ISTANBUL TECHNICAL UNIVERSITY EARTHQUAKE ENGINEERING AND DISASTER MANAGEMENT INSTITUTE
M.Sc. THESIS
JUNE 2018
1-D NUMERICAL ANALYSES OF DYNAMIC SOIL RESPONSE UNDER SURFACE EXCITATIONS
Deniz ÖZ
Earthquake Engineering and Disaster Management Institute Earthquake Engineering Programme
ISTANBUL TECHNICAL UNIVERSITY EARTHQUAKE ENGINEERING AND DISASTER MANAGEMENT INSTITUTE
M.Sc. THESIS
1-D NUMERICAL ANALYSES OF DYNAMIC SOIL RESPONSE UNDER SURFACE EXCITATIONS
Thesis Advisor: Assoc. Prof. Dr. Mehmet Barış Can ÜLKER Deniz ÖZ
(501071223)
Earthquake Engineering and Disaster Management Institute Earthquake Engineering Programme
JUNE 2018
İSTANBUL TEKNİK ÜNİVERSİTESİ DEPREM MÜHENDİSLİĞİ VE AFET YÖNETİMİ ENSTİTÜSÜ
YÜKSEK LİSANS TEZİ
HAZİRAN 2018
YÜZEY YÜKLEMELERİ ALTINDA ZEMİNİN BİR BOYUTLU DİNAMİK DAVRANIŞININ SAYISAL ANALİZLERİ
Tez Danışmanı: Doç. Dr. Mehmet Barış Can ÜLKER Deniz ÖZ
(501071223)
Deprem Mühendisliği Anabilim Dalı Deprem Mühendisliği Programı
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Thesis Advisor : Assoc. Prof. Dr. Mehmet Barış Can ÜLKER ...
Istanbul Technical University
Jury Members : Assoc. Prof. Dr. Mehmet Barış Can ÜLKER ...
Istanbul Technical University
Dr. Barış ERKUŞ ...
Istanbul Technical University
Prof. Dr. İlknur BOZBEY ...
Istanbul University
Deniz ÖZ, a M.Sc. student of ITU Earthquake Engineering and Disaster Management Institute student ID 501071223, successfully defended the thesis entitled “1-D NUMERICAL ANALYSES OF DYNAMIC SOIL RESPONSE UNDER SURFACE EXCITATIONS”, which she prepared after fulfilling the requirements specified in the associated legislations, before the jury whose signatures are below.
Date of Submission : 04 May 2018 Date of Defense : 29 June 2018
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To my family, my fiance and my “Momo”,
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ix FOREWORD
The hardest part of this thesis was deciding to make a brand new start to complete my graduate education a year ago. And the most important part of this process was being accepted by my precious thesis advisor Dr. Ulker. I would like to express my deepest gratitude to him. All the knowledge that I have learned, all the steps that I have made and all of the analysis that I created were completed under his supervision and under his control.
Also, in this process, I would like to say thank you to my family, especially to my father, for offering me their support in every sense, to my beloved fiancé Caner, for his passion and understanding, and to my best friend Pelin, for being by my side in every condition.
Lastly, I would like to offer special thanks to Prof.Dr. Ilhan Sütaş, who, although no longer with us, always believed in my ability to be successful in the academic arena and lead me to go this way. I remember him yearningly and with the highest respect.
May 2018 Deniz ÖZ
(Civil Engineer)
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xi TABLE OF CONTENTS
Page
FOREWORD ... ix
TABLE OF CONTENT ... xi
ABBREVIATIONS ... xiii
SYMBOL LIST ... xv
LIST OF TABLES ... xix
LIST OF FIGURES ... xxi
SUMMARY ... xxiii
ÖZET ... xxv
1. INTRODUCTION ... 1
2.LITERATURE REVIEW ... 5
2.1.Coupled Flow and Deformation ... 5
2.2.1D Theory of Classical Plasticity ... 6
2.3.Analysis of Nonlinear Soil Response ... 8
3.1-D LINEAR DYNAMIC ANALYSES OF COUPLED FLOW AND DEFORMATION……… ... 11
3.1.Introduction ... 11
3.2.Mathematical Formulations ... 11
3.3.Analytical Solutions: 1-D Simplified Responses under Constant Loadings .... 14
3.3.1.1-D quasi-static (QS) response of constant step loading ... 15
3.3.2.1-D partially dynamic (PD) and quasi-static (QS) responses of constant harmonic loading ... 16
3.4.Finite Element Solutions ... 18
3.5.Implicit Newmark- Time Integration ... 19
3.5.1.1-D QS response of constant step loading ... 21
3.5.2.1-D PD and QS responses of constant harmonic loading ... 25
4.CLASSICAL PLASTICITY CONSTITUTIVE MODELS ... 31
4.1.Introduction ... 31
4.2.Elements of Classical Plasticity ... 32
4.2.1.Yield criterion ... 33
4.2.2.Flow rule ... 34
4.2.3.Hardening law ... 35
4.2.4.Derivation of elasto-plastic constitutive matrix ... 36
4.3.Rate-Independent Cyclic Plasticity Models ... 39
4.3.1.1-D elastic-perfectly plastic model ... 39
4.3.2.1-D elastic-isotropic hardening plastic model ... 40
4.3.3.1-D elastic-combined hardening plastic model ... 41
4.4.Analyses of Rate-Independent Cyclic Soil Constitutive Behavior ... 42
4.4.1.Analysis of elastic perfectly plastic model ... 42
