A STUDY ON PILE WORKING UNDER HORIZONTAL LOAD AND SEISMIC LOAD

Ngo Quoc Trinh 12/2012, Using the comparison method to study the interaction problem between the pile and the elastic foundation under horizontal static load, 9th Nationwide Mechanics Co

MINISTRY OF TRAINING AND

EDUCATION

MINISTRY OF CONSTRUCTION

HANOI ARCHITECTURAL UNIVERSITY

NGO QUOC TRINH

A STUDY ON PILE WORKING UNDER

HORIZONTAL LOAD AND SEISMIC LOAD

Major: Civil and Industrial Construction Engineering

Code : 62 58 02 08

SUMMARY OF Eng.D THESIS

HÀ NOI - 2014

The study was completed at:

HANOI ARCHITECTURAL UNIVERSITY

Scientific supervisors:

1 Supervisor: Asst Prof Dr Vuong Van Thanh

Dr Tran Huu Ha

Reviewer 1: Prof DSc Nguyen Dang Bich Reviewer 2: Prof.Ph.D Do Nhu Trang Reviewer 3: Asst Prof Dr Trinh Minh Thu

The thesis will be defended in front of Eng.D Assessment Council at University level held in Hanoi Architectural University

At: date month year 2014

Thesis can be found at:

• National Library of Vietnam

• Library of Hanoi Architectural University

LIST OF WORKS BY AUTHOR

1 Ngo Quoc Trinh (2008), Study on the interaction problem

between shallow foundation and ground deformation, Vietnam

Road and Bridge Magazine

2 Ngo Quoc Trinh, Vuong Van Thanh, Tran Huu Ha (5/2012)

Study on the interaction problem between ground mass and

elastic foundation underhorizontal static load Vietnam Road

and Bridge Magazine

3 Ngo Quoc Trinh, Vuong Van Thanh, Tran Huu Ha (6/2012)

Study on the interaction problem between single pile and

elastic foundation underhorizontal static load Vietnam Road

and Bridge Magazine,

4 Ngo Quoc Trinh, Vuong Van Thanh, Tran Huu Ha (11/2012),

Using the solution of Mindlin to build the interaction problem

between the pile and the elastic foundation under horizontal

static load, Collection of Conference on materials science,

structure and construction technology in 2012 (MSC2012),

Hanoi Architectural University

5 Ngo Quoc Trinh (12/2012), Using the comparison method to

study the interaction problem between the pile and the elastic

foundation under horizontal static load, 9th Nationwide

Mechanics Conference

6 Ngo Quoc Trinh (3/2013), Study on Love wave transmission

problem in the foundation when earthquake occcurs Tranposrt

Journal

INTRODUCTION

1 Background

Though Vietnam is not located in the Ring of Fire of large earthquake areas in the world, it is still affected by strong earthquakes because the territory of Vietnam exists many faults acting complexly such as Lai Chau - Dien Bien fault, Ma River fault, Son La fault, Hong River fault zone, Ca River fault zone (in the history, there was a strong earthquake on 6.8 Richter scale) In order

to design earthquake resistance for buildings, our country now is using some foreign translated standards: TCXDVN 375: 2006; 22 TCN 221-95 ; TCXD 205-1998 ; 22TCN 272- 05; however there is little detailed guidance on calculating the interaction between the building and the foundation

The biggest difficulty when designing pile foundations under horizontal load and seismic load is assessing the interaction between the pile and the foundation Because the interaction between the pile and the foundation is too complex, the current calculating methods are often simplified by the model (Winkler model; continuous elastic model).Therefore, it is difficult to determine the interaction coefficients between the pile and the foundation (spring coefficient, viscosity coefficient), it is difficult to ensure the boundary conditions

as well as endless radiation conditions; and the interaction between the pile and the foundation is insufficient and it is considered only in plane strain problem

From above analysis, the study on the work of the pile, in which the study on the interaction between the pile and the foundation under horizontal load and seismic load is necessary, scientific and practical, contributing a fuller review of the calculating method of the pile foundation of constructions in Vietnam

2 Study objectives

Formulate theoretical methods to study interaction problems between pile – soil foundation and formulate calculating softwares to determine the status of the stress- strain of pile under horizontal load and seismic load

