conditions and solutions It is assumed that the rigid foundation undergoes lateral and rotational motions at the point O, which is located on the vertical line at the center of the foundation (Fig. 3). The origin of the global x–z coordinates is set at O. The origin of the local x–z coordinates is set at the upper left corner of each fundamental cell. The rigid foundation displacement is decomposed into the two displacement modes as shown in Fig. 4, in which the first and second modes are associated, respectively, with the lateral and vertical soil displacements. For the first mode, the displacement compatibility between the soil and foundation can be satisfied conveniently by setting ðX I ðxÞ; Z1I ðza ÞÞ Z ðUx K [z q; 1Þ along the base
(28a)
ðX III ðxa Þ; ZjIII ðzÞÞ Z ðl; Ux K ðzj C zÞqÞ along the side
(28b)
where lzZthe length from O to the foundation base; zjZthe location of the origin of the local z coordinate in the jth layer in Cell III; and Ux and qZlateral translational displacement Ux
O
• •O
d
(a) Mode 1 displacement
•
O,O d
683
and rotation of the foundation at O in the frequency domain, respectively. Similarly, for the second mode, it can be satisfied conveniently by setting ðX I ðxÞ; Z1I ðza ÞÞ Z ðxq; 1Þ along the base
(29a)
ðX III ðxa Þ; ZjIII ðzÞÞ Z ð1; lqÞ along the side
(29b)
where lZhalf-width of the foundation. Other conditions are: (1) the outer lateral ends (xb) are located at infinity and the responses are zero at xb in Cells II and III; (2) the top surface is stress free in Cell III; (3) the soil at the bedrock is fixed in Cells I and II; (4) stress and displacement along xZ0 (xa) are zero in Cell I for the first and second mode displacements, respectively; and (5) the compatibility conditions are satisfied at the layer–layer interface. These boundary conditions (1)–(4) are stated, respectively, as: X II ðxb Þ Z X III ðxb Þ Z 0
ðor fIIb Z fIII b Z 0Þ
pIII a1 ðxÞ Z 0
(30b)
ZJI ðzb Þ Z ZJII ðzb Þ Z 0 pIaj ðzÞ Z 0 or
(30a)
ðor jIbJ Z jIIbJ Z 0Þ
X I ðxa Þ ðor fIa Þ Z 0
(30c) (30d)
With all these conditions and the conditions stated by Eqs. (28a) and (28b) and Eqs. (29a) and (29b), Eqs. (25c) and (25b) become, respectively FX ðN 00II;III ; K 00II;III ; M 00II;III ÞX II;III ðxÞ Z 0
(31a)
FZ ðnj00I;II ; kj00I;II ; mj00I;II ÞZjI;II ðzÞ Z 0
(31b)
j Z 1 wJ I;II
and Eqs. (25a) and (25d) become, respectively ðK 0I K u2 M 0I ÞX I ðxÞ Z pIa1 ðxÞ
(32a)
III III III ðkjIII K u2 mIII j ÞZj ðzÞ Z paj ðzÞ j Z 1 wJ
(32b)
pIa1 ðxÞ
pIII aj ðzÞ
and are pressures acting along It is noted that the base and side of the foundation, respectively. The forms of functions, XI(x) and ZjIII ðzÞ, are given by Eqs. (28a) and (28b) for the first mode and by Eqs. (29a) and (29b) for the second mode. Those of (XII(x), XIII(x)) and ðZjI ðzÞ; ZjII ðzÞÞ are defined by solving, respectively, Eq. (31a) for XII,III(x) and Eq. (31b) for ZjI;II ðzÞ. The expressions of XII,III(x) and ZjI;II ðzÞ are given in Appendix A, in which ZjI;II ðzÞ is given only for a homogeneous individual layer for demonstration purpose. The function ZjI;II ðzÞ for a heterogeneous individual layer can be found in Ref. [16]. 3.3. Soil stiffness matrix and equation of motions of shallow foundation
(b) Mode 2 displacement Fig. 4. Soil displacement decomposed into two mode displacements.
Eqs. (32a) and (32b) indicate that the soil along the foundation faces can be replaced by appropriate Winklertype models defined by Eq. (32a) at the base and by
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
Eq. (32b) at the sides. The displacement of the foundation corner produces a concentrated soil force at the corner due to the reaction from Cell II, as indicated by Eq. (13) at the corner. Therefore, with Eqs. (28a) and (28b) or Eqs. (29a) and (29b), the forces at O associated with the first mode displacement are written for those due to the displacements at the foundation base (Cell I), side (Cell III) and corner (Cell II) as, respectively ( ) ( ) ð [ 1 Px dx Z pðxÞ 0 M K[z ( ) Ux from Cell I (33a) Z ½A q (
Px
) Z
M
J III ð hj X 0
jZ1
( Z ½B (
Px
) ZP
M
( pj ðzÞ
Ux
Kðzj C zÞ
dz
) from Cell III
q
( 1
)
1
(
) Z ½C
Klz
Ux q
(33b)
) from Cell II
(33c)
where Px and MZfrequency-domain lateral force and moment at O, respectively; Ux and qZfrequency-domain lateral displacement and rotation at O, respectively, and " # 1 K[z 0I 2 0I ½A Z [ðK K u M Þ (34a) K[z [2z ½B Z
J III ð hj X jZ1
0
ðkjIII ðzÞ "
Ku
2
mIII j Þ
1
Kðzj C zÞ
Kðzj C zÞ
ðzj C zÞ2
# dz
" 1 K[ # z dZ1II ðzÞ II ½C Z K n1 ðzÞ 2 dz zZ0 K[z [z
(34b)
(34c)
with [Zhalf width of the foundation and hjZthickness of the jth layer in secondary cell. Similarly, the forces at O associated with the second mode displacement are expressed, respectively, as ð[ M Z pðxÞx dx Z Dq from Cell I (35a) 0
ð hj J III X MZ [ pj ðzÞdz Z Eq jZ1
from Cell III
from Cell II
D Z ðK 0I K u2 M 0I Þ
EZ
J III X jZ1
[2
ð hj 0
ð[ 0
x2 dx Z ðK 0I K u2 M 0I Þ
ðkjIII ðzÞ K u2 mIII j ðzÞÞdz
dZ II ðzÞ F Z K[2 nII1 ðzÞ 1 dz zZ0
[3 3
(36a)
(36b)
(36c)
The soil stiffness matrix for a rigid foundation is obtained at O by superimposing the above forces and taking into account the left-hand side of axisymmetry also. Then, the equation of motions of the foundation at O is expressed in the frequency domain as ( ) ( ) Ux Px 2 (37) Z ð½K K u ½MÞ M q where [M]Zmass matrix of the foundation; and [K]Zsoil stiffness matrix expressed by " " ## 0 0 ½K Z 2 ½A C ½B C ½C C (38) 0 D CE CF With Eq. (37), the foundation responses at O can be computed for forces or displacements applied at O of the foundation. The parameters (n, k, m) and (N, K, M) for all secondary cells can be expressed explicitly in closed form after evaluating the integrations in Eqs. (11) and (14), respectively. With these explicit expressions, the above integrations in [K] result in simple closed form expressions also. All of the above formulations are general and applicable for any layered-soil profiles made of either homogeneous or heterogeneous layers. 3.4. Computational procedure I;II I II It is set that dI;II 0=I Z d0=II Z 1 (or Zj ðzÞZ Zj ðzÞ) in connecting Cells I and II herein. This results in Z1II ðza ÞZ Z1I ðza ÞZ 1 according to Eqs. (28a) and (29a). In view of ZJIII ðzb Þ given by Eqs. (28b) and (29b), dII;III 0=II is then automatically defined as dII;III 0=II Z Ux K [z q for the first mode displacement and dII;III 0=II Z [q for the second mode displacement. Eq. (31a) for X(x) and Eq. (31b) for Z(z) are mutually coupled since the parameters (N, K, M) in Eq. (31a) and (n, k, m) in Eq. (31b) are governed, respectively, by the functions Z(z) and X(x). Iterative procedure is adopted to compute the foundation responses. The computational procedure is explained below.