4.4.2.Analysis of elastic isotropic hardening plastic model ... 43
4.4.3.Analysis of elastic combined hardening plastic model ... 44
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5.1-D NONLINEAR DYNAMIC ANALYSES OF SINGLE DEGREE OF
FREEDOM SYSTEMS ... 47
5.1.Introduction ... 47
5.2.Newton-Raphson Iterative Method ... 47
5.3.Nonlinear Transient Analysis of SDOF Solid Model... 52
5.4.Nonlinear Analyses of Elastic-Perfectly Plastic SDOF Solid System ... 58
5.4.1.Transient response under half-sine pulse ... 58
5.4.2.Transient response of a nonlinear frame ... 60
6.1-D NONLINEAR DYNAMIC FINITE ELEMENT ANALYSES OF A SOLID SOIL COLUMN ... 65
6.1.Introduction ... 65
6.2.Finite Element Formulation and Iterative Solution ... 65
6.3.Return Mapping Algorithm ... 68
6.4.Nonlinear Dynamic Finite Element Analyses ... 70
6.4.1.Transient analysis of a soil column under half-sine loading ... 70
6.4.2.Transient analysis of soil column under irregular loading ... 72
6.4.3.Harmonic analysis of elastic-combined hardening plastic soil column .... 74
7.1-D NONLINEAR DYNAMIC FINITE ELEMENT ANALYSES OF A POROUS SOIL COLUMN ... 79
7.1.Introduction ... 79
7.2.Finite Element Formulation and Iterative Solution ... 79
7.3.Nonlinear Dynamic Finite Element Analyses ... 82
8.CONCLUSIONS... 89
9.ONGOING WORKS ... 91
REFERENCES ... 93
APPENDICES ... 99
APPENDIX A 1-D Nonlinear finite element analysis algorithm ... 99
APPENDIX B Return Mapping algorithm ... 100
CURRICULUM VITAE ... 101
xiii ABBREVIATIONS
DOF : Degree of Freedom
EL : Element
FD : Fully Dynamic
FE : Finite Elements
FEM : Finite Element Method MDOF : Multi Degree of Freedom
PD : Partly Dynamic
QS : Quasi Static
SDOF : Single Degree of Freedom
TOL : Tolerance
1-D : One Dimensional 2-D : Two Dimensional 3-D : Three Dimensional
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xv SYMBOL LIST
A : Cross-section area
u p
B , B : Strain-nodal displacement matrix
c : Damping
cv : Consolidation coefficient C : Coupling matrix
Cf : Fluid compressibility matrix Dijkl : Tangent material rigidity De : Elastic constrained modulus Dep : Elasto-plastic modulus d : Nodal displacement vector E : Elasticity modulus
f : Yield Function
ˆf s : Resisting force vector fint : Internal force vector fext : External force vector fy : Yield force
F f : Force vector of fluid phase F s : Force vector of solid phase g : Gravitational acceleration H : Depth of soil / porous medium H : Kinematic hardening modulus H L : Hardening modulus
i : Number of iterations
kz : Vertical permeability coefficient
k : Stiffness
ktan : Tangent stiffness K : Stiffness matrix
xvi Keff : Effective stiffness matrix K : Isotropic hardening modulus Kf : Bulk modulus of fluid K f : Flow matrix
K s : Stiffness matrix of solid L : Length of an element m : Kronecker delta vector
m : Mass
mv : Volumetric compressibility
M : Mass matrix
M s : Mass matrix of solid
Msf : Relative mass matrix of fluid
n : Porosity
n : Number of time step
u p
N , N : Shape function matrices
t : Time
T : Wave period
T : Transformation matrix u : Displacement of solid part u : Velocity of solid part u : Acceleration of solid part
U : Displacement vector
U : Velocity vector
U : Acceleration vector
: Poisson’s ratio
p : Pore fluid pressure P : Pore pressure vector q 0 : Load amplitude q : External loading
q : Kinematic hardening parameter R : External force vector
Reff : Effective force vector Rext : External force vector
xvii Rint : Internal force vector
Rres : Residual force vector S : Degree of saturation
X : Displacement vector of porous medium
X : Velocity vector of porous medium
X : Acceleration vector of porous medium
ij : Kronecker delta
: Damping coefficient
: Derivative matrix
ij : Total strain
x : Strain in the x-direction
e : Elastic strain
p : Plastic strain
: Density of mixture
f : Density of pore fluid
: Lame’s parameter
i j : Total stress 'i j
: Effective stress
0 : Yield stress
: Angular frequency
γ : Newmark parameter
β : Newmark parameter
: Isotropic hardening parameter
: Slip rate
: Domain
1 : Ratio of time for pore fluid flow in the z-direction
2 : Rate of dynamic loading ratio to the compression wave speed
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xix LIST OF TABLES
Page
Table 3.1 : Input parameters for 1-D consolidation analysis. ... 22
Table 3.2 : Input parameters for 1-D Constant Harmonic Loading. ... 25
Table 3.3 : Input parameters for the parametric analyses using both the PD and QS formulations under constant harmonic loading. ... 29