3 Subject and scope of the study

The thesis studies vertical single pile located within an

infinite elastic space under the effect of horizontal static load,

horizontal dynamic load and seismic load

The thesis does not calculate in other foundation models

(elastoplastic, visco-elastic), does not consider the liquefy

phenomenon in the foundation when an earthquake occurs, does not

consider the effect of pore water pressure in saturated foundation and

does not study the limit state problem of pile

4 Study content

Study stress-strain state of the soil mass under horizontal

static load

Study static interaction problem between pile and foundation

under horizontal static load

Study dynamic interaction problem between the pile and the

foundation under horizontal dynamic load and seismic load in the

frequency domain and time domain

Formulate calculating softwares for above cases of study

5 Study method

Formulate theoretical problem by using comparison method

of extreme principle Gauss (hereinafter refering to EP Gauss) when

using the static solution of the semi-infinite elastic space (for all

static interaction problem) and dynamic solution of infinite elastic

space (for the dynamic interaction problem) as comparison system

Using the finite limit element method to solve and basing on the

numerical results obtains the results proving the correctness and

reliability of theoretical calculation

OVERVIEW OF STUDY METHODS ON THE INTERACTION

BETWEEN PILE AND FOUNDATION UNDER

HORIZONTAL LOAD Basing on the annalysis of study methods on the interaction

between pile and foundation under horizontal load, we can draw

some comments as follows:

+ First: It’s dificult to determine stiffness coefficient “linear

springs”, “nonlinear springs” (p-y curve), viscosity coefficient

+ Second: It’s dificult to determine the boundary conditions

at infinity, especially for the wave transmission problem when an earthquake occurs

+ Third: The interaction between the pile and the soil has not been adequately considered, only consider the effect of soil on the pile without considering the effect of pile on the soil

+ Fourth: Mainly study on plane strain problem

From above issues, the author has based on the method of using comparison system of extreme principle Gauss method to formulate the interaction problem between pile and foundation under static load, dynamic horizontal load and seismic load with full consideration of the boundary conditions and endless radiation conditions as well as consideration of the full interaction between the pile and the soil and consider the 3-dimensional problem

Chapter 2 STUDY ON STRESS-STRAIN OF FOUNDATION UNDER

STATIC HORIZONTAL LOAD 2.1 The basic equations and the wave propagation equation of elastic medium

2.1.1 The basic contact of elastic medium 2.1.2 Formulate diferential balance equation and wave propagation equations according to EP Gauss

2.1.2.1 Extreme principle Gauss Extreme principle Gauss (EP Gauss) is an extreme principle

of mechanics stated by Gauss K.F (1777 - 1855) in 1829 with the following content [5][6],[61]: “The motion of the system of points, optionally linked influenced by any force, in every moment occurred

in accordance with the highest possible ability to move but the quality can be done if they are completely free, which means it occurs with the smallest amount of coercion if forced volume measurements in very little time period taken by the total volume of the mass of each point of the square deviation of the position they compared with when they are free”

Expression of forced amount in geometry form of EP Gauss is written as follows:

i i i

m

HereBiCi2is the distance between 2 points Bi and Ci of material

point i with the volume mi Bi is the position which material point i

obtain when moving freely and Ci is the position when that material

point moves with the connection after extremely little time dt

Symbol Σ is the total number of points obtained from the system

EP Gauss is applied for material point system Basing on this

principle, in 1979, Prof.Ph.D Ha Huy Cuong recommended using

Extreme principle Gauss to solve the problem of mechanical

deformation of solid objects

2.1.2.2 Formulate diferential balance equation

Departs from Helmholtz theorem [60], for continuous

environment, establishing three movements: forward motion, motion

and rotation distortion

From EP Gauss for mechanics material points, applying EP

Gauss method for moving elastic deformation elements distributed

3D, the author received three diferential balance equations of elastic

systems (Navier equation) like diferential balance equations

presented in many documents about elastic theory

[26],[46],[53],[60]

2.1.2.3 Formulate wave propagation equation

Applying EP Gauss method for motion of volumetric strain

and rotation like abrasive of distributed elements around the axis of

x, y, z, the author got 4 of wave propagation equations (2.25), (2.33),

(2.36), (2.37)

