Tài liệu A study on pile working under horizontal load and seismic load

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MINISTRY OF TRAINING AND EDUCATION MINISTRY OF CONSTRUCTION The study was completed at: HANOI ARCHITECTURAL UNIVERSITY HANOI ARCHITECTURAL UNIVERSITY Scientific supervisors: 1. Supervisor: Asst. Prof. Dr Vuong Van Thanh Dr. Tran Huu Ha NGO QUOC TRINH Reviewer 1: Prof. DSc Nguyen Dang Bich Reviewer 2: Prof.Ph.D Do Nhu Trang Reviewer 3: Asst. Prof. Dr Trinh Minh Thu A STUDY ON PILE WORKING UNDER HORIZONTAL LOAD AND SEISMIC LOAD Major: Civil and Industrial Construction Engineering Code : 62 58 02 08 The thesis will be defended in front of Eng.D Assessment Council at University level held in Hanoi Architectural University SUMMARY OF Eng.D THESIS At: ......... date ........ month ......... year 2014. Thesis can be found at: HÀ NOI - 2014 • National Library of Vietnam • Library of Hanoi Architectural University 1 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 2 3 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 + Second: It’s dificult to determine the boundary conditions at infinity, especially for the wave transmission problem when an earthquake occurs. 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. Chapter 1 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. + 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: 4 ∑ m .B .C i i 2 i ⇒ min! 5 (2.4) i 2 i Here Bi C is 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). Extended area to examine boundary condition Soil mass c Compared soil mass c P P E1, ν E0, Figure 2.4 Model of problem ofcalculating the elastic soil mass in the infinite elastic half-space E0, ν E0, ν 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: 0 * 0 * 0 * ZV=⌠ ⌡ (σx-σx ) εxdV +⌠ ⌡ (σy-σy ) εydV +⌠ ⌡ (σz-σz ) εzdV V* +⌡ ⌠ V* (τxy-τxy0) γxy dV* +⌡ ⌠ V* V* (τxz-τxz0) γxz dV* +⌡ ⌠ V* (τyz-τyz0) γyz dV* →min V* (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; σx0, σy0, 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: 6 δZV=⌠ (σx-σx0) δ( ⌡V* ∂u ∂v ∂w )dV* +⌠ (σy-σy0) δ( )dV* +⌠ (σz-σz0) δ( )dV* ∂x ∂y ∂z ⌡ ⌡ V* +⌠ ∂u ∂v ) dV* +⌠ (τxy-τxy ) δ ( + ∂y ∂x ⌡ +⌠ (τyz-τyz0) ⌡V* ⌡V* 7 V* 0 ∂v ∂w δ ( + ) dV* = 0 ∂z ∂y (τxz-τxz0) V* σij0 Examined soil A B P ∂u ∂w δ ( + ) dV* ∂z ∂x A B A P P c cP σz0 B (2.52) 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: ∂u ∂u ∂u 0 * 0 * 0 * ⌠ ⌠ ⌠ ⌡V*(σx-σx ) δ( ∂x)dV +⌡V* (τxy-τxy ) δ( ∂y) dV + ⌡V* (τxz-τxz ) δ( ∂z )dV =0 ∂v ∂v ∂v 0 * 0 * 0 * ⌠ ⌠ ⌠ ⌡V*(σy-σy ) δ( ∂y)dV +⌡V* (τxy-τxy ) δ( ∂x)dV +⌡V* (τyz-τyz ) δ( ∂z )dV =0 (2.53) ∂w ∂w ∂w 0 * 0 * 0 * ⌠ ⌠ ⌠ ⌡V*(σz- σz ) δ( ∂z)dV +⌡V* (τxz-τxz ) δ( ∂x)dV + ⌡V* (τyz-τyz ) δ( ∂y)dV =0 Make variational calculations [34] for the (2.53) we received three following equations: ∂σx ∂τxy ∂τxz ∂σx0 ∂τxy0 ∂τxz0 ∂x + ∂y + ∂z = ∂x + ∂y + ∂z ∂σy ∂τxy ∂τyz ∂σy0 ∂τxy0 ∂τyz0 ∂y + ∂x + ∂z = ∂y + ∂x + ∂z (2.54) ∂σz ∂τxz ∂τyz ∂σz0 ∂τxz0 ∂τyz0 + + = + + ∂z ∂z ∂y ∂z ∂z ∂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 σij0 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 σij0 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 σij0. Due to the effect of prestress σz0 on the surface AB of the bottom half, it is necessary to consider the effect of this variable by writing forced amount as follows: ZAB = ⌠ [(σz-σz0) w dΩAB → min (2.55) ⌡ΩAB 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: Z = ZV + ZAB → min (2.56) With constraint σz = 0 on the surface AB. ZV=⌡ ⌠ (σx-σx0) εxdV* +⌡ ⌠ (σy-σ y0) εydV* +⌡ ⌠ (σz-σz0) εzdV* V* +⌠ ⌡ V* V* (τxy-τxy0) γxy dV* +⌠ ⌡ V* V* (τxz-τxz0) γxz dV* +⌠ ⌡ (τyz-τyz0) γyz dV* → min V* (2.57) 8 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: F = ZV + ZAB + λσz → min (2.58) λ = λ(x,y) is a Lagrange factor as a new implicit function of the problem. Extreme conditions of F would be: (2.59) δF = δZV + δZAB + δλσz = 0 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: 9 * Case 1: Put E1 = E0, ν1 = ν0 (a) (b) 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) (a) (b) 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 10 11 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 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 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) (a) (b) (c) 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 senmispace (Mindlin solution). 2.6 Conclusion of chapter 2 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. 