Summary of doctoral thesis: Studying the soil-structure under seismic loading with macro-element

MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF TRANSPORT AND COMMUNICATIONS

HUYNH VAN QUAN

STUDYING THE SOIL-STRUCTURE INTERACTION UNDER SEISMIC LOADING WITH MACRO-ELEMENT

Speciality: Engineering Mechanics

Code: 9520101

SUMMARY OF DOCTORAL THESIS

HA NOI – 2021

This research is completed at: University of Transport and

Communications

Supervisors:

1. Assoc. Prof. Nguyen Xuan Huy

2. Assoc. Prof. Nguyen Trung Kien

Reviewer 1: ………………………………………………

Reviewer 2 ………………………………………………

Reviewer 3: ………………………………………………

This thesis will be defended before Doctoral-Level Evaluation

Council at University of Transport and Communications at …..

hours……Day……Month……Year…….

The thesis can be found at:

1. Library of University of Transport and Communications

2. National Library of Vietnam

1

INTRODUCTION

Research background

Analyzing the response of structure considering to the soil-

structure interaction is not mentioned or only recommended in civil

specifications. Because the structure analysis which simultaneously

includes with superstructure, foundation and soil is so difficult.

Meanwhile, the interaction between foundation and soil is so complex

in case seismic loading.

In recent years, Institute of Geophysics has recorded a lot number

of earthquakes in Vietnam. Earthquake level VIII (MSK-1964) can be

occurred in the north west region of Vietnam. In this level, buildings

can be collapse.

The soil-structure interaction problems have been almost studied

by numerical method in Vietnam. In these researches, the restrains

between structure and soil are replaced by linear springs. In the world,

the soil-foundation interaction is simulated by a macro-element, it is

considered as a modern and efficient solution.

The analysis of soil-structure interaction under seismic loading

needs to be continued studying. Consequently, PhD student’s thesis is

chosen with the title “studying the soil-structure under seismic loading

with macro-element”.

Research Objective

The soil-structure interaction problems under seismic loading are

researched by numerical and experimental methods. In numerical

study, the complication of soil-foundation interaction is simulated by

a proposal macro-element.

2

Research scope

The research scope of this thesis is the lateral displacements and

accelerations. The structure is simulated in lumped-parameter model,

the macro-element, which replaces the system of shallow foundation

and dense sand, ignores the couple of displacement-rock responses. In

experimentation, a laminar soil container is fixed on the shaking table

by the bolted connections, the excitations shake along the length of the

soil container, the properties of soil are kept constant during the testing

process.

Research methodology

Harmonically combining between numerical and experimental

methods.

Scientific and practicality

In analysis, the system of soil and foundation is simulated by

simple macro-element’s method, the analytical results help designers

giving the suitable solutions. In experimentation, the engineers can

observe the prototype responses of the samples, the experimental

results compared with the numerical results.

Research outline

The thesis includes introduction, 4 chapters and conclusion.

Chaper 1. LITERATURE REVIEW OF SOIL-STRUCTURE

INTERACTION

1.1 Overview of soil-structure interaction

In the structural engineer’s point of view, the structure is as a light

leaf and the foundation is a as rigid block. In the geotechnical

engineer’s point of view, the structure is as a rigid block, the

foundation is a as pillow (Figure 1.1).

3

With these points of view, small displacements of foundation affect

the leaf and incremental forces also affect the structure. Therefore, the

system of soil, foundation and superstructure needs to be

simultaneously analyzed.

Figure 1.1. The points of view in structure and foundation’s stiffness (Grrange, 2008) 1.2 The non-linearities of soil-structure interaction under seismic

loading

Geometrical non-linearity, the inertia forces from earthquake

reproduces the moments that overturn the structure. When the

overturning moment is bigger than the resistance moment,

foundation–soil contact area decreases during processing of

foundation rotations and uplift.

Material non-linearity includes all the other non-linearities. These

come from soil yielding under: dead weight, increase of loads and

increase of stresses during uplift (Cremer, 2001).

1.3 simulation’s methods of soil-structure interaction

In order to study soil-structure interaction, three kinds of methods

often be used (Grange, 2008). Direct methods that use a classical finite

element approach, they provide very good results but requiring

numerical expertise, good knowledge of the constitutive laws and cost

in terms of computation. Sub-structuring that aims at decomposing the

4

problem into simpler problems of Kausel’s superposition principle

with kinematics interaction and inertial interaction, this method is

valid only for linear problems. Hybrid method that combines of the

two previous methods and therefore more attractive on the numerical

level, the macro-element approach belongs to this last category.

