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Research ArticleEffect of Soil Structure Interaction on Torsional Response ofStructure Supported by Asymmetric Soil Foundation System

Fangyuan Zhou,1 Xuezhang Wen,2 and Hongping Zhu1

1School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China2College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China

Correspondence should be addressed to Xuezhang Wen; [email protected]

Received 4 August 2015; Revised 30 November 2015; Accepted 2 December 2015

Academic Editor: Nerio Tullini

Copyright © 2016 Fangyuan Zhou et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The torsional response of a structure supported by asymmetric foundation was investigated in this study. Several types of theasymmetric soil foundation system were employed to analyze the effect of soil structure interaction on torsional response ofthe superstructure. It can be concluded from the study that torsional response would be generated for a structure supported byasymmetric soil foundation system under horizontal seismic excitation, and the generated torsional response of the superstructurechanged with the degree of the asymmetry of the foundation.

1. Introduction

In recent decades, several reports have pointed out that struc-ture failures due to torsional response were common duringprevious earthquake [1, 2]. Meanwhile, through numericalanalysis of the structure under seismic excitation, researchersfound that asymmetric structures were subjected to moredamage than symmetric structures. Chandler and Hutchin-son [3] presented a detailed parametric study of the coupledlateral and torsional responses of a partially symmetric singlestorey building subjected to both harmonic and earthquakebase excitation. It was concluded that torsional couplinginduced a significant amplification of earthquake forceswhich should be accounted for in design. de La Llera andChopra [4–6] calculated the value of the accidental eccen-tricity with the equivalent lateral force procedure by studyingthe dynamic response of single and multistory buildingssubjected to torsional ground motion. Nagarajaiah et al. [7,8], using multistory models, concluded that eccentricities inthe isolation system and superstructure both contributed tothe torsional response. Similarly Jangid and Datta [9, 10]found that significant eccentricity of the superstructure couldreduce the effectiveness of the base isolation system. In Tena-Colunga et al. [11–13], parametric study was carried out witheccentricities in both the isolation system and superstructure,

which concluded that eccentricity in the isolation systemmight lead to a torsional response. Most of these studiesassumed base fixity in the structural models, neglecting thesoil structure interaction (SSI).

Some studies showed that soil structure interaction con-siderably influenced the dynamic responses of the structuressubjected to earthquake loading [14–16]. Although currentlythere exist adequate computational capabilities for buildingdesign, SSI analyses were rarely carried out for the seismicdesign of building structures [17].

Neglecting SSI in the seismic response analysis of build-ing structures may result in a false torsional response of thestructure. In many cases, fixed-base analyses cannot predictaccurately the torsional response of building structures. Aswe know the soil properties and the foundation type canchange the dynamic response of the superstructure.Thus, theassumption that the base of the structure is fixed is not real-istic. Therefore, several researchers attempted to incorporatethe flexibility of foundation in asymmetric system models.Among them, Balendra et al. [18] used simple springs toapproximate the foundation impedance in an asymmetricmultistory structure. Chandler and Hutchinson [19, 20]analyzed the seismic responses subjected to different typesof earthquake loading considering soil structure interaction.Sivakumaran et al. [21] and Sivakumaran and Balendra [22]

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 2647675, 14 pageshttp://dx.doi.org/10.1155/2016/2647675

2 Shock and Vibration

S

F

ug(i𝜔)

Figure 1: Structure-foundation-soil system.

inspected the dynamic response of asymmetric multistoreystructure-foundation systems using frequency independentfoundation impedance.

Previous studies mainly focused on the torsionalresponses of structures induced by the asymmetry insuperstructures. Little attention has been paid to thetorsional responses of structures induced by asymmetric soilfoundation system. In this study, the torsional response of astructure supported on asymmetric soil foundation systemwas investigated. Several types of asymmetric soil foundationwere employed to analyze the effect of soil structureinteraction on the torsional response of the superstructure.

2. Analytical Theory

The analysis procedure in this paper was based on the follow-ing basic assumptions:

(1) the soil was assumed to be elastic medium;(2) the foundation remains in close contact with the

surrounding soil; no slippage was allowed at the soilfoundation interface.

For a structure (Figure 1) under seismic excitation, the equa-tion of motion of the structure considering soil structureinteraction can be depicted as formula (1) (the calculationwas performed in the frequency domain, and the term 𝑒

𝑖𝜔𝑡

is omitted in the formula):

[

[𝑆𝑆𝑆] [𝑆𝑆𝐹]

[𝑆𝐹𝑆] [𝑆𝐹𝐹]

]{

{𝑢𝑆}

{𝑢𝐹}

} =

{

{

{

{𝑃𝑆}𝐹

{𝑃𝐹}𝐹

}

}

}

, (1)

where [𝑆] = (1 + 2𝜉𝑖)[𝐾] − 𝜔2[𝑀]. [𝐾], [𝑀], and {𝑢} are the

stiffness matrices, mass matrices, and displacement vector,respectively. Subscripts 𝑆 and 𝐹 represent the superstructureand foundation, respectively. 𝜉 represents the damping ratio.{𝑃𝑆}𝐹 and {𝑃

𝐹}𝐹 stand for the force vector acting at the

superstructure and the foundation, respectively. For theseismic wave input problem, the force vector acting at thesuperstructure {𝑃

𝑆}𝐹= {0}.