Table 4.1 : Input parameters for 1-D linear elastic-perfectly plastic analyses. ... 42
Table 4.2 : Input parameters for 1-D elastic-isotropic hardening plastic analyses .. 43
Table 4.3 : Input parameters for 1-D linear elastic-perfectly plastic analyses. ... 44
Table 5.1 : Material parameters of nonlinear transient analysis. ... 53
Table 5.2 : Nonlinear transient analysis steps. ... 54
Table 5.2 : (continued) : Nonlinear transient analysis steps ... 55
Table 6.1: Input parameters for elastic-linear combined hardening MDOF model ...74
Table 7.1: Input parameters for elastic-linear combined hardening MDOF model ...83
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xxi LIST OF FIGURES
Page Figure 3.1: A saturated porous soil column under constant step loading ... 15 Figure 3.2: A saturated porous soil column under constant harmonic loading... 16 Figure 3.3: Boundary conditions of 1-D idealized models. ... 21 Figure 3.4: a) Pore pressure distribution for kz=1x10-3 m/s b) Pore pressure
response in time for kz=1x10-3 m/s. ... 23 Figure 3.5: a) Pore pressure distribution for kz=1x10-5 m/s. b) Pore pressure
response in time for kz=1x10-5 m/s. ... 24 Figure 3.6: Transient condition and reaching steady state of normalized pore
pressure response at the bottom boundary. ... 26 Figure 3.7: Permeability effect on vertical displacement distribution in depth ... 26 Figure 3.8: Permeability effect on normalized pore pressure distribution in depth. . 27 Figure 3.9: Angular frequency effect on vertical displacement distribution in depth. .
... 27 Figure 3.10: Angular frequency effect on normalized pore pressure distribution. .. 28 Figure 3.11: PD and QS Responses for different and 1 : a) Vertical 2
displacement distribution b) Normalized pore water pressure
distribution. ... 30 Figure 4.1: Material nonlinearity for a) Nonlinear Elasticity b) Elasto-plasticity.... 31 Figure 4.2: a) Elastic-perfectly plastic model b) Elastic-combined linear hardening model.. ... 32 Figure 4.3: Elastic and elasto-plastic zones according to yield criterion.. ... 34 Figure 4.4: a) Isotropic hardening b) Kinematic hardening c) Combined hardening. ..
... 36 Figure 4.5: Linear elastic-perfectly plastic behavior of strain controlled case. ... 43 Figure 4.6: Linear elastic-isotropic hardening behavior of strain controlled case. ... 44 Figure 4.7: Linear elastic-combined linear isotropic and kinematic hardening behavior in a strain-controlled case... 45 Figure 4.8: Linear elastic-combined linear isotropic and kinematic hardening
behavior in a stress-controlled loading.. ... 45 Figure 5.1: Nonlinear force-displacement relationship and full Newton-Raphson iterations. ... 51 Figure 5.2: Nonlinear SDOF bar. ... 52 Figure 5.3: Displacement response of SDOF nonlinear transient analysis. ... 56 Figure 5.4: Velocity response of SDOF nonlinear transient analysis. ... 56 Figure 5.5: Acceleration response of SDOF nonlinear transient analysis. ... 57 Figure 5.6: SDOF Nonlinear transient analysis: a) Displacement time history b) Velocity time history c) Acceleration time history. ... 57 Figure 5.7: Load history and restoring force-deformation relation of SDOF problem
... 58 Figure 5.8: Displacement Response of SDOF problem in time. ... 59
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Figure 5.9: Linear elastic-perfectly plastic stress-strain relationship of the problem. ..
... 59 Figure 5.10: Velocity-time and acceleration-time histories of the SDOF problem .. 60 Figure 5.11: A SDOF frame problem. ... 61 Figure 5.12: The load history and elasto-plastic stiffness relation of the SDOF frame problem. (as in Clough et al., 2003). ... 61 Figure 5.13: Displacement response of SDOF frame problem in time. ... 62 Figure 5.14: Elastic-perfectly plastic stress-strain relationship of the problem
Clough&Penzien (2003). ... 62 Figure 5.15: Velocity and Acceleration responses of SDOF frame problem. ... 63 Figure 6.1: Return mapping concept. ... 69 Figure 6.2: Displacement responses of the top node ... 71 Figure 6.3: Displacement distribution in depth ... 72 Figure 6.4: Original load history of Clough et al (2003) and converted polynomial load history. ... 72 Figure 6.5: Displacement responses of top boundary for Clough et al (2003). ... 73 Figure 6.6: Displacement distribution in normalized depth ... 73 Figure 6.7: Load history of elastic-linear combined hardening MDOF model. ... 74 Figure 6.8: Displacement history of elastic-linear combined hardening soil column..