(35b)
0
M Z P[ Z Fq
where
(35c)
(1) Divide the soil domain into three rectangular secondary cells as shown in Fig. 3. Number these cells located at the base, corner and side of the foundation as I, II,
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
(3)
(4) (5)
(6) 7
(8)
4. Computed results The point O is set at the center of the foundation base and massless foundations are considered in all cases computed. First, a homogeneous ground underlain by bedrock is considered to demonstrate and examine the approach. In this case, Cells I, II, and III are uniform without layering. The lateral responses of massless rigid foundations are computed with j(z)ZZ(z) or Z*(z). The function XII,III(x) is expressed as XII,III(x)ZeKbx (Appendix A). Source problems for an infinite lateral extent of ground require both the real and imaginary parts of b positive for the positive x direction. The values of b at various frequencies are shown in Fig. 5, in which S1 and S2 indicate, respectively, the locations of the first and second natural frequencies of the ground in shear. As shown in the figure, they are always positive for j(z)Z Z*(z). For j(z)ZZ(z), however, its imaginary part becomes negative as frequency increases beyond the frequency a little higher than the fundamental natural frequency of the ground and the computation is terminated when this occurs. The computed soil flexibilities for the above cases are
H/ = 2 = 0.3
d/ d/ d/ d/
1.6
Imaginary
1.2
=0 =2/3 =0 =2/3
X, Z* X, Z
S2
S1 0.8
0.4
0 0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Real Fig. 5. b Values at various frequencies.
shown at various frequencies in Fig. 6 to compare them with those computed by FEM [18]. The frequency parameter a0 in the figure is defined as a0Zu[/vs with vsZsoil shear wave velocity. The figure indicates that the present approximate approach can produce excellent results for j(z)ZZ*(z). Soil flexibilities were previously computed for rigid foundation on a layer overlain by infinitely deep stiff soil as shown in Fig. 7 [19]. In this case, the ground is inhomogeneous and Cells I and II are infinitely long in the z direction. The values computed by the present method and Gazetas [19] are shown in Fig. 8. Gazetas’s formulation is based on the far more elaborated approach, with which it is not possible to have the final expressions in explicit closed form. As seen, the values computed by the two approaches are reasonably close to each other. A foundation of [Z4 m is assumed to rest on the surface of heterogeneous soil underlain by rigid base at depth dZ 2[. The conditions considered for the soil are: vs(z)Z vs(0)(1C1.5z/[) with vsZshear wave velocity of soil (or G(z)ZG(0)(1C1.5z/[)2) and z is the depth from the ground surface; nZ0.25; and DZ0.05. The soil is divided equally into 8 or 16 homogeneous layers as shown in Fig. 9. It is also divided equally into eight heterogeneous layers in which G(z) varies linearly with z within a layer. G(z) in the latter d/ = 2/3 H/ = 2 = 0.3 Flexibility Fx(a0) / Fx(a0=0)
(2)
and III, respectively. Number the layers in each cell from the top layer (jZ1) through the bottom layer (jZ J) in order. Set the location O along the vertical line passing through the center of the foundation. Iteration cycle starts here through the step (8). If this is the first cycle of iteration: assume Z III j ðzÞ based on Z(z)ZUxKqz for the first mode or Z(z)Zq[ for the second mode; and assume Z I;II j ðzÞ based on linear variation with depth from 1 at the top surface to 0 at the bottom surface of the secondary cell for Cells I and II. I;II After the first cycle of iteration, use Z III j ðzÞ and Z j ðzÞ computed at the previous cycle as the assumed ones. Compute (Nj, Kj, Mj) for Cells I, II, and III with Zj(z) assumed at (2), and then (N 0 , K 0 , M 0 ) for Cells I, II, and III with (Nj, Kj, Mj). Compute (N 00 , K 00 , M 00 ) for the assembly of Cells II and III with (N 0 , K 0 , M 0 ). Define XII,III(x) with (N 00 , K 00 , M 00 ) computed at (3), and then XII(x) and XIII(x) with XII,III(x). Compute (nj, kj, mj) for Cells II and III with XII(x) and XIII(x) defined at (4). Compute (nj, kj, mj) for Cell I with X I ðxÞZ U2 K q[2 for the first mode or XI(x)Zqlx/l for the second mode. Compute ðnj00 ; kj00 ; mj00 Þ for the assembly of Cells I and II with (nj, kj, mj). 00 00 00 Define Z I;II j ðzÞ with ðnj ; kj ; mj Þ computed at (5), and I;II I II then Z j ðzÞ and Z j ðzÞ with Z j ðzÞ. Repeat (2)–(6) for the first mode and the second mode displacements. Then, construct [K] and compute Ux and q for applied load to define XI(x) and Z III j ðzÞ. Compare the assumed Zj(z) with the computed Zj(z) for all cells. If the two are close to each other to satisfy the tolerance criteria, the above computed Ux and q are the responses. Otherwise, repeat (2)–(8) for the next cycle.
685
1.6
real imag real imag real imag
1.2 0.8
X, Z* X,Z FEM
0.4
S1
P1
S2
0 0
0.5
1
1.5
2
Frequency a0 Fig. 6. Soil flexibilities for horizontal foundation motion. (Fx(a0)Z0.227 (model), 0.256 (FEM)).
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profiles. The computed soil stiffnesses for the foundation with or without embedment in this soil are shown in Fig. 12. It is seen that the embedment affects the soil stiffness on both its magnitude and frequency dependence, while the difference in variation pattern of G with depth affects primarily on its magnitude.
2
H
H =2
G1
5. Conclusions
G2
The differential equation cell method enables us to develop the expressions for dynamic responses of shallow foundations in simple closed form. Despite rather significant simplifications involved, it computes the soil stiffnesses for partially embedded rigid foundations close to those computed by far more rigorous methods. When the soil is underlain by rigid base, Z*(z) instead Z(z) should be used for the weight function, j(z), in source problems to ensure that soil responses diminish at infinity in x. The parameters (N 0 , K 0 , M 0 ) and (nj, kj, mj) are determined iteratively. The number of iteration did not exceed 10 to achieve reasonably accurate results for the cases computed herein. The soil pressure around the foundation corner increases as approaching to the corner due to singularity at this point. The present approach is capable of including such effects automatically through the concentrated forces at the corners and the pressures along the foundation faces, which are
Fig. 7. Rigid foundation in layered half-space.
distribution is nearly equal to the original G(z). Soil stiffnesses computed for these three cases are shown in Fig. 10 together with those computed by the far more elaborated method [20]. It is seen in the figure that the capability of handling heterogeneity in a subdivided layer increases accuracy. The foundation with or embedment (dZ0 or 0.5[) is considered in two heterogeneous soil profiles (Fig. 11), in which Profile A is defined as G(z)ZG(0)(1C6.45(z/[)0.5) and Profile B is the one considered in Fig. 9. Both G(0) and the average G over the depth are identical between the two G1/G2=0.06 G1/G2=0.25 G1/G2=0.06 G1/G2=0.25
Present method Gazetas 1980 0.6
0.8
real part
Flexibility GF
Flexibility G Fxx
1
0.6 0.4 0.2
real part
0.4
0.2
0 0
0 0
0.2
0.4
0.6
0.8
0.5
-0.2
1
1
1.5
2
Frequency a0
Frequency a0 1 0.6
Flexibility G F
Flexibility G Fxx
0.8
imag. part
0.6
0.4
imag. part
0.4
0.2
0.2 0
0 0
0.2
0.4
0.6
Frequency a0
(a) Horizontal response
0.8
1
0
0.5
1
1.5
Frequency a0
(b) Rocking response
Fig. 8. Responses of a massless foundation in layered half-space.