Thus Navier equations or wave propagation equatations to

study the motion of elastic medium

2.2 The solution for an infinite elastic space and half-infinite

elastic space

2.2.1 The solution for an infinite elastic space (solution of

Kelvin)

2.2.2 The solution for half-infinite elastic space (solution of

Mindlin )

2.3 Formulate the interaction problem between elastic soil mass

and half-infinite elastic space

2.3.1 Comparison system is infinite elastic semi-space

In terms of rectangular soil mass V with elastic parameters

E1, ν1 in the elastic half-space with the elastic parameters E0, ν0 Horizontal force P affects in or outside the ground Considering the comparision system as the infinite half space elastic with the elastic parameters E0, ν0, also under P horizontal force affected as the system needs calculating (Figure 2.5)

Figure 2.4 Model of problem ofcalculating the elastic soil mass in the infinite elastic half-space

Figure 2.5 Comparison system is half-infinite elastic space

Note that on the boundary of the soil mass needed calculating, there is prestress σij affect (figure 2.4) and on the boundary of the soil mass of comparison system, there is prestress

σij0 affects (figure 2.5)

Using prestress state σij0 of the known comparison system to calculate the prestress state σij of the system needed calculating by wrtiting forced density as follows:

Z V =⌡

V*

(σ x -σ x ) ε x dV* +⌡

V*

(σ y -σ y ) ε y dV* +⌡

V*

(σ z -σ z ) ε z dV* +⌡

V*

(τ xy -τxy0 ) γxy dV * +⌡

V*

(τxz-τxz0 ) γxz dV * +⌡

V*

(τyz-τyz0 ) γyz dV * →min

(2.50)

In (2.50), V* is volume of extended soil to consider the boundary conditions; V is volume of soil mass needed calculating (V*> V); εx,

εy,εz, γxy, γxz, γyz are the strain of the soil mass; σx, σy, σz0, τxy0, τxz0,

τyz0 are the stress-strain of definite comparison system according to the solution of (figure 2.5); prestresses σx, σy, σz, τxy, τxz, τyz are the stress-strain of the soil mass of calculated system (figure 2.4) Replace the distortions by the contacts (2.1) EP Gauss considers actual displacement u, v, w in (2.51) as the virtual displacement, i.e whether the distortion is independent on the prestress, the extreme conditions of (2:51) is written as follows:

Extended area to examine boundary condition

P Soil mass

E0,

E1, ν c

Compared soil mass

P

E0,

ν

E0, ν c

δZ V =⌡

V*

(σ x -σ x ) δ( ∂x∂u)dV* +⌡

V*

(σ y -σ y ) δ( ∂y∂v)dV* +⌡

V*

(σ z -σ z ) δ(∂w ∂z)dV*

+⌡⌠

V*

(τ xy -τ xy0) δ (∂u

∂y+

∂v ∂x ) dV

*

+⌡⌠

V*

(τ xz -τ xz0) δ (∂u

∂z +

∂w ∂x ) dV

*

+⌡⌠

V*

(τ yz -τ yz0) δ (∂v

∂z+

∂w ∂y ) dV

*

in which δ is marks obtaining variational

Note that soil mass here contains three implicits u, v, w, so

from (2.52) we obtained 3 equation system:

⌡

V*

(σ

x -σx) δ(∂u

∂x)dV

* +⌡⌠

V*

(τ xy -τxy0 ) δ(∂u

∂y) dV

* + ⌡⌠

V*

(τ xz -τxz0 ) δ(∂u

∂z )dV

* = 0

⌡

V*

(σ y -σ y ) δ( ∂y∂v)dV* +⌡⌠

V*

(τ xy -τ xy0) δ(∂v

∂x)dV

*

+⌡⌠

V*

(τ yz -τ yz0) δ(∂v

∂z )dV

*

=0 (2.53)

⌡

V*

(σ z - σ z ) δ(∂w ∂z)dV * +⌡⌠

V*

(τ xz -τ xz0) δ(∂w

∂x)dV

* + ⌡⌠

V*

(τ yz -τ yz0) δ(∂w

∂y)dV

* = 0

Make variational calculations [34] for the (2.53) we received three

following equations:

∂σx

∂x +

∂τxy

∂y +

∂τxz

∂z =

∂σx

∂x +

∂τxy0

∂y +

∂τxz0

∂z

∂σy

∂y +

∂τxy

∂x +

∂τyz

∂z =

∂σy

∂y +

∂τxy0

∂x +

∂τyz0

∂z (2.54)

∂σz

∂z +

∂τxz

∂z +

∂τyz

∂y =

∂σz0

∂z +

∂τxz0

∂z +

∂τyz0

∂y The right side (2.54) satisfies the balance equation when a

horizontal force P in comparison system causes (Figure 2.4), so the

left side of (2.54) is also the balance equation when a horizontal force

P affecting in the calculated system (Figure 2.3) causes

Thus, by using the comparison system, we obtained three

equilibrium equations of the system needed calculating

2.3.2 Comparison system is infinite elastic space

- Considering the case in which force P effects horizontally

on soil V (Figure 2.8a) AB surface is the free surface

Let horizontal force P acting on the elastic space, using Kelvin

solution to calculate the prestress state σij in it.Because the system

needed calculating lies in half-space (Figure 2.8a), we should only

use the bottom half of infinite space (Figure 2.8b)

(a) (b) (c)

Figure 2.8 The problem model of soil mass under horizontal force

when using the comparison system as infinite elastic space The stress state σij is equivalent to the force P / 2, so we have to put 2 forces P to calculate prestress σij0 according to Kelvin solution In case of the horizontal force P placed at the depth c compared with the free surface, we use 2 forces P symmetrically placed symmetrically through surface AB (Figure 2.8c) When calculating the above diagram, on the surface AB there is also the effect of prestress σz0

Mindlin solution for the elastic half-space under horizontal force P derives from Kelvin solution with calculating diagram as in Figure 2.8c and finds the way to ensure σz0 = 0 on the surface AB The solution obtained is the calculus solution

The author uses the diagram in figure 2.8c to calculate σij Due to the effect of prestress σz0on the surface AB of the bottom half, it is necessary to consider the effect of this variable by writing forced amount as follows:

ZAB =

⌡Ω

AB

[(σz-σz0)wdΩAB → min (2.55)

with ΩAB the surface area of AB

Besides, it is necessary to ensure the condition of σz = 0 on the surface AB In a nutshell, the problem determining the prestress state of soil mass V when using Kelvin solution is written as follows:

With constraint σz = 0 on the surface AB

Z V =⌡

V*

(σx-σ x ) ε x dV* +⌡

V*

(σy-σ y ) ε y dV* +⌡

V*

(σz-σ z ) ε z dV* +⌡

V*

(τ xy -τ xy0) γ xy dV * +⌡

V*

(τ xz -τ xz0) γ xz dV * +⌡

V*

(τ yz -τ yz0) γ yz dV * → min

(2.57)

P

c c

P

Examined soil

0

In (2.57) prestresses σx0, σy0, σz0, τxy0, τxz0, τyz0 are the

prestress state of the comparison system determined according to

Kelvin solution with two forces P (Figure 2.8c) By writing extended

functional Lagrange, we turn constrained extreme problem into

unconstrained extreme problem as follows:

λ = λ(x,y) is a Lagrange factor as a new implicit function of

the problem

Extreme conditions of F would be:

2.4 Solve the problem by using the finite element method

The soil mass to be calculated as well as the soil mass of the

comparison system is divided into rectangular elements ( 3D problem

) having any particle size In order to consider the boundary

conditions on the system to be calculated, the comparison system has

1 more number of elements than the system to be calculated

according to the depth z and direction x, direction y We can use

rectangular elements with 8 nodes [38], but to get a better

approximation, the author used rectangular elements with 20 nodes in

the natural coordinate system with particle size ∆x = ∆y = ∆z = 2 and

using displacements as unknowns

Each node has 3 parameters (unknown) to be determined as

the displacement u according to direction x, v according to direction

y, w according to direction z Thus, the elements has 3 x 20 = 60

displacement parameters (60 unknowns) to be determined Knowing

the displacement of the nodes, the displacements at any point within

the element is determined according to the interpolation function

[39], [60]