For the superstructure, the well-known beam model isone of the most commonly used models for conducting

S

F

S

F E

Structure-foundation system Free field Excavated soil

G+ −=

ug(i𝜔)

ug(i𝜔)

Figure 2: Structure-foundation-soil system based on the substruc-ture method.

comprehensive parametric studies. In the MDOF structuremodels used in this study, each floor was assumed to be alumped mass connected by a beam element. The stiffnessmatrices of the foundation and excavated soil were obtainedby finite element method.

As shown in Figure 2, the structure-foundation-soil sys-tem was separated into three components, that is, structure-foundation, the free field, and excavated soil, using theconcept of the substructure method [23].

The equation of motion of the free field can be depictedas the formula

[𝐾𝐹]𝐺({𝑢𝐹} − {𝑢

𝐺}) = {𝑃

𝐹}𝐺, (2)

where character 𝐺 represents the free field, {𝑃𝐹}𝐺 stands for

the force vector acting at the nodes of the free field.The stiffness matrix [𝐾

𝐹]𝐺 of the free field can be

calculated with the following equations:

[𝐾𝐹]𝐺= [𝐺]−1, (3)

[𝐺] =

[

[

[

[

[

[

[

[𝛿11] [𝛿12] ⋅ ⋅ ⋅ [𝛿

1𝑛]

[𝛿21] [𝛿22] ⋅ ⋅ ⋅ [𝛿

2𝑛]

.

.

.

.

.

. d...

[𝛿𝑛1] [𝛿𝑛2] ⋅ ⋅ ⋅ [𝛿

𝑛𝑛]

]

]

]

]

]

]

]

. (4)

In (4), Green’s function solution for a concentrated load andfor a uniform distributed load acting on a disk (when theexcitation node and reception node coincide with each other)was applied to form the flexibility matrix of the free field [𝛿

𝑖𝑗]

[24] (see Appendix A). At the same time, a paraxial boundarycondition (see Appendix A) [25, 26] was used to simulate thebottom boundary of the layered soil.

The equation of motion of excavated soil can be depictedas the formula

[𝑆𝐹𝐹]𝐸{𝑢𝐹} = {𝑃

𝐹}𝐸, (5)

Shock and Vibration 3

where [𝑆𝐹𝐹]𝐸

= (1 + 2𝜉𝑖)[𝐾𝐹𝐹]𝐸− 𝜔2[𝑀]𝐸. Superscript 𝐸

stands for the excavated soil which has same geometricdimension as the foundation and same physical parametersas the free field.

For the soil structure interaction system,

{𝑃𝐹}𝐹+ {𝑃𝐹}𝐺− {𝑃𝐹}𝐸= 0. (6)

Substituting formulae (1)∼(5) into formula (6) produces

[

[𝑆𝑆𝑆] [𝑆

𝑆𝐹]

[𝑆𝐹𝑆] [𝑆𝐹𝐹]𝐹− [𝑆𝐹𝐹]𝐸+ [𝐺]−1]{

{𝑢𝑆}

{𝑢𝐹}

}

= {

0

{𝐾𝐹}𝐺{𝑢𝐺}

} .

(7)

Formula (7) is the equation of motion for a structure-foundation-soil system under seismic excitation. Throughformula (7) the dynamic response of the structure underseismic excitation can be obtained.

3. Numerical Study

In order to investigate the torsional response of the structureinfluenced by asymmetric soil foundation system, influencesof the following asymmetric soil foundation systems wereanalyzed:

(1) Embedded foundation supported by piles withnonunique pile lengths.

(2) L shape and T shape embedded foundation supportedby piles.

(3) Foundation with an adjacent foundation.

For all the cases discussed in this section, normal incident𝑆𝐻 wave was used as an excitation source; the frequency ofthe excitation wave ranged from 0.1Hz to 10Hz.

A 10-storey structure was used in this section, the first,second, and third horizontal natural frequencies of thesuperstructure were 1.4, 4.2, and 6.8Hz, respectively. Thefirst, second, and third torsional natural frequencies of thesuperstructures were 1.9, 5.7 and 9.4Hz, respectively.