... 75 Figure 6.9: Displacement distribution in depth of elastic-linear combined hardening soil column.. ... 76 Figure 6.10: Stress-strain relationship of elastic-linear combined hardening soil
column. ... 76 Figure 6.11: Velocity time-history of elastic-linear combined hardening soil column.
... 77 Figure 6.12: Acceleration time-history of 50 Elements with elastic-linear combined hardening model. ... 77 Figure 6.13: Converge in time on displacement – time history. ... 78 Figure 7.1: Vertical displacement - time, pore pressure – time and effective stress – time histories of elastic-linear combined hardening porous soil column ... 84 Figure 7.2: Maximum vertical displacement and maximum pore pressure distribution
in normalized depth of elastic-linear combined hardening porous soil column ... 85 Figure 7.3: Pore pressure - time history at the bottom node of elastic-linear
combined hardening porous soil column. ... 85 Figure 7.4: Effective stress - time history at the top node of elastic-linear combined hardening porous soil column. ... 86 Figure 7.5: Maximum effective stress distribution in depth for 20 element analysis of elastic-linear combined hardening porous soil column. ... 86 Figure 7.6: Vertical displacement - time history for different permeability values at the bottom node of elastic-linear combined hardening porous soil column. ... 87 Figure 7.7: Pore pressure - time history for different permeability values at the bottom node of elastic-linear combined hardening porous soil column. 87
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1-D NUMERICAL ANALYSES OF DYNAMIC SOIL RESPONSE UNDER SURFACE EXCITATIONS
SUMMARY
One of the main objectives of solving problems in geotechnical engineering is to evaluate the natural soil response under various external loads due to the fact that all civil engineering structures are located on or inside a soil medium. Under steady state or transient conditions, mechanical changes in a saturated soil system as a result of the changes in drainage conditions or the internal structure, are studied by using the
“theory of coupled flow and deformation” which essentially considers the deformation of solid skeleton and flow of pore fluid simultaneously. The mathematical formulation behind this theory satisfies the coupled dynamic behavior of porous soils employing the poro-elasticity equations. Although, the actual theory proposes what is called a
“fully dynamic (FD) formulation” including all the inertial forces associated with the motion of both solid and fluid parts (i.e. if there is more than one fluid filling in the pores of the porous medium), it is typically preferred to make some simplifiying assumptions in the theory to develop a more practical solution albeit not giving up from the accuracy. Taking the porous soil as a two-phase medium with only a single pore fluid (i.e. as water in the case of soils), these simplified equations are obtained as;
i) Partially dynamic (PD) where the inertial terms associated with the pore fluid are neglected, and ii) Quasi-static (QS) where the inertial terms of both solid and the fluid phases are neglected. The dynamic response of the system is then analyzed by the appropriate form of these equations considering the physical structure and loading situation given in the problem. While employing the complex poroelasticity equations helps better understand the problem of evaluating the actual response of soils in geotechnical engineering, it is of more practical value to initiate related studies by taking the soil medium as a single-phase solid material neglecting the porous-structure in the beginning. In this way, pore water pressure is included externally in the system as an ‘ad-hoc’ way and flow is not taken into account. That said, as it is the easiest way to model the soil behavior, it could be, therefore, pretty much the first approach to obtaining the solution of the dynamic problem, particularly in a one-dimensional (1- D) situation. Along with such a ‘simplified’ approach, one always chooses to develop analytical solutions to the problem because having a direct solution of a problem allows us to obtain exact results. However, most of the time it is not possible to obtain exact mathematical solutions for complex real-world soil-mechanics problems since soils are nonlinear, heterogeneous and anisotropic materials. Therefore, it is of utmost importance to utilize numerical methods to approximate the actual soil behavior and develop accurate solutions. In this thesis, 1-D numerical solutions to the dynamic response of a soil column under surface excitations are developed using the Finite Element Method (FEM). In the first part of the thesis, the soil is considered as a porous medium, and its linear dynamic behavior is examined through 1-D FE analyses using linear bar-elements. The validity of this numerical study is confirmed by the available analytical solutions. Due to the fact that the actual behavior of the soil is inelastic, our attention is turned to the actual nonlinear behavior. Here, since there can be irreversible deformations taking place in the soil under cyclic loading, the “theory of plasticity” is employed to handle the related calculations. To start off, nonlinear elasto-plastic behavior of a solid material is evaluated by the stress-strain relationship which is then implemented in a MATLAB program in terms of a number of basic constitutive
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models. These are: the elastic-perfectly plastic model, elastic-linear isotropic hardening plastic model and the elastic-linear combined isotropic and kinematic hardening plastic model, which are considered to be the basis of the nonlinear part of this study. Then a number of single-degree-of-freedom systems (SDOF) are analyzed using such elasto-plastic behavior of the material. The nonlinear finite element analyses in this section involve solution of the stiffness relation using the ‘Newton- Raphson Method’ and updating its components iteratively at each time-step and calculating and updating the internal force obtained from the elasto-plastic stress-strain relationship as a result of updated degree of freedoms. Then this numerical analysis program, as developed for a SDOF system, is expanded by implementing the equation of motion written in terms of the governing equations of a solid soil element and subsequently of the equations of a porous soil element, into a large-purpose MATLAB code. This is a nonlinear dynamic finite element program developed to analyse a multi degree of freedom soil column under any loading. Our focus was on harmonic response of soil whose stress-strain behavior is governed by elastic-combined hardening plastic model. The FE solution of the nonlinear dynamic analysis of a porous soil column, which is verified for a single element behavior, constitutes the ultimate extent of this thesis study. Interesting results obtained from each chapter are presented at the end of respective sections and summarized in the conclusions. As for future works, it is planned to analyse the dynamic response of the soil with a more realistic nonlinear soil model able to capture the actual cyclic-plastic soil behavior using the gained knowledge on the nonlinear dynamic finite element analysis.