2.5 Check the results and review

2.5.1 Problem using the comparison system as infinite elastic

semi-space

Considering the interaction problem between soil mass V

having elastic parameters E1, ν1 with infinite elastic half space having

elastic parameters E0,ν0 (Figure 2.12) Based on Matlab software, the

author formulated the calulating program Mstatic1 to survey

following cases:

* Case 1: Put E1 = E0, ν1 = ν0

Figure 2.13 Horizontal displacement chart of the soil mass when horizontal force P affects on surface (a) andbottom (b) of the land mass, in case E1 = E0 ; ν1 = ν0

Recognizing that the results calculated by EP Gauss completely coincide with the results of the calculus solution of Mindlin (see Appendix 1)

When changing the volume of block V, even in case that block

V only has 1 element, we still get accurate results

* Case 2: Put ν1 = ν0; E1 ≠ E0 (E1 retained as in case 1, E0 changed of the comparison system)

Figure 2.14 Horizontal displacement chart of the soil mass when horizontal force P affects on surface (a) andbottom (b) of the soil mass, in case ν1 = ν0 ; E1 ≠ E0

Here we find a entirely concurrence between two results according to the EP Gauss solution in case 1 and case 2 When the volume V changes, we still get accurate results as above

Thus, through two survey cases we found that, though the comparison system has the same or different elastic module compared with the elastic module of comparison system to be

calculated, the displacement results of the system to be calculated is

constant This shows the correctness and reliability of theoretical

calculations

2.5.2 The problem of comparison system is infinite elastic

space

Surveying the land mass having E1 = E0, μ1 = μ0 when let the

horizontal force P affect alternately on 3 locations: c = 0 (the free

surface of the land mass), c = 3m c = 5.4 m (bottom of the land mass)

by two calculating programs Mstatic1 (comparison system is the

infinite semi-space); Kstatic1 (comparison system is infinite space)

Hình 2.18 Horizontal displacement chart of the soil mass

calculated according to 2 programs Mstatic1 and Kstatic1 when

horizontal load P affects on the position c=0 (a); c=3m (b); c=5.4m

(c)

Calculating results show that displacement of the land mass when

calculating according to Kstatic1 is approximately equal to the

displacement of the land mass calculating according to Mstatic1 with

the largest error of about 6% and the deeper the force is put

compared with the free surface, the smaller the error between the two

results is and almost overlap Thus, through the numerical solution by

finite element method, we can turn the solution of infinite elastic

space (Kelvin solution) into the solution of infinite elastic

senmi-space (Mindlin solution)

2.6 Conclusion of chapter 2

1- Formulating the interaction problem between the land mass with the remain infinite elastic semi-space under static horizontal load With the conditions of displacement and prestress on the boundary surfaces of the land mass automatically satisfied exactly, there is no need to add extra links (i.e spring links) as the current methods and the condition at infinity is automatically satisfied

2 - Formulating calculating program by using the finite element method in Matlab environment to calculate the land mass Here use the rectangular element with 3-D, 20 nodes Checking the numerical solution, we found a good fit between the calculated results with calculus solutions

3 - Through numerical solution, we can turn the solution of infinite elastic space (Kelvin solution) into solution of half infinite elastic semi-space (Mindlin solution)

Chapter 3 STUDY ON THE INTERACTION PROBLEM BETWEEN PILE AND FOUNDATION UNDER THE HORIZONTAL

STATIC LOAD 3.1 Timoshenko beam theory

Timoshenko beam theory is the bending beam theory considering the horizontal shear strain Beam theory which considers current horizontal shear strain used two implicit functions uc(z); φc(z)

is independent implicit function often leads to Shear locking phenomenon (Shear locking) In the thesis, the author used Timoshenko beam theory but using two implicit functions which are the deflection uc(z) and shear force Q(z) in the pile According to this method, there will be no longer Shear locking phenomenon

3.2 Formulating bending beam problem considering horizontal shear strain according to EP Gauss

By EP Gauss, the author formulated properly the deflection equation of bending beam considering the horizontal shear strain 3.3 Finite element method for beam considering the horizontal shear strain