2
50
100
150
8 homogeneous layers 16 homogeneous layers G(z)=G0(1+1.5z/ )2
40 30
B
10 0 0
10
real part
2
3
4
5
6
Frequency, a 0 imag. part
d = 0.5 d=0
80
A B
60
A
40 20
B
0 0
1
5
2
3
4
5
6
Frequency, a 0
0 0
1
2
3
4
5
6
Frequency, a 0 30
Stiffness Kf /G
1
120
Stiffness K f/G
Stiffness Kf /G
15
A B
100
8 inhomogeneous layers 16 homogeneous layers 8 homogeneous layers Gazetas(1980)
real part
A
d = 0.5 d=0
20
-10
Fig. 9. Distribution of shear modulus with depth.
20
687
50
Shear modulus, G (kN/sq.m.)
0 -1 0 -2 -3 -4 -5 -6 -7 -8
Stiffness K f/G
Depth (m.)
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
8 inhomogeneous layers 16 homogeneous layers 8 homogeneous layers
25 20
imag. part
10 5 0 1
2
3
4
5
6
Frequency, a 0 Fig. 10. Dynamic soil stiffnesses computed for three different cases.
mutually coupled in the formulation. However, it does not provide the detailed distribution of soil pressure acting on the foundation but provides the distribution grossly only in a macroscopic manner. Appendix A. XII;IIIðxÞ and ZI;II j ðzÞ for secondary cells made of homogeneous layers A.1. Expressions of X II;III ðxÞ and ZjI;II ðzÞ
Depth (m.)
Shear modulus, G (kN/sq.m.) 50
(A1a)
XðxÞ Z CeKbx C Debx
(A1b)
100
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 00 K u2 M 00 bZ N 00
150
A G(z)=G0(1+6.45(z/ ))0.5 B G(z)=G0(1+1.5(z/ ))2
B A
Fig. 11. Distribution of shear modulus with depth in Profile A and Profile B.
(A2a)
(A2b)
aj and b are mutually coupled and determined conveniently by iteration [12,14–16]. A.2. Determination of Aj, Bj, C and D The conditions given by Eq. (30a) at xbZN result in CZ 1 and DZ0 for Cells II and III. Therefore, X(x) is in the form of XðxÞ Z eKbx
General solutions of Eqs. (31a) and (31b) are, respectively 0 -1 0 -2 -3 -4 -5 -6 -7 -8
Zj ðzÞ Z Aj sinh aj z C Bj cosh aj z
where Aj, Bj, C, and DZ unknown constants; and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kj00 K u2 mj00 aj Z nj00
15
0
Fig. 12. Dynamic soil stiffnesses for foundations in Profile A and Profile B.
(A3)
Using Eqs. (A1a) and (13) and omitting the common factor xX(x), the expressions for the displacement and stress in the layer j lead to the following expression ( ) Zj ðzÞ Knj Zj0 ðzÞ " #( ) cosh aj z sinh aj z Aj Z Bj Kaj cosh aj z Kaj sinh aj z
(A4)
where Z 0 ðzÞZ dZðzÞ=dz. The origin of z is taken at the upper end of the layer. Since ZJ(zb)Z0 (boundary condition Eq. (30c)), AJ and BJ are expressed from Eq. (A4) as
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ðAJ ; BJ Þ Z AJ ð1; Ktanh aJ hJ Þ
Rewriting (Aj, Bj) as Aj ; Bj Z Aj Aj0 ; Bj0
(A5)
(A6)
Eq. (A4) and the equilibrium and compatibility conditions at the interface of the jK1th and jth layers result in ( 0 ) ( 0) AjK1 Aj Z ½T (A7) jK1=j 0 BjK1 Bj0 where 1 ½TjK1=j Z 2 sinh ajK1 hjK1 K cosh2 ajK1 hjK1 " # KhjK1=j cosh ajK1 hjK1 sinh ajK1 hjK1 KhjK1=j sinh ajK1 hjK1 hjK1=j Z
aj njK1 ajK1 nj
(A8a)
cosh ajK1 hjK1 (A8b)
with hjZ thickness of the jth layer. Substituting ðAJ0 ; BJ0 Þ defined by Eq. (A5) into Eq. (A7), ðAj0 ; Bj0 Þ for jZ1wJ are computed by Eq. (A7) successively in reverse order. Once ðA10 ; B10 Þ is computed, AJ is computed with B10 from the condition Z1(0)Z1 or Z1 ð0Þ Z AJ B10 Z 1
(A9)
With those computed AJ and ðAj0 ; Bj0 Þ, (Aj, Bj) is computed from Eq. (A6).
References [1] Beredugo YO, Novak M. A coupled horizontal and rocking vibration of embedded footings. Can Geotech J 1972;9:477–97. [2] Novak M, Nogami T, Aboul-Ella F. Dynamic soil reactions for plane strain case. J Eng Mech ASCE 1978;104(4):953–9. [3] Nogami T, Chen HS. Dynamic soil stiffnesses at the side of embedded structures with rectangular base. J Eng Mech ASCE 2003;129(8): 963–73. [4] Nogami T, Konagai K. Time-domain flexural response of dynamically loaded single piles. J Eng Mech 1988;114(9):1512–25.
[5] Nogami T, Konagai K, Otani J. Time domain axial response of dynamically loaded single pile. J Eng Mech 1991;112(2):1241–52. [6] Nogami T, Konagai K, Otani J. Nonlinear time domain numerical model for pile group under transient dynamic forces. In: Proceedings of second international conference on recent advances in geotechnical engineering. Soil Dynamics, St Louis, MO. Paper no. 5.51 p. 881–8. [7] Nogami T, Otani J, Konagai K, Chen HC. Nonlinear soil–pile interaction for dynamic lateral motion. J Geotech Eng 1992;118 (1):89–105. [8] Nogami T, Lam Y. A two-parameter layer model for analysis of slab on elastic foundation. J Eng Mech, ASCE 1987;113(9):1041–55. [9] Nogami T, Leung MB. A simplified mechanical subgrade model for dynamic response analysis of shallow foundations. Int J Earthq Eng Struct Dyn 1990;19:1041–55. [10] Nogami T. Simplified subgrade model for the three-dimensional soil– foundation interaction analysis. J Soil Dyn Earthq Eng 1996;15(7): 419–29. [11] Nogami T, Mikami A, Konagai K. A simplified approach for dynamic soil–structure interaction analysis of rigid foundation. In: Geotechnical special publication on observation and modeling in numerical analysis and model tests in dynamic soil–structure interaction problems 64, ASCE p. 26–44. [12] Nogami T, Chen HS. Boundary differential equation method: simplified dynamic soil stiffness for embedded rigid foundations. J Soil Dyn Earthq Eng 2002;22:323–34. [13] Nogami T, Zhu JX, Ito T. First and second order dynamic subgrade models for soil–pile interaction analysis. In: Geotechnical special publication on piles under dynamic loads, no. 34, ASCE p. 187–206. [14] Nogami T. Dynamic soil stiffnesses for partially embedded foundations: formulation with simplified boundary differential equations. Proceedings of international symposium on numerical methods in geomechanics, Roma, Italy 2002. [15] Nogami T. A method for dynamic response analysis of foundations partially embedded in layered inhomogeneous soils. Proceedings of the fifth European conference on numerical methods in geotechnical engineering, Paris, France, September 2002. [16] Nogami T, Mahbub Md AAl. Lateral dynamic soil stiffness for partially embedded foundations in heterogeneous soils. Proceedings of Pacific conference on earthquake engineering, New Zealand, February 2003. [17] Nogami T. Two-parameter layer model for analysis of slab on elastic foundation. J Eng Mech Div ASCE 1987;113(EM9):1279–91. [18] Liang CV. Response of structure in layered soils. PhD dissertation, MIT; 1987. [19] Gazetas G. Static and dynamic displacements of foundations on heterogeneous multilayered soils. Geotechnique 1980;30(2):159–77. [20] Gazetas G. Analysis of machine foundation vibrations: state of the art. J Soil Syn Earthq Eng 1983;2(1):1–42.