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YÜZEY YÜKLEMELERİ ALTINDA ZEMİNİN BİR BOYUTLU DİNAMİK DAVRANIŞININ SAYISAL ANALİZLERİ
ÖZET
Geoteknik mühendisliğindeki problemlerin çözümündeki en temel amaçlardan biri, tüm inşaat mühendisliği yapılarının zemin üzerinde ya da içinde bulunması sebebiyle, çeşitli dış yüklemeler ve iç tepkiler altında doğal zeminin davranışını belirlemektir.
Suya doygun ve gözenekli bir zemin ortamının durağan halinde dış yükler altında vereceği reaksiyon ve yapısında oluşabilecek mekanik değişimler, “birleşik akış ve deformasyon teorisi” kapsamında çalışılır. Teoride, zeminin katı iskeletinin davranışı ve daneler arasındaki zemin suyunun davranışının birbirleri üzerindeki bağıl etkileri ve birbirleri ile hareketi birleşik bir yapıda incelenebilmektedir. Bu birleşik hesap yöntemi ilk olarak bir boyutlu konsolidasyon teorisinde karşımıza çıkar.
Konsolidasyon teorisinin, çok boyutlu ve sıkışabilir boşluk suyu kabulü ile genelleştirilmesi, “birleşik akış ve deformasyon” davranışının tanımlanmasına yol açar. Bu sayede karmaşık zemin modellerinde kullanılan ve zemin iskeletinin hareketi ile boşluk suyu basıncı etkisi birçok problem için çözülebilir hale gelir. Bu teorinin ardındaki matematiksel formülasyon, gözenekli ortamın birleşik dinamik davranışını temsil eden poroelastisite denklemleri aracılığıyla yazılır. Her ne kadar esas teori, hem katı hem de boşluk akışkanlarının (gözeneklerin birden fazla çeşit sıvı ile dolması durumu) hareketi ile ilişkili olsa da ve tüm eylemsizlik kuvvetlerini içeren bir “tam dinamik (FD) formülasyonu” halinde yazılsa da, daha pratik bir çözüm geliştirmek için bazı basitleştirilmiş varsayımlar yapılması tercih edilir. Gözenekli ortamın sadece tek bir sıvı ile dolu olması durumunda iki fazlı olarak ele alınması (örn. katı daneler ve boşluk suyu) ile elde edilen bu basitleştirilmiş denklemler; i) Sıvı fazın ataletinden kaynaklı etkiler ihmal edildiğinde ortaya çıkan kısmi dinamik (PD), ii) Hem katı hem sıvı fazların atalet terimlerinin ihmal edilmesi durumunda yazılan yarı statik (QS) formülasyon şeklindedir. Sistemin dinamik tepkisi, fiziksel yapısı ve yükleme durumu göz önüne alınarak bu denklemlerin uygun formu ile analiz edilmelidir.
Geoteknik mühendisliği problemlerinde, zeminin gerçek davranışının anlaşılması için başlangıçta, zeminin gözeneksiz ve tek fazlı bir katı cisim olarak ele alınması düşünülebilir. Böyle bir durumda zeminde boşluk suyu basıncının harici olarak sisteme etki ettirilip, akışın göz ardı edilmesi tercih edilebilir pratik bir yöntemdir. Bu yöntem, zemin modellemesi açısından en kolay yol olduğu için, özellikle tek boyutlu zemin ortamlarının dinamik çözümlerinde, bir ilk yaklaşım olarak kullanılabilir.