Because there are two implicit functions, displacement function and shear force function of the beam, there are two types of elements: displacement element and shear force element

Displacement element includes 2 nodes, each node has two unknown

displacement and rotation, shear force element includes 3 nodes, each

node has one unknown shear force And ground element is

rectangular element with 20 nodes, each node has three

displacements u, v, w

3.4 Formulating interaction problem between single pile with

foundation under horizontal static load

3.4.1 In case of using comparison system as infinite elastic

semi-space

(a) System to be calculated (b) System to be compared

According EP Gauss, forced density Z of the problem includes

two components: Z = Zd + Zc → min

Zd: forced account considering prestress state of the land

mass of the comparison system which affects on the land mass

containing piles of the system to be calculated (2.50)

Zc considering the forced amount of bending pile considering

horizontal shear strain γc in the pile

Zc = ⌡

l

Mχcdz + ⌡

l

The condition ensuring the simultaneous work of the pile

under the horizontal force compared with the foundation is that the

horizontal displacement of the pile uc is equal to the horizontal

displacement of the foundation u at the heart of the pile

We have: uc(z, xc, yc) = u(z, xc, yc) (3.49)

We can lead the constrained extreme problem to unconstrained

extreme problem by using Lagrange factor λ (z) The function λ (z) is

implicit function to be calculated which changes according to the

length of the pile The Lagrange extended function F is now written

as follows:

F= Zd + Zc + ⌡l λ (z) (uc-u)dz → min (3.50)

Extended area to examine boundary condition P

Land mass contains pile

Pile

E 0 , ν 0

E1, ν1

P

E0, ν0

E0, ν0

Comparative soil mass

3.4.2 In case of using comparison system as infinite elastic space

According EP Gauss, forced density Z of the problem includes two components: Z = Zd + Zc → min (3.53)

Zd forced account considering prestress state of the land mass

of the comparison system which affects on the land mass containing piles of the system to be calculated: Zd = ZV + ZAB;

ZV is the forced amount to calculate land mass V (formula 2.50);

ZAB is the forced amount considering surface condition AB of the bottom half of the land mass: ZAB =

⌡Ω

AB

(σz-σz0)wdΩAB (3.57)

Zc is forced amount (motion) of bending pile considering horizontal shear strain γc in piles (formula 3.46)

Constrained conditions uc(z, xc, yc) = u(z, xc, yc) and σz = 0

on the free surface

We can lead the constrained extreme problem (3.53) to unconstrained extreme problem by using Lagrange factor λ as follows:

F = Zd + Zc+⌡l λ1 (z) (uc-u)dz + ⌡Ω

AB

λ2 (x,y) σz d ΩAB → min (3.59)

3.5 Surveying some cases to test the reliability of the calculating program

3.5.1 Compare the results when let the elastic modules of comparison system be diferent

Figure 3.9 Horizontal displacement diagram (a), bending moment (b) of the pile calculated according to two cases that the comparison system has E0 = 10MPa; E0 = 20MPa

Realizing that the results of two cases are the same.Thus, the Such internal force displacement of pile in the system to be calculated does

not depend on the elastic module of the comparison systems, which

proves that the algorithms provided is absolutely right

3.5.3 Survey the problem compared with the method of

Zavriev (1962) based on the local deformation foundation model

[16]

The author used input parameters of example V.5 in [16]

calculated according to the method of Zavriev to calculate according

to EP Gauss, then compared their results with each other

Table 3.5 Displacement value, maximum bending moment according

to the method of Zavriev and EP Gauss

Result

Method

Maximum displacement on the head of the pile (m)

Maximum bending moment (kN.m) Pile under the

load P, M

Pile under the load P

Pile under the load P, M

Pile under the load P

Comment: The displacement on the head of the pile, the

maximum bending moment calculated according to the method of

Zavriev is approximately equal to the results of The displacement on

the head of the pile, the maximum bending moment calculated

according to Gauss (about 4.1% error)

3.5.4 Survey the problem compared with the method of Poulos

(1971) basing on continuous elastic foundation model [50]