Macro-element is a global element but it also contains all properties

in local scale. The macro-element is firstly researched and published

by PhD student in Vietnam and we have called “phần tử vĩ mô” in

Vietnamese name.

1.4 Summary

Comments:

1. Under seismic loading, the response of soil-foundation

interaction is so complex, it costs a lot of computation. Macro-element

is a best solution for this problem. A lot of macro-element models have

been introduced by researchers, such as cyclic model (Nova 1991,

Cremer 2001, Grange 2009, Chatzigogos 2009), seismic model

(Paolucci 1997, Paolucci 2008, Figini 2012). However, they should be

continued to more studying. In Vietnam, the macro-element’s contents

have not been studied by any one yet.

2. The simulation and experimental results in literature only shown

the settlements and vertical displacements of system.

Problem statements: (i) Proposing a macro-element with the

material-geometric coupling for seismic analysis of shallow

foundation. (ii) Experimental study on the soil-structure interaction

model with and without superstructure. (iii) This thesis focuses on the

lateral acceleration and lateral displacements of system in both simulation and experimentation.

5

Chapter 2. SIMULATING THE SOIL-STRUCTURE

INTERACTION UNDER SEISMIC LOADING WITH

MACRO-ELEMENT

2.1 Structure of the macro-element model

+) Spatial macro-element (Figure 2.1):

Forces (2.1)

Displacements (2.2) 𝑭 = {𝐻𝑥 𝐻𝑦 𝑀𝑥 𝑀𝑦 𝐻𝑧}𝑇 𝒖 = {𝑢𝑥 𝑢𝑦 𝜃𝑥 𝜃𝑦 𝑢𝑧}𝑇

Figure 2.1. Generalized variables of forces and displacements in spartial macro-element (Grange, 2009)

Figure 2.2. Generalized variables of forces and displacements in planar macro-element (Chatzigogos, 2009)

+) Planar macro-element (Figure 2.2):

Forces (2.3)

Displacements (2.4) 𝑸 = {𝑁 𝑀 𝑉}𝑇 𝒒 = {𝑣 𝜃 𝑢}𝑇

2.2 The yield function and flow rule

The yield function 𝑓(𝐹𝐹) proposed by Nova (1991) and the plastic

flow rule 𝑔(𝐹𝐹) proposed by Cremer (2001) are employed:

+) Spatial macro-element:

2 + ℎ𝑦

2 + 𝑚𝑥

6

2 + 𝑚𝑦 2) + 𝜒2(𝑚𝑥

2 − 𝜉2(1 − 𝜉)2𝛽 2 + 𝑚𝑦

(2.5) { 2) + 𝜉2 − 1

𝑓(𝐹𝐹) = ℎ𝑥 𝑔(𝐹𝐹) = 𝜆2(ℎ𝑥 2 + ℎ𝑦 + Planar macro-element:

(2.6) { 𝑓(𝐹𝐹) = ℎ2 + 𝑚2 − 𝜉2(1 − 𝜉)2𝛽 𝑔(𝐹𝐹) = 𝜆2ℎ2 + 𝜒2𝑚2 + 𝜉2 − 1

ℎ𝑥 = 𝐻𝑥/𝜇𝑁𝑚𝑎𝑥, ℎ𝑦 = 𝐻𝑦/𝜇𝑁𝑚𝑎𝑥, ℎ = 𝑉/𝜇𝑁𝑚𝑎𝑥, 𝑚𝑥 = 𝑀𝑥/ 𝜓𝐵𝑥𝑁𝑚𝑎𝑥, 𝑚𝑦 = 𝑀𝑦/𝜓𝐵𝑦𝑁𝑚𝑎𝑥, 𝑚 = 𝑀/𝜓𝐵𝑁𝑚𝑎𝑥, 𝜉 = 𝑁𝐹/𝑁𝑚𝑎𝑥. 𝐵 the effective foundation width; 𝐵𝑥, 𝐵𝑦 the effective foundation width in 𝑥, 𝑦 directions. 𝑁𝑚𝑎𝑥 = 𝑞𝑚𝑎𝑥𝑆 is the ultimate bearing capacity under vertical center load, 𝑆 is the effective foundation area, 𝑞𝑚𝑎𝑥 is the ultimate compression stress of soil. 2.3 The stiffness matrix of macro-element

In step of nth is determined by the following formula of Paolucci

(1997).