3.1. Embedded Foundation Supported by Piles with NonuniquePile Lengths. Plane sketch of the employed embedded foun-dation supported by piles of the superstructure was depictedin Figure 3. The dimension of the embedded foundation was72m in length (2𝐷) and 54m in width (2𝐶) and burieddepth of the foundation (𝐸) was 4m. Piles were defined bydiameter𝑑 (1.5m), center-to-center spacing between adjacentpiles 𝑆 (9.0m), and length 𝐿. The material parameters of thepile were as follows: density = 2.4 t/m3; moduli of elasticityof the pile = 2.1 × 104MPa; Poisson ratio of the pile = 1/6.The soil condition surrounding the foundation was shownin Figure 4. To ensure the accuracy of the thin layer methodat low frequency, the relationship between the wavelength 𝜆

and the depth of the soil 𝐻 should be 𝜆/𝐻 ≤ 4. Meanwhile,to ensure the accuracy of the thin layer method at high

2D

2C

E

4

56

d

S

L

S

3(z)

2(y)

1(x)

Figure 3: Sketch of the embedded foundation supported by piles.

Soil parameters

Soil profile

Paraxial boundary

Pile length

𝜌: density of the soil layersVS: shear velocity of the soil layers�: Poisson ratio𝜉: critical damping ratio

𝜉 = 0.03� = 0.45

VS = 150m/s𝜌 = 1.5 t/m3

20 × 1m

30 × 1m

50 × 2m

25 × 4m

①

①

①

Figure 4: Soil condition.

frequency, the relationship between the wavelength 𝜆 and thedepth of the soil layer 𝐻

1should be 𝜆/𝐻

1≥ 6 [27]. In this

paper, the frequency domain of the excitation varied from0.1Hz to 10Hz; thus, the soil profile shown in Figure 4 wasemployed.

In order to investigate the impact of irregular soil founda-tion system on the torsional response of the superstructure,the following cases as shown in Figure 5 were investigated:

(1) Case 1: embedded foundation supported by piles withunique pile length.

(2) Cases 2∼3: embedded foundation supported by pileswith nonunique piles lengths (nonunique piles weredistributed along 𝑥-axis).

(3) Cases 4∼5: embedded foundation supported by pileswith nonunique piles lengths (nonunique piles weredistributed along 𝑦-axis).

The following ratio was employed to investigate the torsionalresponse of the superstructure: 𝑈

6𝐷/𝑈𝑔. Here 𝑈

6is the

torsional response of the structure; 𝑈𝑔is the lateral free field

displacement.Torsional response of the roof floor (RF) of the super-

structure for Cases 1∼5 is depicted in Figure 6. It can beobserved that when the excited direction was perpendicularto 𝑥-axis, the torsional responses of the roof floor in Case2 and Case 3 were generated, and the torsional response of


L

E

2D

S

1 1

Sketch of cross section 1_12C

3(z) 2(y)2(y)

1(x)1(x)

Case 1: L = 16m(a) Embedded foundation supported by pileswith unique pile length

2DL1 1

2C

S

Sketch of cross section 1_1

3(z) 2(y)

2(y)

1(x)1(x)

L1

D1 = 27m

Case 2: L = 16m, L1 = 14m; Case 3: L = 16m, L1 = 8m

(b) Embedded foundation supported by piles with nonunique pile length(nonunique piles distributed along 𝑥-axis)

L1 1

2C

S

Sketch of cross section 1_1

2D

3(z)

2(y) 2(y)

1(x)1(x)

L1

D1 = 18m

Case 4: L = 16m, L1 = 14m; Case 5: L = 16m, L1 = 8m

(c) Embedded foundation supported by piles with nonunique pile length(nonunique piles distributed along 𝑦-axis)

Figure 5: Embedded foundation supported by piles.

Case 1Case 2

Case 3

URF

(6)D

/Ug

0.0

0.3

0.6

2 4 6 8 100Frequency (Hz)

(a) Excited perpendicular to 𝑥 direction

Case 1Case 4

Case 5

URF

(6)D

/Ug

0.0

0.3

0.6


(b) Excited parallel to 𝑥 direction

Figure 6: Torsional response of the roof floor of the superstructure.


U(6)/Ug

(10−

3ra

d/m

)

0.0

0.3

0.6


Case 1Case 2

Case 3

(a) Excited perpendicular to 𝑥 directionU(6)/Ug

(10−

3ra

d/m

)

0.0

0.3

0.6


Case 1Case 4

Case 5

(b) Excited parallel to 𝑥 direction

Figure 7: Foundation input motion of the embedded foundation supported by piles.

(a) Case 6: L shape (b) Case 7: L shape

(c) Case 8: T shape (d) Case 9: T shape (e) Case 10: T shape

Figure 8: L shape and T shape embedded foundation supported by piles.

the roof floor in Case 4 and Case 5 occurred when the exciteddirection was parallel to the 𝑥-axis. However, the torsionalresponse of the roof floor in Case 1 cannot be observed underboth directions of horizontal excitation.Through Figures 6(a)and 6(b), it can be seen that the torsional response of the rooffloor increased with the length difference of the piles.