Böylesi basitleştirilmiş bir yaklaşımın yanı sıra, zemin problemleri için analitik çözümlerin geliştirilmesi her zaman için tercih edilmektedir. Bir problem için analitik çözümün olması, problemde istenilen bilinmeyenlere neredeyse kesin cevap verecek sonuçlar elde etmemizi sağlar. Ancak ne yazık ki böylesi matematiksel çözümlerin doğrusal olmayan, heterojen ve anizotropik davranış özelliği gösteren kompleks zeminler için elde edilebilmesi pek mümkün değildir. Bu sebeple nümerik çalışmalar vasıtasıyla bahsedilen gerçek zemin davranışına ulaşmada yaklaşımlar yapılması, geoteknik mühendisliği problemlerinin çözümünde oldukça önemli bir yer tutar. Bu tez çalışmasında, Sonlu Elemanlar Yöntemi kullanılarak yüzey yüklemeleri altındaki bir boyutlu bir zemin kolonunun dinamik davranışına uygun sayısal çözümler geliştirilmiştir.
Tezin ilk kısmında, zemin gözenekli bir ortam olarak ele alınmış, doğrusal dinamik davranışı incelenmiştir. Suya doygun bir boyutlu bir zemin kolonunun, harmonik
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yüzey yüklemesi altındaki kısmi dinamik ve yarı statik davranışı, sayısal olarak analiz edilmiş, düşey deplasmanlar ve normalize edilmiş boşluk suyu basıncının zaman ve derinlik ile değişimleri elde edilmiştir. Bahsedilen problemler için daha önceden referanslarda belirtildiği üzere geliştirilmiş olan analitik çözümlerin varlığı, bu sayede hesaplanan kesin sonuçların, yapılan sayısal modelleme çalışmasının doğruluğunu teyid etmek adına kullanılması açısından önem teşkil etmektedir. Bu sayede sayısal olarak elde edilen deplasman ve boşluk suyu basınçlarının analitik sonuçlarla uyumu, geliştirilen sayısal modellerin ve onların bilgisayara aktarımının doğruluğunu kanıtlamıştır. Böylece, ilerleyen dönemde yapılan sayısal modelleme çalışmalarının da temelini atmıştır.Buraya kadar bahsedilen bu çalışmalar bir boyutlu suya doygun poroz elemanın doğrusal elastik davranışı için modellenmiş ve analiz edilmiştir. Lakin zeminin doğadaki gerçek davranışı, gerek malzeme açısından gerek fiziksel açıdan, doğrusallıktan uzaktır. Bu durumda sistemde kalıcı deformasyonlar oluşmaktadır.
Malzeme açısından doğrusal olmayan davranış; malzemenin gerilme-şekil değiştirme ilişkisi tarafından belirlenir ve çeşitli teorik zemin modelleri ile sayısal analizlere yansıtılır. Bu modellerin temelini klasik plastisite teorisi oluşturmaktadır ve akla gelen ilk model de elastik-mükemmel plastik modeldir.Bu tez çalışmasının ikinci aşamasında malzeme açısından plastik davranış ve bu davranışı sağlayan gerilme şekil değiştirme ilişkisi kavramları incelenmiştir. Zemin modellemesi olarak, plastisite teorisinin temelini oluşturan ve doğrusal olmayan davranış analizinde başlangıç çalışmaları için kullanılması uygun kabul edilen elastik-mükemmel plastik ve elastik- doğrusal pekleşen plastik modelleri ele alınmıştır. Malzemenin gerilme-şekil değiştirme davranışının elde edilmesi için bir bilgisayar programı geliştirilmiş, çevrimsel yük altında gerilme kontrollü ve çevrimsel bir deplasman altında da şekil değiştirme kontrollü olarak analizler gerçekleştirilmiştir. Elastik ve elasto-plastik malzeme davranışları da elde edilen grafikler üzerinden değerlendirilmiştir.
Analizlerin doğruluğunu sağlamak adına, her hesap adımı kendi içerisinde çok daha küçük adımlara bölünerek ilerlenmiş, bu sayede malzemenin elastikten elasto-plastik davranışa geçiş aşamaları, hata payını minimuma çekecek şekilde hesaplanmıştır.
Çalışma daha sonra malzeme açısından elasto-plastik davranışın bir boyutta tek serbestlik dereceli sistemlere uygulanışı üzerine devam etmiştir. Hareket denkleminin belirtilen sistemler için çözümü, uzay ve zaman tanım alanında elasto-plastik davranışı içerecek şekilde gerçekleştirilmiştir. Zincir şekilde birbirini etkileyen doğrusal olmayan hesap adımları, belirli bir güncelleme ve iterasyon algoritması üzerinden sürdürülmektedir. Seçilen zemin modelinin yönettiği bu süreçte, yinelemeli hesaplamalar sonucu %0.5’in altında bir hata payı için analiz sürdürülmüş, bu hata oranının altında kalınan her değer, gerçeğe yakınsak kabul edilerek bir sonraki yük adımına geçilmesi uygun görülmüştür. Analizin tamamı yüzeysel bir yükleme süreci altında, tek serbestlik dereceli sistemlerin zaman ve uzay boyutundaki elasto-plastik dinamik davranışını kapsamaktadır. Hesaplanan deplasmanların zamana bağlı değişimi, ilgili problemlerin referans alınan kaynaklardaki çözümleriyle karşılaştırılarak tezin ilgili bölümü tamamlanmıştır. Bu sayede oluşturulan sayısal analiz yönteminin tek serbestlik dereceli bir sistemin elasto-plastik malzeme davranışı için dinamik olarak doğru analiz sonuçları verdiği kanıtlanmıştır. Devamında, tek serbestlik dereceli sistemler için oluşturulmuş sayısal analiz programı, sonlu elemanlar yöntemi ile geliştirilerek, çok serbestlik dereceli sistemlere kanalize edilmiştir. Burada bir zemin kolonu dikkate alınmıştır. Sayısal çalışma ile çok serbestlik dereceli katı bir zemin kolonunun elasto-plastik dinamik davranışı modellenmiştir. Analizlerin doğru ilerleyebilmesi adına bu bölümde elasto-plastik davranış için bölgesel bir yinelemeli algoritma kullanılmış, bu adımlarda da hata oranının %0.5’in altında kaldığı durumlar
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yeterli yakınsak kabul edilip, bir sonraki alt-adımlamaya geçiş sağlanmıştır.