The author used the input parameters of the example 6.10 in [50]

calculated according to the method of Poulos to calculate according

to EP Gauss and then compared their results with each other

Table 3.6 Maximum displacement value on the head of the pile

according to the method of Poulos and EP Gauss

Result

Method

Pile under the loadP,

M Pile under the loadP

The displacement on the head of the pile calculated according to the

method of Poulos is nearly equal to the displacement on the head of

the pile calculated according to EP Gauss (12.7% error)

3.5.5 Survey the problem compared with Kim's research results based on the method of using p-y curve [45]

The author used the input parameters in the study of soft pile of Kim [45] to formulate the KstaticPLs software calculated according

to EP Gauss then compared their results with each other

(a)

(b)

Figure 3.14 Horizontal displacement chart, bending moment of piles calculated according to KstaticPLs (a); Kim, O’Neill, Matlock [45] (b) under the horizontal force effects in turn: 200kN, 400kN, 600kN, 800 kN

Comment: Results of displacement, bending moment of the problem based on the author's solution (KstaticPLs) are consistent with findingf of Kim, O'Neill, Matlock in the different cases of putting force both in shape, values and position that reaches the maximum value, minimum value, bending point

3.6 Survey the parameters affecting the working of single pile under horizontal static load

3.6.1 Survey the change of pile length in homogeneous elastic foundation

Survey the short piles, long piles with reinforced concrete cross section (40x40) cm with the elastic module Ec = 30.000MPa Piles

have 2 different lengths: l =4m and l=16m and under the horizontal

force P = 20kN on the head of the pile (Figure 3.8)

(a) L = 4m (b) L = 16m

Figure 3.18 Chart horizontal displacement, bending moment of pile length

L= 4m (a); L = 16m(b)

Survey results consistent with the results calculated by Matlock

and Reese (1956); Zavriev (1962);Broms (1964) for short pile and

long pile However, according to the author's methodology, just a

computer program can get results directly consider both short pile

and long pile without sorting through piles step short, long pile; the

single assumption simplification in calculations

3.6.2 Single pile examination on hard rock layer

The survey of single pile by reinforced concrete by section

(30x30) cm, length l = 6m with elastic modulus Ec = of 30,000 MPa,

Poisson's ratio νc = 0.25 Pile effects of horizontal force P = 100kN

at the poles Calculation in two cases: uniform piles in the

background; foot pile is plugged in tight limestone with 0.6 m

thickness

Figure 3.19 Horizontal displacement diagram (a), bending moment (b) of

the pile in uniform elastic foundation and is located in the

elastic, hard truth

Comment: when pile leans on the hard rock, horizontal displacement at pile foot is zero and turning point does not appear near the pile foot (Figure 3.19a), and bending moment in piles near the foot up from the pile in uniform background (Figure 3.19b) Thus, the method of comparison used Gauss system can also calculate similar pile against pile (pile tip is contraindicated in hard rock)

3.8 Conclusion of chapter 3

1 - By using method of comparative system has been built by the bending beam deflection equation under transverse shear strain

2 - Develop a fully interactive problem between the pile and soil Therefore, no need to add additional links contained soil piles at the edge, so this approach not only ensures the boundary conditions on the surface of the soil containing piles but also ensure the boundary conditions at infinity, boundary conditions between pile and soil

3 - The problems were solved with the use of the Kelvin and Mindlin solution as comparative system to solve the problem on the horizontal force placed at the top of the pile, the pile foot or in different depths in the range including outside piles Thereby the construction of pile load problem will be research content in the next chapter of the thesis

4 - Results of the problem are compared with the results of a number of traditional methods helps improve the reliability of calculation theories

Chapter 4 STUDY INTERACTION PROBLEM BETWEEN PILE AND FOUNDATION UNDER HORIZONTAL LOAD

AND SEISMIC LOAD 4.1 Solution of pulse units of infinite elastic space 4.2 Hysteretic damping coefficient of soil materials

In the calculation of construction as well as foundation always consider the energy consumption in the fluctuations process and energy consumption which is described by viscous drag Viscous drag is by viscous drag coefficient multiple with velocity

Under the loads, the foundation may appear deformed plastic, but plastic deformation does not depend on the frequency of the load,

so this time instead of the usual viscous drag coefficient, people often