The above phenomenon can be explained as follows.When the embedded foundation was supported by piles withdifferent pile lengths the mass center and the rigidity centerof the soil foundation system did not coincide with eachother; thus eccentricity existed in the soil foundation system.The foundation input motion of the embedded foundationsupported by piles in Cases 1∼5 is depicted in Figure 7. Itcan be concluded that in Case 1 (symmetric soil founda-tion system) torsional foundation input motion was notgenerated, but in Cases 2∼5 the torsional foundation inputmotion was generated due to the asymmetric soil foundationsystem.Therefore, the asymmetric soil foundation system can

induce the torsional foundation input motion; thus for thestructure supported by asymmetric soil foundation system,the torsional response of the superstructure is of concern.

3.2. L Shape and T Shape Embedded Foundation Supported byPiles. Figure 8 shows the L shape (Cases 6∼7) and T shape(Cases 8∼10) embedded foundations supported by piles. Thediameter and length of the piles used in this sectionwere 1.5mand 16m, respectively. The distance between the adjacentpiles was 9m. The parameters of the superstructure andthe soil were the same as those in Section 3.1. The plandimension of the L shape and T shape embedded foundationsupported by piles can be obtained through the distancebetween the two adjacent piles as shown in Figure 8. Thetorsional response ofCases 6∼7 is comparedwith that of Case1 depicted in Figure 9(a). It can be observed from Figure 9(a)that for a 10-storey structure supported on L shape embeddedfoundation supported by piles, torsional response of the roof


Case 1Case 6

Case 7 Case 1Case 6

Case 7

URF

(6)D

/Ug

0.0

0.3

0.6

URF

(6)D

/Ug

0.0

0.3

0.6



Excited parallel to x-axis Excited perpendicular to x-axis

(a) L shape

Case 1Case 8

Case 9Case 10

Case 1Case 8

Case 9Case 10

URF

(6)D

/Ug

0.0

0.3

0.6

URF

(6)D

/Ug

0.0

0.3

0.6




(b) T shape


floor was generated regardless of the horizontal excitationdirection, and it increased with the irregular degree of theL shape foundation. The torsional responses in Cases 8∼10are compared with that of Case 1 depicted in Figure 9(b).From Figure 9(b), it can be concluded that the torsionalresponse phenomenon can be observed in Cases 8∼10 whenthe excitation direction was parallel to 𝑥-axis. However, if theexcitation direction was perpendicular to 𝑥-axis, there wastorsional response in Case 10 but none in Case 8 and Case

9 because both the mass center and the rigidity center of thesoil foundation system were on the 𝑦-axis.

3.3. Foundation with an Adjacent Foundation. The influenceof an adjacent structure (namedAS) on the torsional responseof the targeted structure (named TS) is addressed in thissection. The parameters of the soil were the same as thosedepicted in Figure 4. The dimension of the embedded foun-dation was 60m in length (2𝐷) and 60m in width (2𝐶)


4

56

TS AS

10-storey 10-storey 10-storey 10-storey 10-storey10-storey

TS AS TS AS

3(z)2(y)

1(x)

Case 11: S = 6m; 𝛽 = 0∘ Case 12: S = 30m; 𝛽 = 0∘ Case 13: S = 150m; 𝛽 = 0∘

S = 6m S = 30m S = 150m

(a) Variation of distance 𝑆

Case 11

Case 14

AS

ASAS

TS

Case 15

2(y)

1(x)

S = 6m

𝛽

S√2S

(b) Variation of angle 𝛽 (plan sketch)

Figure 10: Structure with an adjacent structure.

and the buried depth of the foundation (𝐸) was 4m. Thematerial parameters of the pile were the same as those inSection 3.1.Thediameter and the length of the pileswere 1.5mand 16m, respectively.The distance between the two adjacentpiles was 12m. Figure 10(a) shows the sketch paragraph ofa 10-storey structure (TS) adjacent to a 10-storey structure(AS) with varying distance (𝑆) between the two adjacentfoundations. Figure 10(b) shows the sketch paragraph of a 10-storey building (TS) adjacent to a 10-storey structure (AS)with varying placed angle (𝛽) between the two adjacentfoundations. The following cases are discussed: Case 11: 𝑆 =

6m, 𝛽 = 0∘; Case 12: 𝑆 = 30m, 𝛽 = 0

∘; Case 13: 𝑆 = 150m,𝛽 = 0∘; Case 14: 𝑆 = 6m, 𝛽 = 45

∘; Case 15: 𝑆 = 6m, 𝛽 = 90∘.