Sonuçların teyidi, sonlu elemanlar modelinde tek eleman seçimi sonucu meydana gelen deplasmanların, bir önceki bölümde tek serbestlik dereceli sistemler için elde edilen deplasman değerleri ile kabul edilebilir ölçüde uyuşması ile sağlanmıştır.
Doğruluğu teyid edilen bu sonlu elemanlar analizi, katı bir zemin kolonu için genelleştirilmiş, tüm kolona ait düşey deplasmanların zamanla ve derinlikle değişimleri sunulmuştur.Bu tezin amacı; suya doygun bir boyutlu bir zemin kolonunun, çoğu tekrarlı yüzeysel yüklemeler altında doğrusal olmayan dinamik davranışının sayısal yöntemlerle analiz edilmesidir. Bu amaca yönelik olarak her türlü yükleme altında poroz zemin kolonunun lineer olmayan davranışını çözümleyen bir program yazılmış, harmonik yükleme tipi ele alınmıştır. Detaylı bir şekilde sürdürülen çalışmalar birleştirilerek meydana getirilen bir boyutlu sayısal zemin modelinin analizleri yapılmış, sonuçlar zemindeki deplasmanlar, boşluk suyu basıncı ve efektif gerilmenin derinlikle ve zamandaki değişimleri cinsinden çıkarılmıştır. Suya doygun ve poroz bir zemin kolonunun gerilme-şekil değiştirme davranışı elastik-doğrusal ve birleşmiş pekleşen plastik modeli ile tanımlanmış, bu model birleşik akış ve deformasyon teorisi üzerine kurulmuş sonlu elemanlar programına aktarılarak zemin kolonunun sayısal analizleri zaman ve mekan tanım alanında elde edilmiştir. Sonuçlar elastik-doğrusal pekleşen böyle bir modelin zeminin tekrarlı yük altındaki doğrusal olmayan dinamik davranışını gerçekte gözlenen doğrulukta modelleyemediği yönündedir. Bu açıdan ileride, suya doygun bir boyutlu bir poroz zeminin doğrusal olmayan dinamik davranışını hesaplamak üzere gerçekte deneylerle gözlenen zemin davranışını modelleyecek bir bünye modelinin aktarımıyla sonlu elemanlar modelinin geliştirilmesi hedeflenmektedir.
1 1. INTRODUCTION
One of the main objectives of solving problems in geotechnical engineering is to evaluate the natural soil response under various external loads. Unfortunately, deriving exact solutions is usually not possible for complex soil problems and it is needed to make drastic simplifications to the problem in terms of material properties and geometrical domain or use sophisticated numerical methods to obtain only approximate solutions to it. The objective of this thesis is to develop numerical models and accurate solutions to the one-dimensional (1-D) dynamic response of soil column under surface excitations. That is, the coupled flow and deformation response of a soil medium is to be modeled with a numerical model and perform multi degree of freedom (MDOF) nonlinear dynamic analyses to understand the basic features of soil response leading to a possible failure condition. The Finite Element Method is employed to develop numerical models and the governing equations of poroelasticity are discretized in temporal and spatial domains. Soil elemental level constitutive equations are included in the system of governing equations through a number of basic plasticity models including a more realistic one capable of tackling the actual cyclic behavior of soils. Subsequently, analyses are performed combining these components according to the governing laws of soil mechanics.
Following a brief summary of the works in literature in Chapter 2, the PD and QS response of a soil column under constant surface loadings is analyzed numerically in Chapter 3. Then vertical displacements and normalized pore pressure responses in time and depth are obtained. The existence of analytical solutions as developed previously by the thesis supervisor, allows this numerical work to be verified with those exact solutions. Here, the linear elastic behavior of a saturated porous bar element is used.
Since in the actual behavior of soils there exists irreversible deformations occurring in the system under cyclic loads, material nonlinear behavior is to be determined by stress-strain relationship. Such relationships are frequently reflected in the analyses in terms of various constitutive soil models. Therefore, in Chapter 4, various features of the Theory of Classical Plasticity are presented in terms of the necessary concepts of
2
plastic behavior such as yielding, flow rule and stress-strain relationship. As for soil models, elastic-perfectly plastic and elastic-linear combined hardening plastic models, which are considered to be the basics of the plasticity theory, are studied in this chapter.