𝐹 = 𝑭𝑛 𝑭𝑛+1

𝐹 + 𝑲𝐹(𝒙𝑛+1 − 𝒙𝑛) The cross-coupled elastic terms are neglected for a surface

(2.7)

foundation, the off-diagonal elastic terms are responsible for the

volumetric behavior.

(2.8) 𝑲𝐹0 =

0 𝑘𝑦0 0 0 0 The elastic stiffness matrix 𝑲𝐹 = 𝑲𝐹0. 0 𝑘𝑥0 0 0 0 0 𝑘𝑟𝑦0 0 0 0 [ 0 0 0 0 𝑘𝑧0]

(2.9) 𝑲𝐹0 = [ ] 𝑘0 0 0 0 0 𝑘𝑣

0 0 𝑘𝑟𝑥0 0 0 0 𝑘𝑟 0 𝑘𝑥0(𝑘0), 𝑘𝑦0, 𝑘𝑧0(𝑘𝑣) và 𝑘𝑟𝑥0, 𝑘𝑟𝑦0(𝑘𝑟) are the equivalent elastic spring coefficients of the soil-foundation system defined by Gazetas

(1991):

2𝐺𝐿

0,2𝐺𝐿

𝐵

7

(1 −

)

𝐾𝑦0 =

(2 + 2,050,85); 𝐾𝑥0 = 𝐾𝑦 −

2−𝑣

0,75−𝑣

𝐿

2𝐺𝐿

(0,73 + 1,540,75)

1−𝑣

(2.10)

𝐾𝑧0 = 0,25

0,25

𝐺

𝐿

𝐵

𝐺

𝐿

)

(2,4 + 0,5

)

]

𝐾𝑟𝑥0 =

) ; 𝐾𝑟𝑦0 =

0,75 ( 𝐼𝑏𝑥

0,75 [3 ( 𝐼𝑏𝑦

1−𝑣

𝐵

𝐿

1−𝑣

𝐵

{

The resulting in a reduction of the effective foundation width (𝐵′) that can be expressed as follows: 𝐵′ = 𝐵(1 − 𝛿)

(2.11) 𝛿 can be interpreted as a degradation parameter defined in the range

0 ≤ 𝛿 < 1. Substituting Equation (2.11) into the approximate static

stiffness formulas for a rectangular footing, and supposing that the

square-shaped foundation base is in full contact with the soil prior to

the seismic excitation, the reduced stiffness factors for each vibration

′

mode are obtained (Paolucci, 2008):

′

𝑘𝑥0(𝑦0) (2.12) {

= 𝑘𝑥0(𝑦0)[0,74(1 − 𝛿)0,35 + 0,09 + 0,17(1 − 𝛿)] = 𝑘𝑟𝑥0(𝑦𝑜)[(1 − 0,2𝛿)(1 − 𝛿)2] 𝑘𝑟𝑥0(𝑦0) ′ = 𝑘𝑣0[0,66(1 − 𝛿)0,25 + 0,34(1 − 𝛿)] 𝑘𝑣0

1+

𝛿1 1 𝛿2𝜃𝑝

(2.13) 𝛿(𝜃𝑝) =

Where 𝛿1 =0.75 and 𝛿2 =5000/rad are model parameters related to

𝑛

(2.14) the ultimate value of 𝛿 and to the degradation speed, respectively. ′ | 𝜃𝑝 = ∑ |∆𝜃𝑛 − ∆𝑀𝑛/𝑘𝑟 The stiffness matrix is calculated following:

(2.15) 𝑲𝐹′ =

0 0 0 𝑘′𝑟𝑦0 0 𝑘′𝑥0 0 0 0 0 [ 0 0 0 0 𝑘′𝑧0]

(2.16) ] 𝑲𝐹′ = [ 0 𝑘′𝑦0 0 0 0 𝑘′0 0 0 0 0 𝑘′𝑟𝑥0 0 0 0 𝑘′𝑟 0 0 0 𝑘′𝑣

8

−1

The soil behavior is assumed to be linear elastic until the failure surface in Equation 4 is reached. When the failure surface is reached, the plastic flow occurs when 𝑓(𝑭) ≥ 0 and 𝑑𝑓(𝑭) = 0. The elastic stiffness matrix will be reduced by a differential value 𝑑𝑲𝐹 for initial value. This is calculated as a function of the elastic stiffness matrix 𝑲𝐹0 and the derivatives of the yield and plastic potential functions (Paolucci, 1997):

𝑇 )

𝑇 )