Torsional responses of the roof floor in Cases 11∼13compared with that in Case 1 (excited direction was perpen-dicular to the 𝑥-axis) are depicted in Figure 11(a). It can beobserved that torsional response of the superstructure couldbe induced by the existence of the adjacent structure, andit increased with the decrease of the distance 𝑆. The abovephenomenon can be explained as follows. As the movementof the soil around the foundation of the targeted structurecan be restricted by the presence of an adjacent structure,motion characteristic of the soil around the targeted structureat different spatial location may be impacted in differentextent. Thus, the torsional response of the roof floor of thetargeted structure can be observed under horizontal seismicexcitation.

In order to investigate the effect of the placed angle 𝛽 ofthe adjacent structure to the torsional response of the existingstructure, comparison results between Case 11, Case 14, andCase 15 are depicted in Figure 11(b). It can be observed that

when the excitation direction was parallel to the 𝑥-axis, thetorsional responses of the roof floor of the TS in Cases 14and 15 were generated, and if the excitation direction wasperpendicular to the 𝑥-axis, the torsional responses of theroof floor of TS in Cases 11 and 14 occurred. Thus, it can beconcluded that the torsional response of TS can be influencedby the placed angle of the adjacent structure (AS).

Figure 12 shows the response of a structure adjacent toanother structure with varying storey numbers. The firsthorizontal natural frequencies of the four-, six-, and eight-floor structures were 3.2, 2.3, and 1.7Hz, respectively. Andthe first torsional natural frequencies of the four-, six-, andeight-floor structures were 4.5, 3.2, and 2.3Hz, respectively.Symbol 4 6 was used to represent a 4-storey structure (TS)adjacent to a 6-storey structure (AS). Figure 13(a) shows thetorsional response of the TS affected by the adjacent structurewith varying storey numbers in Case 19; it can be concludedthat the torsional response of the roof of the structure canbe affected by the adjacent structure, and when the targetedstructure was adjacent to a structure with same structuraland foundation configuration (Case 16 (4 4)), the torsionaldynamic response of the targeted structure may be moresevere than the other cases (Case 16 (4 6) and Case 16 (4 8)).Meanwhile, through Figure 13(a), it can also be observedthat around the horizontal natural frequency of the adjacentstructure the torsional response of the targeted structurewas generated. In order to investigate the torsional dynamicresponse of the targeted structure affected by the horizontalnatural frequency of the adjacent structure, two cases shownin Figures 12(b)∼12(c) are discussed here. As it has beendescribed that the first horizontal natural frequencies of


Case 1Case 11

Case 12Case 13

URF

(6)D

/Ug

0.0

0.3

0.6


(a) Variation of distance 𝑆 (excited perpendicular to 𝑥-axis)

Case 11Case 14

Case 15Case 11Case 14

Case 15

URF

(6)D

/Ug

0.0

0.3

0.6

URF

(6)D

/Ug

0.0

0.3

0.6




(b) Variation of angle 𝛽


the four- and six-floor structures coincided with the firsttorsional natural frequencies of the six- and eight-floorstructures. Figures 13(b)∼13(c) show the torsional response ofthe structure affected by the adjacent structure; when the hor-izontal natural frequency of the adjacent structure coincidedwith the torsional natural frequency of the targeted structure,the torsional dynamic response of the targeted structure wasmore significant as can be seen in Figures 13(b) and 13(c).

4. Conclusion

The effects of soil structure interaction on torsional responseof structure supported on asymmetric foundation underhorizontal seismic excitation were assessed in this study. Theresults of the study may lead to the following conclusions:

(1) The torsional response can be generated for thestructure supported on asymmetric foundation under


(a1) TS: 4-floor, AS: 4-floor (4_4) (a2) TS: 4-floor, AS: 6-floor (4_6) (a3) TS: 4-floor, AS: 8-floor (4_8)

TSTSTS TSASAS

4

563(z)

2(y)

1(x)

S = 6m S = 6m S = 6m

(a) Case 16: 4-floor structure (TS) adjacent to structures

(b1) TS: 6-floor, AS: 4-floor (6_4) (b2) TS: 6-floor, AS: 6-floor (6_6)

TSTS TS ASASAS

(b3) TS: 6-floor, AS: 8-floor (6_8)

S = 6m S = 6m S = 6m

(b) Case 17: 6-floor structure (TS) adjacent to structures

(c1) TS: 8-floor, AS: 4-floor (8_4) (c2) TS: 8-floor, AS: 6-floor (8_6) (c3) TS: 8-floor, AS: 6-floor (8_8)

TS TS TSAS AS ASS = 6m S = 6m S = 6m

(c) Case 18: 8-floor structure (TS) adjacent to structures

Figure 12: Existing structure adjacent to 4-, 6-, and 8-storey structure.

horizontal seismic excitation due to the eccentricity inthe soil foundation system.