A new method for formulation of dynamic responses of shallow foundations in simple general form T. Nogami*, A.Al Mahbub, S.H. Chen Department of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore, Singapore 117576 Accepted 1 November 2004
Abstract A single uniform rectangular area, either homogeneous or heterogeneous, is considered in a soil medium (fundamental cell). Two governing ordinary differential equations in special form are developed for the fundamental cell. The ground supporting a partially embedded foundation is divided into a number of coarse rectangular areas (secondary cells). Each secondary cell is treated as either a single fundamental cell for homogeneous ground or a stack of fundamental cells for inhomogeneous ground. Differential equations for the assembly of secondary cells are formed with those for the fundamental cells. These equations lead to the soil responses in each cell expressed in simple closed form. They also lead to the convenient treatment of soil with appropriate Winkler-type models along the foundation faces and concentrated forces acting at the foundation corners. With them, the foundation responses are finally expressed in simple closed form. The approach is demonstrated for various cases and confirmed to produce the results reasonable enough for civil engineering use. q 2005 Elsevier Ltd. All rights reserved. Keywords: Foundation; Inhomogeneous soil; Vibrations; Computational method
1. Introduction Simple expressions for the dynamic responses of foundations were presented previously. Novak and his colleagues [1,2] treated the soil at the side of foundation as a stack of mutually uncoupled thin layers. The stiffnesses of an individual layer at the foundation were formulated from vibrations of a horizontal, massless, rigid, circular slice of the foundation contained in a horizontal layer of unit thickness [2]. Those for foundations with rectangular base area were also formulated recently [3]. For the time-domain analysis, frequency-dependent soil stiffnesses at the side formulated in this manner were idealized as frequencyindependent spring–mass–dashpot systems for circular foundations [4,5]. Non-linear mechanism was introduced further in the side soil stiffnesses [6,7]. Nogami and his colleagues used another approach for shallow foundations [8–12] and deep foundations [13]. In * Corresponding author. Address: 13250 Kibbings Road, San Diego, CA 92130, USA. Tel.: C1 858 755 3952. E-mail address: [email protected] (T. Nogami).
0267-7261/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2004.11.013
this approach, the medium was idealized as a system of springs and closely spaced one-dimensional columns with distributed mass, in which the springs interconnect the two adjacent columns. Thus, these springs produce coupling among the columns, enabling the model to closely simulate the behaviour of a continuous medium. Without springs, the model is similar to Novak’s model. The above second simplified approach results in two governing ordinary differential equations in convenient form, to describe the dynamic behavior of a rectangular area in the plane-strain medium. Using these two equations as a base, a special method was used to develop simple closed form expressions for the foundation responses [14–16]. The method was called later the differential equation cell method (DECM). This paper refines DECM and demonstrates its application to shallow foundations in general soil profile in a systematic manner. The major emphasis of the present paper is placed on simplicity of the final formulations and generality to use them for any inhomogeneous soil profiles. Linear visco-elastic soils are assumed as commonly assumed in simple expressions proposed by many people.
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2. Differential equations for assembly of cells 2.1. Fundamental differential equations for a single cell A soil medium is assumed to be a horizontally layered visco-elastic medium, in which an individual layer is assumed to be either homogeneous or heterogeneous. Its material properties are defined by complex Lame`’s constants (l* and G*, in which G* is the complex shear modulus) and unit mass (r). With the Cartesian coordinates x and z taken, respectively, in the horizontal and vertical directions, the complex moduli dependent on z are defined herein as ðG ðzÞ; l ðzÞÞ Z ð1 C 2DiÞðGðzÞ; lðzÞÞ (1) pffiffiffiffiffiffi where iZ K1; and DZfrequency-independent damping factor of the medium. Linear variations of the moduli with z are stated as ðGðzÞ; lðzÞÞ Z ð1 C Cðz K z0 ÞÞðG0 ; l0 Þ
(2)
where z0Zreference location in z0 and l0ZG(z) and l(z) at z0, respectively; CZconstant (0 for homogeneous material); and denoting n as Poisson’s ratio lðzÞ l 2n Z 0 Z GðzÞ G0 1 K 2n
(3)
It is noted that Eq. (5) is also the equation of motion of a column in a system of closely spaced columns that are interconnected by lateral springs distributed along the side [9,17]: where kc and ks correspond, respectively, to the complex stiffnesses of the column and spring per length of the column; and mc corresponds to the mass per length of the column. The non-dimensional parameters, kc and ks , are dependent rather uniquely on n and very little on soil profile [9,17]. A rectangular cell is considered in a continuous medium, in which the sides of the cell are parallel to the x–z coordinates (Fig. 1). The displacement of the medium in the cell is assumed to be expressed in the form of uðx; zÞ Z XðxÞZðzÞ
(7)
Substituting Eq. (7) into Eq. (5) and denoting f(x) as a weight function, the Galerkin method for weighted residual over x in the cell yields ð xb 2 d XðxÞ ks ðzÞ fðxÞdx ZðzÞ 2 xa dx ð xb d dZðzÞ kc ðzÞ C XðxÞfðxÞdx dz dz xa ðx b C mc XðxÞfðxÞdx ZðzÞ Z 0
(8)
Equations of motion of the medium in the frequency xa domain are then stated for the plane-strain conditions as 2 3 2 v v2 v v 9 v v ( ) 8 G C l < Kru2 ux ðx; zÞ = 6 ðl C 2G Þ vx2 C vz G vz 7 ux ðx; zÞ vz vx vxvz 6 7 Z 6 7 2 : Kru2 u ðx; zÞ ; 4 v2 5 u ðx; zÞ v v v v v z z l ðl C C C 2G Þ G G vz vz vxvz vz vx2 vz where ux and uzZfrequency-domain displacements in the horizontal and vertical directions, respectively; and uZ circular frequency. Off-diagonal terms couple the lateral and vertical soil motions. It is known that ignorance in the minor component of these motions affects the computed foundation responses very little. Ignoring the off-diagonal terms in Eq. (4) and slightly modifying, the following equation is obtained v vuðx; zÞ v vuðx; zÞ k ðzÞ k ðzÞ C Cmc u2 uðx; zÞ Z 0 vx s vx vz c vz (5) where uZeither ux or uz; and according to Refs. [9,17] ðks ðzÞ; kc ðzÞ; mc Þ Z ðAs ðzÞks ; Ac ðzÞ; kcr; m c Þ
(6a)
with ks , kc and m c Znon-dimensional factors [9], and ( * ðl ðzÞ C 2G* ðzÞ; G* ðzÞÞ for ux ðAs ðzÞ; Ac ðzÞÞ Z (6b) ðG* ðzÞ; l* ðzÞ C 2G* ðzÞÞ for uz
(4)
pa a, a
pa( )
b, a
pb
ux a, b
h
b, b
p
b
G( ), ( ),
D
pa a, a
b, a
uz
pa( ) a, b
pb
h
b, b
pb G( ), ( ),
D
homogeneous or heterogeneous cell Fig. 1. Fundamental cell subjected to tractions.