12 13 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 Extended area to examine 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 = ⌠ (σz-σz0) w dΩAB (3.57) ⌡ΩAB 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: Land mass contains pile P boundary condition Comparative soil mass P Pile E1, ν1 E0, ν0 E0, ν0 E0, ν0 (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 = ⌠ (3.46) ⌡ Mχcdz + ⌠ ⌡ Qγcdz l F = Zd + Zc +⌠ ⌡ΩAB λ2 (x,y) σz d ΩAB → min (3.59) ⌡l λ1 (z) (uc-u)dz + ⌠ 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 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. (3.49) We have: uc(z, xc, yc) = u(z, xc, yc) 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 + ⌠ (3.50) ⌡ λ (z) (uc-u)dz → min l a) b) 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 14 15 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 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. Result Method Zavriev EP Gauss Maximum displacement on the head of the pile (m) Pile under the Pile under the load P, M load P 0,0116 0,0093 0,0097 Maximum bending moment (kN.m) Pile under the Pile under load P, M the load P 123,9 80,571 80,340 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 Maximum displacement on the head of the pile (cm) Result Method Poulos EP Gauss Pile under the load P, M 5,8 (a) Pile under the load P 4,2 4,7 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) (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 16 17 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). 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 (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. (a) (b) 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. 18 19 use some hysteretic damping coefficient and this coefficient indicates more accurate than soil properties compared to viscous drag cω coefficient c: 2ζh= k (4.15) ζh is called the hysteretic damping coefficient 4.3 The solution of the problem of dynamics 4.3.1 Earthquake data El Centro, 1940 and Discrete Fourier Transform (DTF). The authors used data from the El Centro earthquake in 1940 to study the problem of dynamic of pile foundation under earthquake loads. 4.3.2 Duhamel integral in the time and frequency domain To calculate Duhamel integral in the time domain is often calculated in the frequency domain. In this thesis, the author will use the following diagram to calculate: p(t) FFT Cx(f) IFFT x Cy(f) y(t) Ch(f) The above diagram can be understood: firstly, using fast Fourier transform (FFT) to transform the force in time domain p (t) over the frequency domain Cx (f), then using the methods of comparison system of extreme principle Gauss to determine the spectral response of the pile Ch (f), and then inverse fast fourier transform (IFFT) to have the results in time domain y (t). 4.4 Develop dynamics interaction problems of the pile under horizontal load D'Alembert principle applied to the problems of the construction dynamics. It is based on conditions at equilibrium of the static force which adds inertia inertia to set in the volume. Thus, according to extreme principle Gauss, forced density function of the dynamics problem of piles in elastic semi-space is written as follows: 0 0 * * Z=⌠ ⌡ (σx-σx ) εxdV +⌠ ⌡ (σy-σy ) εydV +⌠ ⌡ +⌠ ⌡ V* 0 (σz-σz ) εzdV* + V* (τyz-τyz0) γyz dV* V* (fy-fy0)v dV* + V* +⌠ ⌡ ⌠ ⌡V* V* 0 (τxy-τxy ) γxy 0 * dV* +⌠ ⌡V* (τxz-τxz ) γxz dV +⌠ ⌡l Mχcdz + ⌠ ⌡l Qγcdz +⌠ ⌡ (fx-fx0) u dV* V* 0 * ⌠ ⌡V* (fz -fz )w dV (4.25) With the constraint conditions as well as static interaction problem between the pile and foundation is presented in Chapter 3. When solving the problems in stable domain, meaning that there are no initial conditions. Using the method of finite element is similar to the problem of static interactions between piles and foundation are presented in chapter 3. In this section, 20 nodes rectangular elements are used, each node has three displacements (u, v, w) in three axes x, y, z. An element has a volume of 1 and is splitted into 8 nodes at the corner. 4.5 Survey fluctuations of the pile of soil and piles under horizontal load 4.5.1 Survey Fluctuation of the pile of soil - The case does not mention hysteretic damping coefficient (a) (b) Figure 4.7 The chart of horizontal displacement of surface layers, the bottom layer of soil under dynamic load for frequencies from 0.5 to 30 Hz, 0.5 Hz frequency step (a); frequency range from 1.0 to 60 Hz, frequency step (b)1.0 Hz Comment: When surveying with different frequency range, the value of the vibration amplitude are equal at the same location on the frequency. - The case mentions hysteretic damping coefficient 20 (a) 21 Solving density function directly (4.29) will receive the results of the vibration amplitude of pile of soil following the frequency. The author examines three cases: elastic modulus of upper layer is equal with lower elastic modulus (Figure 4.13a); elastic modulus of upper layer is smaller than lower elastic modulus (Figure 4.13b); modulus increases in the upper soil layer, the lower elastic modulus is unchanged (Figure 4.13c). (b) Figure 4.8 The chart of horizontal displacement of surface layers, the bottom layer of soil under dynamic load for frequencies from 0.5 to 30 Hz, 0.5 Hz frequency step (a); frequency range from 1.0 to 60 Hz, frequency step (b)1.0 Hz Comment: fluctuation reduced by several dozen of times compared to not consider viscous, because viscous drag of materials appears. The value of the vibration amplitude changes up and down are relative frequencies, especially in the foundation. 