(2) For the structure supported by embedded-pile foun-dation with different pile lengths, the torsionalresponse of the roof floor will be induced underhorizontal seismic excitation as the mass center andthe rigidity center of the soil foundation system withnonunique pile lengths do not coincide with eachother, and the induced torsional response increaseswith the increase in the difference of pile lengths.Torsional response of the roof floor can also beobserved for the structure supported by L shape or Tshape embedded-pile foundation.

(3) As the movement of the soil around the foundationof the targeted structure can be restricted by thepresence of an adjacent structure, motion charac-teristics of the soil around the targeted structure atdifferent spatial locationmay be impacted in differentextent by the adjacent structure. Thus, the torsionalresponse of the roof floor of the targeted structurecan be observed under horizontal seismic excitation.Especially when the horizontal natural frequency ofthe adjacent structure coincides with the torsionalnatural frequency of the targeted structure, the tor-sional dynamic response of the targeted structure canbe affected significantly.


4_44_6

4_8

URF

(6)D

/Ug

0.0

0.5

1.0


(a) Case 16: excited perpendicular to 𝑥 direction

6_46_6

6_8

URF

(6)D

/Ug

0.0

0.5

1.0


(b) Case 17: excited perpendicular to 𝑥 direction

8_48_6

8_8

URF

(6)D

/Ug

0.0

0.5

1.0


(c) Case 18: excited perpendicular to 𝑥 direction



R

S

z

y

x R

1

SExcitation point

Reception point

N

Layer (1)

Layer (i)

Layer (N− 1)

r cos𝜃

...

...

...

(x0, y0)

(x0, y0)

(x, y)(r, 𝜃) ux

ux

uyuz

uz

PxPx

PzPz

Py

Figure 14: Geometric relationship between the excitation point and reception point in the thin layered method.

Appendix

A. Formulation of Green’s Function Used inthe Thin Layer Method and Expression ofthe Paraxial Boundary Condition

The displacement solution of a point located at 𝑅 surface forthe load acting at 𝑆 surface (see Figure 14) includes

(1) point load solution (𝑟 > 0);(2) disk load solution (𝑟 → 0, pile tip);(3) ring load solution (𝑟 → 0, pile shaft).

A.1. Point Load Solution. Let 𝑢𝑥, 𝑢𝑦, and 𝑢

𝑧displace com-

ponents at elevation 𝑅 for a point load acting at elevation 𝑆.Green’s function for point load is

{{

{{

{

𝑢𝑥

𝑢𝑦

𝑢𝑧

}}

}}

}𝑅

=[

[

[

𝑉1cos 2𝜃 + 𝑉

2𝑉1sin 2𝜃 𝑉

4cos 𝜃

𝑉1sin 2𝜃 −𝑉

1cos 2𝜃 + 𝑉

2𝑉4sin 𝜃

−𝑉3cos 𝜃 −𝑉

3sin 𝜃 𝑉

5

]

]

]

{{

{{

{

𝑃𝑥

𝑃𝑦

𝑃𝑧

}}

}}

}𝑆

,

(A.1)

where

𝑉1= −

1

2𝜋

2𝑁

∑

𝑘=1

𝑋𝑅𝑘𝑋𝑆𝑘

𝐷𝑘𝛼

𝛼2

𝑘𝐹1(𝛼𝑘𝑟)

+

1

4𝜋

𝑁

∑

𝑘=1

𝑌𝑅𝑘𝑌𝑆𝑘

𝐷𝑘𝛽

𝐹1(𝛽𝑘𝑟) .

𝑉2=

1

2𝜋

2𝑁

∑

𝑘=1


𝐷𝑘𝛼

𝛼2


+

1

4𝜋

𝑁

∑

𝑘=1


𝐷𝑘𝛽

𝐹2(𝛽𝑘𝑟)

𝑉3=

1

𝜋

2𝑁

∑

𝑘=1

𝑍𝑅𝑘𝑋𝑆𝑘

𝐷𝑘𝛼

𝛼2


𝑉4=

1

𝜋

2𝑁

∑

𝑘=1

𝑋𝑅𝑘𝑍𝑆𝑘

𝐷𝑘𝛼

𝛼2


𝑉5=

1

𝜋

2𝑁

∑

𝑘=1

𝑍𝑅𝑘𝑍𝑆𝑘

𝐷𝑘𝛼

𝛼2


𝐹1 (𝑧) = −

2

𝑧2− 𝑖

𝜋

𝑧

𝐻(2)

1(𝑧) + 𝑖

𝜋

2

𝐻(2)

0(𝑧)

𝐹2 (𝑧) = −𝑖

𝜋

2

𝐻(2)

0(𝑧)

𝐹3 (𝑧) = −𝑖

𝜋

2

𝐻(2)

0(𝑧)