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
Integrating the first term by parts, Eq. (8) results in d dZðzÞ nðzÞ K C ðkðzÞ K u2 mÞZðzÞ dz dz
FZ ðn; k; mÞZðzÞ Z fðxa Þpa ðzÞ K fðxb Þpb ðzÞ
u1, (n1, k1, m1), (N1, K1, M1) u2, (n2, k2, m2), (N2, K2, M2)
1 2
hj uj, (nj, kj, mj), (Nj, Kj, Mj)
j
Z fðxa Þpa ðzÞ K fðxb Þpb ðzÞ or in compact form
J
(9)
where xa and xbZleft and right ends of the cell in the x coordinate, respectively; pa,b(z)Zthe traction acting along z at (xa,b,z), expressed as dXðxÞ (10) pa;b ðzÞ Z K ks ðzÞZðzÞ dx xa;b
681
uJ, (nJ, kJ, mJ), (NJ, KJ, MJ)
(a) Assembly of vertically connected cells hj 1
u1, (n1, k1, m1), (N1, K1, M1)
2
L
u2, (n2, k2, m2), (N2, K2, M2)
j
L
J
uj, (nj, kj, mj), (Nj, Kj, Mj)
(b) Assembly of horizontally connected cells
and Fig. 2. Assembly of vertically or horizontally connected cells.
ðnðzÞ; kðzÞ; mÞ ð xb dXðxÞ dfðxÞ Z ; mc XðxÞfðxÞ dx kc ðzÞXðxÞfðxÞ; ks ðzÞ dx dx xa (11) Similarly, substituting Eq. (7) into Eq. (5) and using j(z) as a weight function, the Galerkin method for weighted residual over z in the cell yields d2 XðxÞ KNðzÞ C ðKðzÞ K u2 MÞXðxÞ dx2
Xj ðxÞ Z djK1=j XjK1 ðxÞ
djK1=j Z
ZjK1 ðzb Þ Zj ðza Þ
(15a)
(15b)
where djK1 / jZfactor; d0 / 1Zreference factor (assumed to be d0 / 1Z1); and X0(x) and Z0(z)Zreference functions. In addition, the force equilibrium condition at this interface leads to
Z jðza Þpa ðxÞ K jðzb Þpb ðxÞ or in compact form FX ðN; K; MÞXðxÞ Z jðza Þpa ðxÞ K jðzb Þpb ðxÞ
the bottom, as shown in Fig. 2a. Between the jK1th and jth fundamental cells, the compatibility conditions along x lead to
(12)
where za and zbZupper and lower ends of the cell in the z coordinate, respectively; pa,b(x)Ztraction acting along x at (x,za,b), expressed as dZðzÞ pa;b ðxÞ Z K kc ðzÞXðxÞ (13) dz za;b
paj ðxÞ K pbjK1 ðxÞ Z 0
Successive applications of Eqs. (15a) from jZ1 through j in order result in Xj ðxÞ Z d0=j X0 ðxÞ
ð zb dZðzÞ djðzÞ Z ;mc ZðzÞjðzÞ dz ks ðzÞZðzÞjðzÞ;kc ðzÞ dz dz za (14) Eqs. (9) and (12) are the fundamental differential equations for a single cell (fundamental cell). The weight functions are selected as (f(x),j(z))Z(X(x),Z*(z)), in which Z*(z) is the conjugator of Z(z) [12]. For the soil profiles considered herein, integrations yield closed form expressions for (n, k, m) and (N, K, M). 2.2. Differential equations for assembly of cells J cells of equal length in x are assumed to be connected vertically in series and are numbered in order from the top to
(17a)
where
and ðN;K; MÞ
(16)
d0=j Z
j Y
djK1=j
(18)
jZ1
Multiplying Eq. (12) by d0/j and replacing Xj(x) with d0/jX0(x), Eq. (12) for the jth cell can be rewritten as FX ðd20=j Nj ; d20=j Kj ; d20=j Mj ÞX0 ðxÞ Z d0=j jj ðza Þpaj ðxÞ K d0=j jj ðzb Þpbj ðxÞ
(19)
Since d0=jK1 ZjK1 ðzb ÞZ d0=j Zj ðza Þ, summing up Eq. (19) from jZ1 through J and denoting X0(x)ZX(x) result in a single differential equation for X(x). With this and Eq. (9), the differential equations for the assembly are FX ðN 0 ; K 0 ; M 0 ÞXðxÞ Z d0=1 j1 ðza Þpa1 ðxÞ K d0=J jJ ðzb ÞpbJ ðxÞ
(20a)
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FZ ðnj ; kj ; mj ÞZj ðzÞ Z fðxa Þpaj ðzÞ K fðxb Þpbj ðzÞ (20b) j Z 1 wJ where
J X N 0; K 0; M 0 Z ðd0=j Þ2 ðNj ; Kj ; Mj Þ
(21)
jZ1
Next, J rectangular cells of equal length in z are assumed to be connected horizontally in series and are numbered from the left to the right in order as shown in Fig. 2b. Similarly, the following relations exist in this connection: Zj ðzÞ Z djK1=j ZjK1 ðzÞ Z d0=j Z0 ðzÞ djK1=j Z
XjK1 ðxb Þ Xj ðxa Þ
paj ðzÞ K pbjK1 ðzÞ Z 0
(22a) (22b) (22c)
Then, the differential equations for the assembly are expressed as
Fig. 3. Foundation partially embedded in soil and soil medium divided into a number of secondary cells.