𝜕𝑔 𝜕𝑭

𝜕𝑓 𝜕𝑭

𝜕𝑓 𝜕𝑭

𝜕𝑔 𝜕𝑭

(2.17) 𝑑𝑲𝐹 = 𝑲𝐹0 ( ) ( 𝑲𝐹0 ( 𝑲𝐹0 [( )]

Thus, the elastoplastic stiffness matrix of the proposed macro- element in step of 𝑛th is determined by the following formula:

− 𝑑𝑲𝐹 𝑲𝐹 = 𝑲𝐹′

(2.18) 2.4 Simulating the soil-structure interaction under seismic loading

with macro-element

The system of soil and foundation simulated by macro-element

under seismic loading (Figure 2.3(a)) with equation: 𝑴𝒙̈ + 𝑪𝒙̇ + 𝑭𝐹 = 𝑷

(2.19) The system of soil, foundation and superstructure simulated by

(a) System of soil and foundation

(b) System of soil, foundation and superstructure

(2.20) macro-element under seismic loading (Figure 2.3(a)) equation: 𝑴𝒙̈ + 𝑪𝒙̇ + 𝑭𝑆 + 𝑭𝐹 = 𝑷

Figure 2.3. Dynamic mode1 for non-linear analyses (Paolucci, 2008)

9

2.5 The Newmark time integration scheme

Denoting by the subscript 𝑛, the quantities calculated at time 𝑡 =

𝑛∆𝑡, the motion form of Newmark time integration scheme can be

rewritten as (Chopra, 1995):

- The system of soil-foundation (Paolucci, 1997):

1−2𝛽 2𝛽

[ ] 𝒙𝑛+1 + 𝑭𝑛+1(𝒙𝑛+1) = 𝑷𝑛+1 + 𝑴 [ 𝒙̈ 𝑛 +

𝑴 𝛽(∆𝑡)2 + 𝒙̇ 𝑛∆𝑡+𝒙𝑛 𝛽(∆𝑡)2 ] + 𝑪 [(

𝑪𝛾 𝛽∆𝑡 𝛾 2𝛽

𝛾 − 1) 𝒙̈ 𝑛∆𝑡 + ( 𝛽

𝛾 𝛽∆𝑡

(2.21) − 1) 𝒙̇ 𝑛 + 𝒙𝑛]

𝑪𝛾 𝛽∆𝑡

1−2𝛽 2𝛽

[ 𝒙̈ 𝑛 +

𝛾 𝛽∆𝑡

(2.22) − 1) 𝒙̇ 𝑛 + 𝒙𝑛] + 𝑲𝑆] 𝒙𝑛+1 + 𝑭𝑛+1(𝒙𝑛+1) = 𝑷𝑛+1 + 𝑴 [ 𝛾 − 1) 𝒙̈ 𝑛∆𝑡 + ( 𝛽

- The system of soil-foundation-superstructure: 𝑴 𝛽(∆𝑡)2 + 𝒙̇ 𝑛∆𝑡+𝒙𝑛 𝛾 𝛽(∆𝑡)2 ] + 𝑪 [( 2𝛽 2.6 Application

A system of soil and foundation with the squared shape is analyzed:

width 3m, height 1,6m; dense sand, total height 12m with the

equivalent parameters in table 2.1. Earthquake is time history

acceleration of El-centro (1940). The numerical results compared by

the direct method with CyclicTP software.

Values 2,30 × 109 3,62 × 109 2,17 × 106 34,56 × 103 2,45 × 105 0,43 4

Parameters 𝑘𝑟 (Nm/rad) 𝑐0 (Ns/m) 𝑐𝑣 (Ns/m) 𝐽 (kgm2) 𝜇 𝜉 

Values 2,85 × 109 4,82 × 106 2,90 × 106 3,33 × 104 0,682 0,95 6

Table 2.1. Numerical model parameters of soil-foundation used in

dynamic analyses Parameters 𝑘0 (N/m) 𝑘𝑣 (N/m) 𝑐𝑟 (Ns/m) 𝑚0 (kg) 𝑁𝑚𝑎𝑥 (kN) 𝜓 𝜆

10

Lateral displacements (mm)

Lateral accelerations (𝑚/𝑠2)

CyclicTP

CyclicTP

Error (%)

Error (%)

13,32

Proposed macro- element 12,35

7,85

4,654

Proposed macro- element 4,652

0,04

Table 2.2. Error of lateral acceleration and displacement

Summary of Chapter 2

The typical results in this chapter:

- Prosing a macro-element with the material-geometric coupling

for seismic analysis of shallow foundation.

- The proposed macro-element has been applicated to simulating

the response of the systems of soil and structure.