𝐷𝑘𝛼

= − {𝑋}𝑇

𝑘[𝐸𝑆] {𝑋}𝑘

− {𝑍}𝑇

𝑘[𝐸𝑃] {𝑍}𝑘

+ 𝛼2

𝑘{𝑋}𝑇

𝑘[𝐴𝑃] {𝑋}𝑘

+ 𝛼2

𝑘{𝑍}𝑇

𝑘[𝐴𝑆] {𝑍}𝑘

𝐷𝑘𝛽

= {𝑌}𝑇

𝑘[𝐴𝑆] {𝑌}𝑘

𝑟 = √(𝑥 − 𝑥0)2+ (𝑦 − 𝑦

0)2

𝜃 = 𝑎 tan(𝑦 − 𝑦0

𝑥 − 𝑥0

)

(A.2)

𝐻(2)

V (𝑧) is the second Hankel function of order V. 𝛼𝑘is the

𝑘th eigenvalue of characteristic equation (A.7). 𝛽𝑘is the 𝑘th

eigenvalue of characteristic equation (A.8). 𝑋𝑘, 𝑍𝑘are the

corresponding eigenvector of 𝛼𝑘. 𝑌𝑘is the corresponding

eigenvector of 𝛽𝑘.𝑋𝑅𝑘

is the 𝑅th component of the eigenvec-tor 𝑋𝑘. 𝑍𝑅𝑘

is the 𝑅th component of the eigenvector 𝑍𝑘. 𝑌𝑅𝑘

is 𝑅th component of the eigenvector 𝑌𝑘.


A.2. Disk Load Solution. Let 𝑢𝑥, 𝑢𝑦, and 𝑢

𝑧displace com-

ponents at elevation 𝑅 for a disk load acting at elevation 𝑆.Green’s function for disk load is then

{{

{{

{

𝑢𝑥

𝑢𝑦

𝑢𝑧

}}

}}

}𝑅

=

[

[

[

[

𝑉6(𝑟 = 𝑟

0)

𝑉6(𝑟 = 𝑟

0)

𝑉7(𝑟 = 𝑟

0)

]

]

]

]

{{

{{

{

𝑃𝑥

𝑃𝑦

𝑃𝑧

}}

}}

}𝑆

,

(A.3)

where

𝑉6=

1

2𝜋

2𝑁

∑

𝑘=1


𝐷𝑘𝛼

𝛼2


+

1

4𝜋

𝑁

∑

𝑘=1


𝐷𝑘𝛽

𝐹4(𝛽𝑘𝑟)

𝑉7=

1

𝜋

2𝑁

∑

𝑘=1


𝐷𝑘𝛼

𝛼2


𝐹4 (𝑧) = −

2

𝑧2− 𝑖

𝜋

2

𝐻(2)

1(𝑧) .

(A.4)

𝑟0is the radius of the disk.

A.3. Ring Load Solution. Consider the following:

{{

{{

{

𝑢𝑥

𝑢𝑦

𝑢𝑧

}}

}}

}𝑅

=

[

[

[

[

𝑉2(𝑟 = 𝑟

0)

𝑉2(𝑟 = 𝑟

0)

𝑉5(𝑟 = 𝑟

0)

]

]

]

]

{{

{{

{

𝑃𝑥

𝑃𝑦

𝑃𝑧

}}

}}

}𝑆

,

(A.5)

where

𝑉2=

1

2𝜋

2𝑁

∑

𝑘=1


𝐷𝑘𝛼

𝛼2


+

1

4𝜋

𝑁

∑

𝑘=1


𝐷𝑘𝛽

𝐹2(𝛽𝑘𝑟)

𝑉5=

1

𝜋

2𝑁

∑

𝑘=1


𝐷𝑘𝛼

𝛼2


𝐹2 (𝑧) = −𝑖

𝜋

2

𝐻(2)

0(𝑧) .

(A.6)

𝑟0is the radius of the ring.

A.4. Characteristic Equation of the Thin Layer Model asShown in Figure 14. The characteristic equations of the thinlayer model (shown in Figure 1) for Rayleigh (in-plane) andLove (anti-plane) waves are expressed by (A.7) and (A.8),respectively:

(𝛼2[

[𝐴𝑃]

[𝐴𝑆]

] + 𝛼[

[𝐵]

[𝐵]𝑇

]

+ [

[𝐸𝑆]

[𝐸𝑃]

])[

𝑋

𝑍

] = {0} ,

(A.7)

(𝛽2[𝐴𝑆] + [𝐸

𝑆]) {𝑌} = {0} , (A.8)

where [𝐴𝑆], [𝐴𝑃], [𝐸𝑆], [𝐸𝑃], and [𝐵] are the matrices assem-

bled by [𝐴𝑆](𝑖), [𝐴

𝑃](𝑖), [𝐸𝑆](𝑖), [𝐸𝑃](𝑖), and [𝐵](𝑖), respectively,

in the manner presented in

[𝐴] =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

𝐴(1)

11𝐴(1)

12

𝐴(1)

21𝐴(1)