4.5.2 Survey transmitted Love wave in foundation Love wave has to satisfy the equation of shear wave: ∂2v G1 ∂2v ∂2v if 0≤ z ≤H (4.28a) ∂t2 = ρ1( ∂x2+ ∂z2) 2 2 2 ∂ v G2 ∂ v ∂ v if z ≥ H (4.28b) ∂t2 = ρ2( ∂x2+ ∂z2) x z H ρ1, G1 Surface layer ρ 2 , G2 Half-space Figure 4.11 The diagram illustrates the layer of softer surface soil (G1/ρ1 < G2/ρ2) located on elastic half-space conditions for Love waves to exist [46]. Graduate student considers the problems that only has shear stress τyz (reviewed in the horizontal plane yx) and τyx (reviewed in the vertical plane yz). Therefore, with these shear stresses, there are not volumetric strains that only shear strain in the plane yx and yz. According to extreme principle Gauss: Z=⌡ ⌠ V* * 0 ⌠ ⌡V* (fy-fy )v dV (τyx-τyx0) γyx dV* +⌡ ⌠ +⌠ ⌡ V* (τyz-τyz0) γ yz dV* +⌡ ⌠ (fz -fz0)w dV* → min V* (fx-fx0) u dV* + V* (4.29) (a) (b) Figure 4.13 Horizontal displacement diagram following frequency v (c) Comments: - When surveying stiffness of two similar soil layers, meaning that the velocity of shear wave cutting the upper soil layer is equal with the velocity of shear wave cutting the lower soil layer, so surface oscillation amplifier phenomenon has not appear. - When surveying stiffness of upper soil layer, recording that this value is smaller than the stiffness of lower soil layer, so surface oscillation amplifier phenomenon appears. - When surveying, elastic modulus of upper layer increases so the surface oscillation amplitude decreases. Therefore, surface oscillation amplifier phenomenon depends on the stiffness of upper soil layer, the weaker upper soil layers are, the larger fluctuating surfaces are. To obtain reliable results, many different cases are needed to survey, then is treatment of statistical data, and then the amplification coefficient is used in calculating earthquake. 4.5.3 Survey fluctuations of single pile Survey a pile with length l = 10 m, cross-sectional area (30x30) cm, Ec = 20000 MPa, located in the ground with Ed = 10Mpa, ν = 0.3 under the effects of dynamic loads at the butt of pile with frequency range from 0, 1 to 6 Hz, the frequency step is 0.1. - The case does not mention hysteretic damping coefficient 22 23 (c) (a) (d) (b) Figure 4.15 The displacement diagram with frequency at the location of head of pile, middle and butt of pile (a). Horizontal displacement diagram following the length of piles at frequencies of 4.9 Hz (b) Figure 4.18 Horizontal displacement diagram over time at the location of head and butt of pile (a). Horizontal displacement diagram (b), shear force (c), moment (d) following the length of pile at time 3,12s. - Survey the tremor of time of earthquake T = 10,24s - The case mentions hysteretic damping coefficient (a) (b) Figure 4.16 The displacement diagram with frequency at the location of head of pile, middle and butt of pile (a). Horizontal displacement diagram following the length of piles at frequencies of 5.2 Hz (b) Comment: the amplitude of oscillation can be determined following the frequency of pile. Thereby can determine the amplitude of oscillation following the length of piles at the frequency which has the largest amplitude. 4.6 Survey fluctuations of pile under dynamic loads of soil Using accelerogram of El Centro 1940 earthquakes to examine: - Survey the tremor of time of earthquake T = 5.12s (a) (b) (c) (d) Figure 4.19 Horizontal displacement diagram over time at the location of head and butt of pile (a). Horizontal displacement diagram (b), shear force (c), moment (d) following the length of pile at time 8,24s - Survey the tremor of time of earthquake T = 20,48s (a) (b) (a) (b) 24 (b) 25 (d) Figure 4.20 Horizontal displacement diagram over time at the location of head and butt of pile (a). Horizontal displacement diagram (b), shear force (c), moment (d) following the length of pile at time 18,48s Comment: When calculating the time t = 10.24s of earthquakes shows the result of vibration amplitude, the biggest internal force of pile, it causes the most detrimental, so in this case has chosen to design the pile. It can be extended to research on different types of soil, the pile group, multiple spectral acceleration, etc, to receive generic and more accurate conclusions, as the basis for the calculation of seismic resistant design for pile foundation of construction. 