- The Newmark time integration schemes were set up for the

analytical model of soil-structure interaction, equations (2.21) and

(2.22).

- Numerical analyst for a simple model, the results compared to the

CyclicTP, the errors of lateral acceleration and displacement are

7,85% and 0,04%, respectively.

Chapter 3. EXPERIMENTAL STUDY ON SEISMIC

RESPONSES OF STRUCTURE CONSIDERING SOIL-

STRUCTURE INTERACTION

3.1 Prototype characteristics

The prototype of the experimental tests is a soil-structure system

with dimensional characteristics illustrated in figure 3.1: deck’s

weight 120 tons, effective height ℎ =12,5m, hollow foundation with

squared shape 𝐵=5m and 2m in high; soil medium is dense sand; the

soil lateral boundaries and bedrock depth have been selected to be

11

and respectively.

5B=55m=25m

2,5B=2,55m=12,5m,

(Anastasopoulos 2012, Tabatabaiefar 2016). Checking the slenderness

ratio: ℎ/𝐵 =12,5/5=2,5<3.

Figure 3.1. Dimensions of the prototype

3.2 Setting-up the geometric scaling model

Dimensional characteristics of scale model considering different

scaling factors, basing on the existing dimensions of R202(UTC) are

𝑊’ (m) 15

𝐷’ (m) 12,5

𝐵’ (m) 5

𝐿’ (m) 25

0,25

0,1

1,25

0,75

0,625

1,25

0,3

dimensions Prototype Scale 𝜆 = 1: 20 Minimum Maximum

- 𝐵 = 0,25

0,75 𝐿∗ = 1,85 𝑊∗ = 1,50 𝐷∗ = 0,7

- ℎ𝑓 = 0,1

22m, the scale factor 𝜆 =1:20 (𝑛=20) is selected. Hence, foundation’s dimensions: 𝐵 × 𝐵=25cm×25cm, ℎ𝑓=10cm (figure 3.2(a)). Table 3.1. Container dimensions ℎ′𝑓 (m) 2

(a) System of soil-foundation

(b) System of soil-foundation- superstructure

12

Figure 3.2. Pictures of tests The experimental soil, the yellow sand of Lo river was filled in the

is 18mm, 𝐷50=0,42mm, uniformity coefficient

soil container, width 0,7m. The soil properties were obtained by LAS- XD381: 𝐷𝑟=82%, 𝜌 = 2,68 𝑔/𝑐𝑚3 và 𝜑 = 42,6𝑜, maximum grain is 4,67, size permeability coefficient is 2,69 × 10−4cm/s.

Superstructure, weight 120000𝑘𝑔/𝑛3 = 120000𝑘𝑔/203 = 150𝑘𝑔, height 0,25m. Column is a short steel beam with H100 cross-

section connecting the two massive blocks, height 0,4m (figure

3.2(b)).

3.3 Shaking loads

According to TCVN 9386:2012 specification, maximum

earthquake acceleration in the west north of Vietnam: ag=

0,12g÷0,24g. In this thesis, were used for the experiments with maximum acceleration increased: 0,5𝑚/𝑠2, 1,0𝑚/𝑠2, 1,5𝑚/𝑠2, 2,0𝑚/𝑠2 and 2,5𝑚/𝑠2 in soil-foundation model test series; 0,1𝑚/𝑠2, 0,2𝑚/𝑠2, 0,4𝑚/𝑠2, 0,8𝑚/𝑠2, 1,4𝑚/𝑠2 and 2,0𝑚/𝑠2 in soil- foundation-superstructure model test series. The tests were conducted

13

in one horizontal direction. The motion records were derived from the

Tolmezzo earthquake (Friuly, Italia).

3.4 Sensor set-up

In test of soil-foundation system, two accelerometers placed on the

foundation model, figure 3.3(a). In test of soil-foundation-

superstructure system, an accelerometer and a displacement sensor

(a) System of soil-foundation

(b) System of soil-foundation- superstructure

placed on the top of the superstructure, figure 3.3(b).

Figure 3.3. Location of gauges

3.4 The experimental results

3.4.1 The systems of soil-foundation

Table 3.2 summaries the errors of maximum value of acceleration

between foundation and shaking table.