22+ 𝐴(2)

11𝐴(2)

21

𝐴(2)

12𝐴(2)

22+ 𝐴(3)

11

d

𝐴(𝑁−3)

22+ 𝐴(𝑁−2)

11𝐴(𝑁−2)

12

𝐴(𝑁−2)

21𝐴(𝑁−2)

22+ 𝐴(𝑁−1)

11𝐴(𝑁−1)

12

𝐴(𝑁−1)

21𝐴(𝑁−1)

22+ 𝐴(𝑁)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

, (A.9)


where 𝐴(𝑖) is the matrix corresponding to the 𝑖th soil layerand𝐴(𝑁) is matrix corresponding to the boundary condition.Consider the following:

[𝐴𝑆](𝑖)=

𝐺𝑖𝐻𝑖

6

[

2 1

1 2

] ,

[𝐴𝑃](𝑖)=

(𝜆𝑖+ 2𝐺𝑖)𝐻𝑖

6

[

2 1

1 2

] ,

[𝐸𝑆](𝑖)= [𝐺𝑆](𝑖)− 𝜔2[𝑀](𝑖),

[𝐺𝑆](𝑖)=

𝐺𝑖

𝐻𝑖

[

1 −1

−1 1

] ,

[𝑀](𝑖)=

𝜌𝑖𝐻𝑖

6

[

2 1

1 2

] ,

[𝐸𝑃](𝑖)= [𝐺𝑃](𝑖)− 𝜔2[𝑀](𝑖),

[𝐺𝑃](𝑖)=

(𝜆𝑖+ 2𝐺𝑖)

𝐻𝑖

[

1 −1

−1 1

] ,

[𝐵](𝑖)=

[

[

[

[

−

(𝜆𝑖− 𝐺𝑖)

2

(𝜆𝑖+ 𝐺𝑖)

2

−

(𝜆𝑖+ 𝐺𝑖)

2

(𝜆𝑖− 𝐺𝑖)

2

]

]

]

]

,

(A.10)

where𝜔 is the excited frequency of the load and𝜆𝑖,𝐺𝑖,𝐻𝑖, and

𝜌𝑖are the Lame constant, shear elastic modulus, thickness,

and density of the 𝑖th soil layer, respectively.𝛼𝑘and {𝑋𝑘

𝑍𝑘}𝑇 (𝑘 = 1, 2 . . . , 2𝑁) represent the

eigenvalue and eigenvector, respectively, of characteristicequation (A.7).

𝛽𝑘and {𝑌

𝑘}𝑇 (𝑘 = 1, 2 . . . , 𝑁) represent the eigenvalue

and eigenvector, respectively, of the characteristic equation(A.8).

A.5. Paraxial Boundary

(1) The boundary condition of (A.8) is as follows:

𝐴𝑁

𝑆= −𝑖

𝐺𝑁

2

𝑉𝑆𝑁

𝜔

,

𝐸𝑁

𝑆= 𝑖𝜔𝜌

𝑁𝑉𝑆𝑁.

(A.11)

(2) The boundary condition of (A.7) is shown as follows:

𝐴𝑁

𝑆= 𝑖

𝐺𝑁

2

𝑉𝑆𝑁

𝜔

𝜂2

𝑁(𝜂𝑁− 2)

= 𝑖

𝜆𝑁+ 2𝐺𝑁

2

𝑉𝑆𝑁

𝜔

(𝜂𝑁− 2) ,

𝐴𝑁

𝑃= 𝑖

𝐺𝑁

2

𝑉𝑆𝑁

𝜔

(1 − 2𝜂𝑁) ,

𝐵𝑁= 𝐺𝑁(2 − 𝜂

𝑁) ,

𝐸𝑁

𝑆= 𝑖𝜔𝜌

𝑁𝑉𝑆𝑁,

𝐸𝑁

𝑃= 𝑖𝜔𝜌

𝑁𝑉𝑃𝑁

,

(A.12)

where

𝜂𝑁=

𝑉𝑃𝑁

𝑉𝑆𝑁

,

𝜆𝑁=

2]𝑁

1 − 2]𝑁

𝐺𝑁,

𝐺𝑁= 𝜌𝑁𝑉2

𝑆𝑁.

(A.13)

𝜆𝑁, 𝐺𝑁, 𝜌𝑁, ]𝑁, 𝑉𝑆𝑁, and 𝑉

𝑃𝑁are the Lame constant,

shear elastic modulus, density, Poisson ratio, 𝑆-wavevelocity, and𝑃-wave velocity of the boundary, respec-tively.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to gratefully acknowledge the finan-cial support for this research provided by the NationalNatural Science Foundation of China (Grant no. 51308243),the National Basic Research Program (973) of China (Grantno. 2011CB013800), and the Fundamental Research Funds forthe Central Universities: HUST (Grant no. 2015TS122).

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