4.7 Conclusion of Chapter 4 1 - Develop the problem of interactive dynamics of the pile and the soil under horizontal dynamic loads and dynamic loads of soil that is allowed to mention the boundary conditions at infinity as well as the mechanical impedance conditions (radiation conditions) of the pile of soil and therefore do not need to add the spring coefficients, coefficient of viscosity as other methods. 2 - To consider the plastic deformation of the soil, the author put the hysteretic damping coefficient in the calculation and found that the influence of the coefficient of damping to the vibration amplitude is significant, particularly in the specific oscillation frequency. 3- Through numerical solution shows clear the surface oscillation amplifier phenomenon when Love waves transmitted from the ground up, in accordance with the theory of Love wave propagation. 4 - Using the accelerogram of a real earthquake (El Centro 1940) as input parameters to examine the problem of interactive dynamics of the pile under the dynamics load of soil. Using the convolution integral Duhamel provides solution in the frequency domain, then Fourier transform, reverse (IFFT) provides results in the time domain. CONCLUSIONS AND RECOMMENDATIONS * The main results achieved: By using the method of compare system of extreme principle Gauss in solving problem in the study of interactive dynamics of the pile and the soil under horizontal dynamic loads and dynamic loads of soil, author received some main results as follows: 1. Through numerical solution using the finite element method, Kelvin solution about Mindlin solution can be provided, that means getting the solution of elastic infinite semi-space from the solution of elastic infinite space with load placed in any position. 2. Formulating the static interactive problem, dynamic interaction between piles with soil under static load, horizontal dynamic load at any position. Using the finite element method with soil tobe 3-D element, 20 buttons; pile with 2 button elements for displacement and 3 buttons for shear force to solve. This method automatically satisfies the boundary conditions at infinity, the soil boundary conditions as well as conditions between pile and soil, i.e no need to add extra links as springs, viscous box on the contact surface between the pile ground, on the border of the land mass storing piles. In addition, the study also investigates the parameters affecting the work of the piles such as pile length, stiffness of the pile, piles placed on a layer of hard rock and the effect of piles to the work of the soil. 3. In the calculation of the construction dynamics and earthquake, the viscosity coefficient is offen considered. In this thesis, the author does not use the ordinary viscosity coefficient but use the hysteretic damping or dry friction coefficient. This coefficient allows the consideration of the phenomenon of plastic deformation of the ground when needed . 4. Formulating Love wave problem transmitted from the ground up to soil layer by considering love waves in the horizontal plane and in vertical plane. Based on the numerical solution of the finite element method, phenomenon of fluctuating surface amplification vertically to the wave propagation is studied, consistently with the theory of Love wave propagation. 5. Formulating the interactive dynamic problem of the pile under the seismic load. Using the convolution integral Duhamel provides solution in the frequency domain, then Fourier transform, reverse 26 (IFFT) provides results in the time domain. Use the acceleration map of a real earthquake (El Centro 1940) as input parameters to survey, identify displacement parameters, torque, shear force of the pile at any time. 6. Based on Matlab programming language to develop the calculating software program to serve the case studies and surveys: Mstatic1; Kstatic1; MstaticP1; KstaticP1; KstaticPLs; KdynaS; KdynaL; KdynaP; KdynaPE. * These issues need for further research 1. The dissertation build the problem of interaction problem between single pile and ground in the elastic domain, which is the basis for building extensive research to further examine the problem of the special properties of the ground and construction such as viscoelasticity, liquefaction phenomenon, soil properties change when the load changes, the phenomenon of "space" (GAP), etc. 2. Expanding research problems of simultaneous interactions of pile-soil-building system, group of piles, hard rise pile cap foundation, soft rise pile cap foundation, high rise pile cap foundation.
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