FX ðN 0I ; K 0I ; M 0I ÞX I ðxÞ Z jIa1 pIa1 ðxÞ K jIbJ pIbJ ðxÞ
(25a)
FX ðNj ; Kj ; Mj ÞXj ðxÞ Z jj ðza Þpaj ðxÞ K jj ðzb Þpbj ðxÞ (23a) j Z 1 wJ 0
0
I I I;II II II I;II Z dI;II 0=I fa paj ðzÞ K d0=II fb pbj ðzÞ j Z 1 wJ
0
FZ ðn ; k ; m ÞZðzÞ Z d0=1 f1 ðxa Þpa1 ðzÞ K d0=J fJ ðxb ÞpbJ ðzÞ
(23b)
J X
ðd0=j Þ2 ðnj ; kj ; mj Þ
(24)
(25b)
FX ðN 00II;III ; K 00II;III ; M 00II;III ÞX II;III ðxÞ III III II;III II II Z dII;III 0=III ja1 pa1 ðxÞ K d0=II jbJ pbJ ðxÞ
where d0/j is expressed by Eq. (18) and ðn 0 ; k 0 ; m 0 Þ Z
FZ ðnj00I;II ; kj00I;II ; mj00I;II ÞZjI;II ðzÞ
(25c)
III III III III III III III FZ ðnIII j ; kj ; mj ÞZj ðzÞ Z fa paj ðzÞ K fb pbj ðzÞ j
jZ1
Z 1 wJ III
3. Application to shallow foundation problems 3.1. Differential equations for assembly of secondary cells A partially embedded rigid foundation is considered in a layered inhomogeneous soil medium underlain by bedrock as shown in Fig. 3a. Axisymmetry is assumed with respect to the vertical line at the center of the foundation and thus only the shaded portion is considered for formulation. The soil domain is divided into three rectangular areas (secondary cells) as shown in Fig. 3b, in which each layer within the secondary cell forms a fundamental cell. Fundamental cells are connected vertically in secondary cells. In each secondary cell as an assembly of fundamental cells, it is set that Zj(za)ZZjK1(zb) and d0/1Z1. Cells I and II are mutually connected horizontally and Cells II and III are vertically. Thus, from Eqs. (20a) and (20b) and Eqs. (23a) and (23b), the differential equations for the assembly of secondary cells can be written as
(25d)
where the quantities with superscripts I,II or II,III indicate the quantities common in Cells I and II or in Cells II and III; jZlayer number; JI,II and JIIIZJ for Cells I and II and for Cell III, respectively, ðfa;b ; ja;b ÞZ ðfðxa;b Þ; jðza;b ÞÞ and ðnj00I;II ; kj00I;II ; mj00I;II Þ 2 I I I I;II 2 II II II Z ðdI;II 0=I Þ ðnj ; kj ; mj Þ C ðd0=II Þ ðnj ; kj ; mj Þ
ðN 0I ; K 0I ; M 0I Þ Z
JI X
ðNjI ; KjI ; MjI Þ
(26a)
(26b)
jZ1
ðN 00II;III ; K 00II;III ; M 00II;III Þ 2 0III 0III 0III II;III 2 Z ðdII;III 0=III Þ ðN ; K ; M Þ C ðd0=II Þ
!ðN 0II ; K 0II ; M 0II Þ I;II ðZjI ; ZjII Þ Z ZjI;II ðdI;II 0=I ; d0=II Þ
(26c) (26d)
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688 II;III ðX II ; X III Þ Z X II;III ðdII;III 0=II ; d0=III Þ
(26e)
with X I ðxb Þ I;II ; d Þ Z 1; ðdI;II 0=I 0=II X II ðxa Þ II;III ðdII;III 0=II ; d0=III Þ
Z
ZJIII ðzb Þ Z1II ðza Þ
;1
(27a) (27b)
3.2. Implementation of boundary conditions and solutions It is assumed that the rigid foundation undergoes lateral and rotational motions at the point O, which is located on the vertical line at the center of the foundation (Fig. 3). The origin of the global x–z coordinates is set at O. The origin of the local x–z coordinates is set at the upper left corner of each fundamental cell. The rigid foundation displacement is decomposed into the two displacement modes as shown in Fig. 4, in which the first and second modes are associated, respectively, with the lateral and vertical soil displacements. For the first mode, the displacement compatibility between the soil and foundation can be satisfied conveniently by setting ðX I ðxÞ; Z1I ðza ÞÞ Z ðUx K [z q; 1Þ along the base
(28a)
ðX III ðxa Þ; ZjIII ðzÞÞ Z ðl; Ux K ðzj C zÞqÞ along the side
(28b)
where lzZthe length from O to the foundation base; zjZthe location of the origin of the local z coordinate in the jth layer in Cell III; and Ux and qZlateral translational displacement Ux
O
• •O
d
(a) Mode 1 displacement
•
O,O d
683
and rotation of the foundation at O in the frequency domain, respectively. Similarly, for the second mode, it can be satisfied conveniently by setting ðX I ðxÞ; Z1I ðza ÞÞ Z ðxq; 1Þ along the base
(29a)
ðX III ðxa Þ; ZjIII ðzÞÞ Z ð1; lqÞ along the side
(29b)
where lZhalf-width of the foundation. Other conditions are: (1) the outer lateral ends (xb) are located at infinity and the responses are zero at xb in Cells II and III; (2) the top surface is stress free in Cell III; (3) the soil at the bedrock is fixed in Cells I and II; (4) stress and displacement along xZ0 (xa) are zero in Cell I for the first and second mode displacements, respectively; and (5) the compatibility conditions are satisfied at the layer–layer interface. These boundary conditions (1)–(4) are stated, respectively, as: X II ðxb Þ Z X III ðxb Þ Z 0
ðor fIIb Z fIII b Z 0Þ
pIII a1 ðxÞ Z 0
(30b)
ZJI ðzb Þ Z ZJII ðzb Þ Z 0 pIaj ðzÞ Z 0 or
(30a)
ðor jIbJ Z jIIbJ Z 0Þ
X I ðxa Þ ðor fIa Þ Z 0
(30c) (30d)
With all these conditions and the conditions stated by Eqs. (28a) and (28b) and Eqs. (29a) and (29b), Eqs. (25c) and (25b) become, respectively FX ðN 00II;III ; K 00II;III ; M 00II;III ÞX II;III ðxÞ Z 0
(31a)
FZ ðnj00I;II ; kj00I;II ; mj00I;II ÞZjI;II ðzÞ Z 0
(31b)
j Z 1 wJ I;II
and Eqs. (25a) and (25d) become, respectively ðK 0I K u2 M 0I ÞX I ðxÞ Z pIa1 ðxÞ
(32a)
III III III ðkjIII K u2 mIII j ÞZj ðzÞ Z paj ðzÞ j Z 1 wJ
(32b)
pIa1 ðxÞ
pIII aj ðzÞ
and are pressures acting along It is noted that the base and side of the foundation, respectively. The forms of functions, XI(x) and ZjIII ðzÞ, are given by Eqs. (28a) and (28b) for the first mode and by Eqs. (29a) and (29b) for the second mode. Those of (XII(x), XIII(x)) and ðZjI ðzÞ; ZjII ðzÞÞ are defined by solving, respectively, Eq. (31a) for XII,III(x) and Eq. (31b) for ZjI;II ðzÞ. The expressions of XII,III(x) and ZjI;II ðzÞ are given in Appendix A, in which ZjI;II ðzÞ is given only for a homogeneous individual layer for demonstration purpose. The function ZjI;II ðzÞ for a heterogeneous individual layer can be found in Ref. [16]. 3.3. Soil stiffness matrix and equation of motions of shallow foundation
(b) Mode 2 displacement Fig. 4. Soil displacement decomposed into two mode displacements.