50% embedment depth

100% embedment depth

Test

0% embedment depth Error to shaking table (%) 45,20

Error to shaking table (%) 44,10

Error to 0% embedment depth (%) -0,76

Error to shaking table (%) 57,00

Error to 0% embedment depth (%) 8,13

T12

49,60

34,20

-10,29

56,93

4,90

T13

16,15

22,65

5,60

23,75

6,54

T14

Table 3.2. The error of acceleration in soil-foundation test model

50% embedment depth

100% embedment depth

Test

0% embedment depth Error to shaking table (%) 18,80

T15

Error to shaking table (%) 24,08

Error to 0% embedment depth (%) 4,44

Error to shaking table (%) 11,76

Error to 0% embedment depth (%) -5,93

14

3.4.2 The systems of soil-foundation-superstructure

Test T25 (𝑎𝑚𝑎𝑥 = 1,4𝑚/𝑠2): because the superstructure got high displacement (27,80mm), structure toppled at 6,17s in 0% embedment depth case (figure 3.4). Test T26 (𝑎𝑚𝑎𝑥 = 2,0𝑚/𝑠2), because the superstructure got high displacement (24,58mm), structure toppled at

)

m m

l

( t n e m e c a p s i D

Time (s)

) 2 s /

m

l

( n o i t a r e e c c A

Time (s)

Figure 3.4. Responses of superstructure in test T25

4,35s in 0% embedment depth case (figure 3.5).

(a)

(b)

(c) Figure 3.5. Residual foundations at the end of seismic excitation in T26 with different embedment depths: (a) 0%, (b) 50%, (c) 100%

)

m m

l

( t n e m e c a p s i D

Time (s)

15

) 2 s /

m

l

( n o i t a r e e c c A

Time (s)

Figure 3.6. Responses of superstructure in test T26 Table 3.3. The maximum values of superstructure’s displacements in

16

50% embedment depth

100% embedment depth

0% embedment depth

Test

Value (mm)

Value (mm)

Value (mm)

0,6290 1,336 4,707 9,878

0,5446 0,9768 3,478 8,629

0,4662 0,9136 2,997 7,489

Error to embedment depth (%) -25,88 -31,62 -36,33 -9,97

Error to embedment depth (%) -13,42 -26,89 -26,11 -12,64 Toppled Toppled

T21 T22 T23 T24 T25 T26

soil-foundation-superstructure test model

Table 3.4. The maximum values of superstructure’s acceleration in

50% embedment depth

100%embedment depth

0% embedment depth

Test

Value (𝑚/𝑠2)

0,182 0,222 0,357

Value (𝑚/ 𝑠2) 0,175 0,256 0,460

Error to 0% embedment depth (%) -3,85 15,32 28,85

Value (𝑚/ 𝑠2) 0,265 0,394 0,728

Error to 0% embedment depth (%) 45,60 77,48 103,92

T21 T22 T23

soil-foundation-superstructure test model

50% embedment depth

100%embedment depth

0% embedment depth

Test

Value (𝑚/𝑠2)

T24 T25 T26

0,457 0,9579 1,229

Value (𝑚/ 𝑠2) 0,976 1,352 1,242

Error to 0% embedment depth (%) 113,57 41,14 9,72

Value (𝑚/ 𝑠2) 1,107 1,263 1,427

Error to 0% embedment depth (%) 142,23 31,85 11,83

17

Summary of Chapter 3

- The experimental specimens and soil container were designed.

They are suitable to macro-element model, soil-structure interaction

models and the existing shaking table of University of Transport and

communications. The properties of Lo river sand which filled in the

soil container were defined in lab.

- The shaking table tests were conducted with: earthquake

excitations in long direction, acceleration amplitudes increase.

- The results of soil-foundation system test shown: the maximum

accelerations of foundation are higher than accelerations of shaking

table, the maximum accelerations of foundation are different from

various embedment depths.

- The results of soil-foundation-superstructure system test shown:

the displacements of superstructure are reduced and the accelerations

of superstructure are decreased by embedment depth of foundation.

So, embedment depth of foundation affects to response of

superstructure.

18

Chapter 4. SEISMIC ANALYST OF STRUCTURES WITH

MACRO-ELEMENT

4.1 Seismic responses of soil-foundation systems

The numerical model parameters of soil-structure system in table

4.1 are put into equation (2.22). The errors of test T13-00 (-7,58%)

and T14-00 (-8,70%) are all smaller than 10% (table 4.2).