Eqs. (32a) and (32b) indicate that the soil along the foundation faces can be replaced by appropriate Winklertype models defined by Eq. (32a) at the base and by
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
Eq. (32b) at the sides. The displacement of the foundation corner produces a concentrated soil force at the corner due to the reaction from Cell II, as indicated by Eq. (13) at the corner. Therefore, with Eqs. (28a) and (28b) or Eqs. (29a) and (29b), the forces at O associated with the first mode displacement are written for those due to the displacements at the foundation base (Cell I), side (Cell III) and corner (Cell II) as, respectively ( ) ( ) ð [ 1 Px dx Z pðxÞ 0 M K[z ( ) Ux from Cell I (33a) Z ½A q (
Px
) Z
M
J III ð hj X 0
jZ1
( Z ½B (
Px
) ZP
M
( pj ðzÞ
Ux
Kðzj C zÞ
dz
) from Cell III
q
( 1
)
1
(
) Z ½C
Klz
Ux q
(33b)
) from Cell II
(33c)
where Px and MZfrequency-domain lateral force and moment at O, respectively; Ux and qZfrequency-domain lateral displacement and rotation at O, respectively, and " # 1 K[z 0I 2 0I ½A Z [ðK K u M Þ (34a) K[z [2z ½B Z
J III ð hj X jZ1
0
ðkjIII ðzÞ "
Ku
2
mIII j Þ
1
Kðzj C zÞ
Kðzj C zÞ
ðzj C zÞ2
# dz
" 1 K[ # z dZ1II ðzÞ II ½C Z K n1 ðzÞ 2 dz zZ0 K[z [z
(34b)
(34c)
with [Zhalf width of the foundation and hjZthickness of the jth layer in secondary cell. Similarly, the forces at O associated with the second mode displacement are expressed, respectively, as ð[ M Z pðxÞx dx Z Dq from Cell I (35a) 0
ð hj J III X MZ [ pj ðzÞdz Z Eq jZ1
from Cell III
from Cell II
D Z ðK 0I K u2 M 0I Þ
EZ
J III X jZ1
[2
ð hj 0
ð[ 0
x2 dx Z ðK 0I K u2 M 0I Þ
ðkjIII ðzÞ K u2 mIII j ðzÞÞdz
dZ II ðzÞ F Z K[2 nII1 ðzÞ 1 dz zZ0
[3 3
(36a)
(36b)
(36c)
The soil stiffness matrix for a rigid foundation is obtained at O by superimposing the above forces and taking into account the left-hand side of axisymmetry also. Then, the equation of motions of the foundation at O is expressed in the frequency domain as ( ) ( ) Ux Px 2 (37) Z ð½K K u ½MÞ M q where [M]Zmass matrix of the foundation; and [K]Zsoil stiffness matrix expressed by " " ## 0 0 ½K Z 2 ½A C ½B C ½C C (38) 0 D CE CF With Eq. (37), the foundation responses at O can be computed for forces or displacements applied at O of the foundation. The parameters (n, k, m) and (N, K, M) for all secondary cells can be expressed explicitly in closed form after evaluating the integrations in Eqs. (11) and (14), respectively. With these explicit expressions, the above integrations in [K] result in simple closed form expressions also. All of the above formulations are general and applicable for any layered-soil profiles made of either homogeneous or heterogeneous layers. 3.4. Computational procedure I;II I II It is set that dI;II 0=I Z d0=II Z 1 (or Zj ðzÞZ Zj ðzÞ) in connecting Cells I and II herein. This results in Z1II ðza ÞZ Z1I ðza ÞZ 1 according to Eqs. (28a) and (29a). In view of ZJIII ðzb Þ given by Eqs. (28b) and (29b), dII;III 0=II is then automatically defined as dII;III 0=II Z Ux K [z q for the first mode displacement and dII;III 0=II Z [q for the second mode displacement. Eq. (31a) for X(x) and Eq. (31b) for Z(z) are mutually coupled since the parameters (N, K, M) in Eq. (31a) and (n, k, m) in Eq. (31b) are governed, respectively, by the functions Z(z) and X(x). Iterative procedure is adopted to compute the foundation responses. The computational procedure is explained below.
(35b)
0
M Z P[ Z Fq
where
(35c)
(1) Divide the soil domain into three rectangular secondary cells as shown in Fig. 3. Number these cells located at the base, corner and side of the foundation as I, II,
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
(3)
(4) (5)
(6) 7
(8)
4. Computed results The point O is set at the center of the foundation base and massless foundations are considered in all cases computed. First, a homogeneous ground underlain by bedrock is considered to demonstrate and examine the approach. In this case, Cells I, II, and III are uniform without layering. The lateral responses of massless rigid foundations are computed with j(z)ZZ(z) or Z*(z). The function XII,III(x) is expressed as XII,III(x)ZeKbx (Appendix A). Source problems for an infinite lateral extent of ground require both the real and imaginary parts of b positive for the positive x direction. The values of b at various frequencies are shown in Fig. 5, in which S1 and S2 indicate, respectively, the locations of the first and second natural frequencies of the ground in shear. As shown in the figure, they are always positive for j(z)Z Z*(z). For j(z)ZZ(z), however, its imaginary part becomes negative as frequency increases beyond the frequency a little higher than the fundamental natural frequency of the ground and the computation is terminated when this occurs. The computed soil flexibilities for the above cases are
H/ = 2 = 0.3
d/ d/ d/ d/
1.6
Imaginary
1.2
=0 =2/3 =0 =2/3
X, Z* X, Z
S2
S1 0.8
0.4
0 0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Real Fig. 5. b Values at various frequencies.
shown at various frequencies in Fig. 6 to compare them with those computed by FEM [18]. The frequency parameter a0 in the figure is defined as a0Zu[/vs with vsZsoil shear wave velocity. The figure indicates that the present approximate approach can produce excellent results for j(z)ZZ*(z). Soil flexibilities were previously computed for rigid foundation on a layer overlain by infinitely deep stiff soil as shown in Fig. 7 [19]. In this case, the ground is inhomogeneous and Cells I and II are infinitely long in the z direction. The values computed by the present method and Gazetas [19] are shown in Fig. 8. Gazetas’s formulation is based on the far more elaborated approach, with which it is not possible to have the final expressions in explicit closed form. As seen, the values computed by the two approaches are reasonably close to each other. A foundation of [Z4 m is assumed to rest on the surface of heterogeneous soil underlain by rigid base at depth dZ 2[. The conditions considered for the soil are: vs(z)Z vs(0)(1C1.5z/[) with vsZshear wave velocity of soil (or G(z)ZG(0)(1C1.5z/[)2) and z is the depth from the ground surface; nZ0.25; and DZ0.05. The soil is divided equally into 8 or 16 homogeneous layers as shown in Fig. 9. It is also divided equally into eight heterogeneous layers in which G(z) varies linearly with z within a layer. G(z) in the latter d/ = 2/3 H/ = 2 = 0.3 Flexibility Fx(a0) / Fx(a0=0)
(2)
and III, respectively. Number the layers in each cell from the top layer (jZ1) through the bottom layer (jZ J) in order. Set the location O along the vertical line passing through the center of the foundation. Iteration cycle starts here through the step (8). If this is the first cycle of iteration: assume Z III j ðzÞ based on Z(z)ZUxKqz for the first mode or Z(z)Zq[ for the second mode; and assume Z I;II j ðzÞ based on linear variation with depth from 1 at the top surface to 0 at the bottom surface of the secondary cell for Cells I and II. I;II After the first cycle of iteration, use Z III j ðzÞ and Z j ðzÞ computed at the previous cycle as the assumed ones. Compute (Nj, Kj, Mj) for Cells I, II, and III with Zj(z) assumed at (2), and then (N 0 , K 0 , M 0 ) for Cells I, II, and III with (Nj, Kj, Mj). Compute (N 00 , K 00 , M 00 ) for the assembly of Cells II and III with (N 0 , K 0 , M 0 ). Define XII,III(x) with (N 00 , K 00 , M 00 ) computed at (3), and then XII(x) and XIII(x) with XII,III(x). Compute (nj, kj, mj) for Cells II and III with XII(x) and XIII(x) defined at (4). Compute (nj, kj, mj) for Cell I with X I ðxÞZ U2 K q[2 for the first mode or XI(x)Zqlx/l for the second mode. Compute ðnj00 ; kj00 ; mj00 Þ for the assembly of Cells I and II with (nj, kj, mj). 00 00 00 Define Z I;II j ðzÞ with ðnj ; kj ; mj Þ computed at (5), and I;II I II then Z j ðzÞ and Z j ðzÞ with Z j ðzÞ. Repeat (2)–(6) for the first mode and the second mode displacements. Then, construct [K] and compute Ux and q for applied load to define XI(x) and Z III j ðzÞ. Compare the assumed Zj(z) with the computed Zj(z) for all cells. If the two are close to each other to satisfy the tolerance criteria, the above computed Ux and q are the responses. Otherwise, repeat (2)–(8) for the next cycle.
685
1.6
real imag real imag real imag
1.2 0.8
X, Z* X,Z FEM
0.4
S1
P1
S2
0 0
0.5
1
1.5
2
Frequency a0 Fig. 6. Soil flexibilities for horizontal foundation motion. (Fx(a0)Z0.227 (model), 0.256 (FEM)).
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
profiles. The computed soil stiffnesses for the foundation with or without embedment in this soil are shown in Fig. 12. It is seen that the embedment affects the soil stiffness on both its magnitude and frequency dependence, while the difference in variation pattern of G with depth affects primarily on its magnitude.
2
H
H =2
G1
5. Conclusions
G2
The differential equation cell method enables us to develop the expressions for dynamic responses of shallow foundations in simple closed form. Despite rather significant simplifications involved, it computes the soil stiffnesses for partially embedded rigid foundations close to those computed by far more rigorous methods. When the soil is underlain by rigid base, Z*(z) instead Z(z) should be used for the weight function, j(z), in source problems to ensure that soil responses diminish at infinity in x. The parameters (N 0 , K 0 , M 0 ) and (nj, kj, mj) are determined iteratively. The number of iteration did not exceed 10 to achieve reasonably accurate results for the cases computed herein. The soil pressure around the foundation corner increases as approaching to the corner due to singularity at this point. The present approach is capable of including such effects automatically through the concentrated forces at the corners and the pressures along the foundation faces, which are
Fig. 7. Rigid foundation in layered half-space.
distribution is nearly equal to the original G(z). Soil stiffnesses computed for these three cases are shown in Fig. 10 together with those computed by the far more elaborated method [20]. It is seen in the figure that the capability of handling heterogeneity in a subdivided layer increases accuracy. The foundation with or embedment (dZ0 or 0.5[) is considered in two heterogeneous soil profiles (Fig. 11), in which Profile A is defined as G(z)ZG(0)(1C6.45(z/[)0.5) and Profile B is the one considered in Fig. 9. Both G(0) and the average G over the depth are identical between the two G1/G2=0.06 G1/G2=0.25 G1/G2=0.06 G1/G2=0.25
Present method Gazetas 1980 0.6
0.8
real part
Flexibility GF
Flexibility G Fxx
1
0.6 0.4 0.2
real part
0.4
0.2
0 0
0 0
0.2
0.4
0.6
0.8
0.5
-0.2
1
1
1.5
2
Frequency a0
Frequency a0 1 0.6
Flexibility G F
Flexibility G Fxx
0.8
imag. part
0.6
0.4
imag. part
0.4
0.2
0.2 0
0 0
0.2
0.4
0.6
Frequency a0
(a) Horizontal response
0.8
1
0
0.5
1
1.5
Frequency a0
(b) Rocking response
Fig. 8. Responses of a massless foundation in layered half-space.
2
50
100
150
8 homogeneous layers 16 homogeneous layers G(z)=G0(1+1.5z/ )2
40 30
B
10 0 0
10
real part
2
3
4
5
6
Frequency, a 0 imag. part
d = 0.5 d=0
80
A B
60
A
40 20
B
0 0
1
5
2
3
4
5
6
Frequency, a 0
0 0
1
2
3
4
5
6
Frequency, a 0 30
Stiffness Kf /G
1
120
Stiffness K f/G
Stiffness Kf /G
15
A B
100
8 inhomogeneous layers 16 homogeneous layers 8 homogeneous layers Gazetas(1980)
real part
A
d = 0.5 d=0
20
-10
Fig. 9. Distribution of shear modulus with depth.
20
687
50
Shear modulus, G (kN/sq.m.)
0 -1 0 -2 -3 -4 -5 -6 -7 -8
Stiffness K f/G
Depth (m.)
T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
8 inhomogeneous layers 16 homogeneous layers 8 homogeneous layers
25 20
imag. part
10 5 0 1
2
3
4
5
6
Frequency, a 0 Fig. 10. Dynamic soil stiffnesses computed for three different cases.
mutually coupled in the formulation. However, it does not provide the detailed distribution of soil pressure acting on the foundation but provides the distribution grossly only in a macroscopic manner. Appendix A. XII;IIIðxÞ and ZI;II j ðzÞ for secondary cells made of homogeneous layers A.1. Expressions of X II;III ðxÞ and ZjI;II ðzÞ
Depth (m.)
Shear modulus, G (kN/sq.m.) 50
(A1a)
XðxÞ Z CeKbx C Debx
(A1b)
100
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 00 K u2 M 00 bZ N 00
150
A G(z)=G0(1+6.45(z/ ))0.5 B G(z)=G0(1+1.5(z/ ))2
B A
Fig. 11. Distribution of shear modulus with depth in Profile A and Profile B.
(A2a)
(A2b)
aj and b are mutually coupled and determined conveniently by iteration [12,14–16]. A.2. Determination of Aj, Bj, C and D The conditions given by Eq. (30a) at xbZN result in CZ 1 and DZ0 for Cells II and III. Therefore, X(x) is in the form of XðxÞ Z eKbx
General solutions of Eqs. (31a) and (31b) are, respectively 0 -1 0 -2 -3 -4 -5 -6 -7 -8
Zj ðzÞ Z Aj sinh aj z C Bj cosh aj z
where Aj, Bj, C, and DZ unknown constants; and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kj00 K u2 mj00 aj Z nj00
15
0
Fig. 12. Dynamic soil stiffnesses for foundations in Profile A and Profile B.
(A3)
Using Eqs. (A1a) and (13) and omitting the common factor xX(x), the expressions for the displacement and stress in the layer j lead to the following expression ( ) Zj ðzÞ Knj Zj0 ðzÞ " #( ) cosh aj z sinh aj z Aj Z Bj Kaj cosh aj z Kaj sinh aj z
(A4)
where Z 0 ðzÞZ dZðzÞ=dz. The origin of z is taken at the upper end of the layer. Since ZJ(zb)Z0 (boundary condition Eq. (30c)), AJ and BJ are expressed from Eq. (A4) as
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T. Nogami et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 679–688
ðAJ ; BJ Þ Z AJ ð1; Ktanh aJ hJ Þ
Rewriting (Aj, Bj) as Aj ; Bj Z Aj Aj0 ; Bj0
(A5)
(A6)
Eq. (A4) and the equilibrium and compatibility conditions at the interface of the jK1th and jth layers result in ( 0 ) ( 0) AjK1 Aj Z ½T (A7) jK1=j 0 BjK1 Bj0 where 1 ½TjK1=j Z 2 sinh ajK1 hjK1 K cosh2 ajK1 hjK1 " # KhjK1=j cosh ajK1 hjK1 sinh ajK1 hjK1 KhjK1=j sinh ajK1 hjK1 hjK1=j Z
aj njK1 ajK1 nj
(A8a)
cosh ajK1 hjK1 (A8b)
with hjZ thickness of the jth layer. Substituting ðAJ0 ; BJ0 Þ defined by Eq. (A5) into Eq. (A7), ðAj0 ; Bj0 Þ for jZ1wJ are computed by Eq. (A7) successively in reverse order. Once ðA10 ; B10 Þ is computed, AJ is computed with B10 from the condition Z1(0)Z1 or Z1 ð0Þ Z AJ B10 Z 1
(A9)
With those computed AJ and ðAj0 ; Bj0 Þ, (Aj, Bj) is computed from Eq. (A6).
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