Value 202,68 × 106 338,48 × 106 1,26 × 103 15 28,05 0,43 4

Parameters 𝑘𝑟 (Nm/rad) 𝑐0 (Ns/m) 𝑐𝑣 (Ns/m) 𝐽 (kgm2) 𝜇 𝜉 

Value 201,74 × 105 1,34 × 105 2,42 × 105 90,625 × 10−3 0,682 0,95 6

Parameters 𝑘0 (N/m) 𝑘𝑣 (N/m) 𝑐𝑟 (Ns/m) 𝑚0 (kg) 𝑁𝑚𝑎𝑥 (kN) 𝜓 𝜆

Table 4.1. Numerical model parameters used in dynamic analyses

Figure 4.1. The zooms of time history acceleration

19

Table 4.2. Errors of acceleration’s maximum values in soil-

T12-00 19,21

T13-00 -7,58

T14-00 -8,70

T15-00 -14,41

foundation test Test Error (%)

4.2 Seismic responses of soil-foundation-superstructure systems

ℎ (m)

Parameter 𝑚1 (kg)

𝑘1 (N/m)

𝑐1 (Ns/m) 0

𝐽 (kgm2) 50,12

𝑁𝑚𝑎𝑥/ 𝑁 16,64

150

0,575

120,88× 105

Table 4.3. Numerical model parameters used in dynamic analyses

27,8𝑚𝑚−23,64𝑚𝑚 27,8𝑚𝑚

× 100%=14,96%. With

Value The numerical model parameters of soil-structure system in tables 4.1 and 4.2 are put into equation (2.23). With test T25-00 (𝑎𝑚𝑎𝑥 = 1,4𝑚/𝑠2), the error of displacement’s maximum value between simulation (23,64mm) and experimentation (27,80mm) at toppled moment: test T26-00 (𝑎𝑚𝑎𝑥 = 2,0𝑚/𝑠2), the error of displacement’s maximum value between simulation (22,67mm) and experimentation (24,58mm): 24,58𝑚𝑚−22,67𝑚𝑚 24,58𝑚𝑚

× 100%=7,77%.

Figure 4.2. Responses in time history of T26-00

Table 4.4. Errors of maximum values in soil- foundation-

Test

Measurement

T25-00

T26-00

Toppled

Displacement Acceleration

T21-00 T22-00 T23-00 -7,07% 8,71% 5,84% -4,29% -4,31% 4,03%

T24-00 -10,11% -13,97%

-5,77%

-12,28%

superstructure test

20

4.3. Effects of 𝑲𝑺 to the responses of superstructure

In Chapter 2, thesis has set-up the Newmark time integration

scheme, equation (2.23). However, with the same model, but Paolucci

[ ] 𝒙𝑛+1 + 𝑭𝑛+1(𝒙𝑛+1) = 𝒑𝑛+1 + 𝑴 [ 𝒙̈ 𝑛 + (1997, 2008) wrote the Newmark time integration scheme as: 1−2𝛽 2𝛽

𝛾 − 1) 𝒙̈ 𝑛∆𝑡 + ( 𝛽

𝛾 𝛽∆𝑡

𝑴 𝛽(∆𝑡)2 + 𝒙̇ 𝑛∆𝑡+𝒙𝑛 𝛽(∆𝑡)2 ] + 𝑪 [(

𝑪𝛾 𝛽∆𝑡 𝛾 2𝛽

(4.1) − 1) 𝒙̇ 𝑛 + 𝒙𝑛]

(a) T21-00

(b) T24-00 Figure 4.3. Time histories of superstructure’s acceleration

𝑴 𝛽(∆𝑡)2 +

𝑪𝛾 𝛽∆𝑡

The stiffness 𝑲𝑆 in equation (4.1) was not mentioned, while the + 𝑲𝑆] 𝒙𝑛+1. Coming first term of equation (2.23) wrote: [ up nest, thesis surveys the effects of 𝑲𝑆 (the column stiffness 𝑘1 in experimentation) to the lateral acceleration and displacement in test T21-00 and T24-00.

21

The acceleration results from experimentation, present Newmark

scheme and Paolucci’s equation are shown in figure 4.3, these

diagrams are suitable at important points and general shapes. In figure

4.4, Paolucci’s displacement diagrams linearly increase and are far different from other results.

(a) T21-00

(b) T24-00 Figure 4.4. The displacements in time history of superstructure

Summary of Chapter 4

(i) Two models which were tested in have been defined in

numerical analysis. (ii) The systems of soil-foundation and soil-

foundation-superstructure are numerically simulated under seismic

loading. In these simulations, the seismic loadings are the time

histories of shaking which got in Chapter 3. (iii) The numerical results

in soil-foundation test shown: the errors of maximum acceleration

22

between simulation and experimentation in test T13-00 is -7,58%, test

T14-00 is -8,70%, test T15-00 is -14,41%. The numerical results in

soil-foundation test shown: the errors of maximum acceleration

between simulation and experimentation are all less than 15%. (iv) The Newmark time integration scheme with stiffness 𝑲𝑆 is suitable to and displacements of accelerations analyzing lateral the

superstructure.

CONCLUSION AND RECOMMENDATION

I. Conclusions

From the detailed results, the following can be concluded:

- In this thesis, a new macro-element for modelling the behavior of

soil-shallow foundation interaction under seismic loading have been

represented. The proposed macro-element considered simultaneously

the effect of material and geometric nonlinearities on the response of

soil-foundation.

- In the experimentations of soil-foundation system: the

accelerations of foundation are all higher than the accelerations of

shaking table (T14-00 is 16,15%, T15-00 is 18,80%; T14-50 is

22,65%, T15-50 is 24,08%; T14-100 is 23,75%, T15-100 is 11,76%);

the accelerations of foundation are increased by embedment depth of

foundation (T14-50 is 5,60%, T15-50 is 4,44%, T14-100 is 6,59%;

T25-50 is 41,14%, T26-50 is 9,72% T25-100 is 31,85%, T26-100 is

11,83%).

- The responses of foundation were gotten from analyses of soil-

foundation system by proposed macro-element and CyclicTP

software: the error of maximum displacement is 7,85%, the error of

23

maximum acceleration is 0,04%; the simulations with proposed

macro-element are simple and save computational cost.

- The comparison between simulation and experiment results

shown that this model was suit for simulating the couple of material

and geometric behaviors of shallow foundation under seismic loading.

The errors of maximum acceleration value in soil-foundation test:

T13-00 is -7.58%, T14-00 is -8,70%, T15-00 is -14,41%. The errors

of maximum value in soil-foundation-superstructure test,

displacements of T25-00 is 14,96% and T26-00 is 7,77%,

accelerations of T25-00 is -5,77% and T26-00 is -12,28%.

- The detailed numerical results from proposed Newmark time

integration scheme and Paolucci (1997, 2008) with test T21-00 and test T24-00 shown: the present scheme with appearance of 𝑲𝑆 is suitable to analyzing the lateral acceleration and displacement of

superstructure.

New contributions of the thesis:

- The proposed macro-element for modelling the behavior of soil-

shallow foundation interaction under seismic loading have been

presented. The proposed macro-element considered simultaneously

the effect of material and geometric nonlinearities on the response of

soil-foundation.

- In this work, shake table simulations were performed using a

scaled model to investigate the responses of soil-structure interactions

with varying excitations and different embedment depths.

- The new Newmark time integration schemes were set-up for soil-

structure interaction analysis.

24

- In this thesis, the numerical and experimental results are lateral

accelerations and displacements which not mentioned in past

researches.

II. Recommendations

- Interesting extensions of the macro-element model can among

others deal with: toppling limit of structure, other types of foundation

and soil medium, coupling of displacement-rock responses.

- Interesting extensions of the experimental model can among

others deal with: the uplift and non-linear material of soil-foundation

interfaces, other response of structure (displacement and acceleration

of foundation, settlement, rocking, …), …

PUBLISHCATIONS

1. Huỳnh Văn Quân, Nguyễn Xuân Huy và Nguyễn Trung Kiên, Ứng

xử của kết cấu chịu tác dụng động đất có xét đến tương tác phi

tuyến đất nền-kết cấu, Tuyển tập công trình khoa học Hội nghị Cơ

học toàn quốc lần thứ X, Học viện Kỹ thuật quân sự, Hà Nội, 8-

9/12/2017, Tập 3, tr. 918-925.

2. Huỳnh Văn Quân, Nguyễn Xuân Huy và Nguyễn Trung Kiên

(2018), Mô hình phi tuyến hình học biến dạng nền trong phân tích

ứng xử kết cấu chịu tải trọng động đất, Tạp chí Khoa học Giao

thông Vận tải, 66, tr. 3-11.

3. Van Quan Huynh, Xuan Huy Nguyen, Trung Kien Nguyen,

Seismic analysis of structures considering geometrical non-

linearity of soil-structure interaction by spatial macro-element,

International Conference on Sustainability in Civil Engineering,

University of Transport and Communications, Vietnam, 24-

25/12/2018, pp. 379-383.

4. Van Quan Huynh, Xuan Huy Nguyen, Trung Kien Nguyen (2020),

A macro-element for modelling the non-linear interaction of soil-

shallow foundation under seismic loading, Civil Engineering

Journal, 6(4), pp. 714-723. DOI:

https://www.civilejournal.org/index.php/cej/article/view/2120.