These are considered necessary preliminary studies to work on prior to solving the actual soil column problem using a more realistic soil model. In this chapter, a computer program was developed in MATLAB to obtain the stress-strain relationship of the material at the elemental level. In this program, the related aspects of these two models are written in terms of governing constitutive equations, which are, then, integrated at the Gauss integration points using the ‘implicit integration method with a radial return mapping algorithm’. In order to ensure the accuracy of the analyses, each integration step, or also called the ‘load step’, is progressed by dividing it into smaller substeps, which is also named as the ‘substepping method’. Overall, Chapter 4 represents the foundations of the elasto-plastic soil behavior in the context of material constitutive response. In Chapter 5, application of the knowledge on simple material nonlinear behavior through the abovementioned two basic models, is done on a number of single degree of freedom (SDOF) systems in 1-D. The main purpose of Chapter 5 is to conduct SDOF nonlinear dynamic analyses by making use of the elasto- plastic dynamic behavior presented in Chapter 4 at the material level. This way, it becomes possible to build a structure of topics which are necessary to create a flow enabling a smooth transition from one chapter to another. The results of the nonlinear analyses completed in this chapter are compared with the calculated DOF in time obtained by various sources in the references. Chapter 6 presents the multi-degree of freedom (MDOF) solutions of the dynamic response of a soil column under various dynamic loads. Here, FE models are developed by the classical Finite Element Method (FEM) using multiple 1-D bar elements. A separate MATLAB code is developed for this purpose. Firstly, the time history variations, in terms of displacement, velocity and acceleration, and the displacement variations in the soil depth, are verified with the results of the SDOF problems presented in the previous chapter. Then, the FE numerical solution of a soil column under harmonic surface loading is analyzed by using an elastic-linear combined hardening plastic model whose details are given in Chapter 4.
The main objective of this thesis is that, the numerical analyses of 1-D nonlinear dynamic behavior of a saturated soil column under surface loadings are made by
3
utilizing a more realistic theoretical model to govern the soil dynamic behavior. As for this purpose, the seventh chapter includes both the knowledge gathered on dynamic finite element analysis of 1-D saturated response of a soil column and the knowledge associated with the material point analysis of soil elasto-plastic stress-strain behavior.
It represents conceptually unite all the previous chapters to form a more realistic one as far as the actual soil response. While the former employs the theory of coupled flow and deformation as to take the soil as a porous medium, the latter uses elastic – linear combined hardening plastic model which is considered capable of capturing the plastic deformations in the soil leading to a permanent reduction in effective stress. That way, problems such as soil liquefaction or shear-induced softening can be solved with such a numerical model. A MATLAB code is developed to analyse the nonlinear behavior of 1-D porous soil column under harmonic loadings. The results are given as graphs which includes time histories of vertical displacement, pore pressure and effective stress and they are discussed in final section of related chapter.
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5 2. LITERATURE REVIEW
2.1. Coupled Flow and Deformation
The mastermind for the growth of poro-elasticity was the solution to the problem of soil consolidation. The first attitude of this issue was a phenomenological method by Terzaghi (1925, 1943), who considered soil to be laterally constraint, therefore allowing continuous uniaxial deformations. He developed the analytical description for one-dimensional case considering the Darcy’s law for the flow of a fluid through a permeable medium, and his own concept of the effective stress principle. In the theory, Terzaghi assumes that pore fluid is incompressible and individual soil particle compressibilities can be essentially neglected. A general multi-dimensional theory of elastic deformation of fluid permeated porous media was first proposed by Biot (1941), in which some allowance to volumetric deformation in both phases is involved.
Moreover, the increase in fluid flux per unit volume as a function of change in pore pressure is presented. In Biot (1955), the concept is expanded to the overall anisotropic elastic situation along with the inclusion of dynamic response in Biot (1956). Some other additions in terms of nonlinear elasticity are given in Biot (1978). Some applications of Biot’s linear system which are appropriate for some of the problems in soil mechanics is suggested by Verruijt (1969). Rice et al. (1976) modified the formulation with undrained constants. Hence, the difference between drained and undrained conditions is presented. Additions to the usage of nonlinear system are given by Zienkiewicz et al. (1980) and Prevost (1980, 1982). A general handling of Biot’s theory including the nonlinear material behavior and large deformations is established in Coussy (1991, 2004), where the theory is represented using a thermodynamics framework. The mechanical response of both the fluid and the solid phases are calculated by coupling solid deformation and pore fluid flow. Hence the concept is more commonly referred to as “coupled flow and deformation” since then, as opposed to ‘multi-dimensional consolidation’ which is equally accurate. The material is typically assumed isotropic here, and is exposed to quasi-static deformations under isothermal conditions. Porosity refers only to the associated pore space, though, the
6
presence of insulated spaces or cracks within the solid skeleton is not excluded. Solid phase is deformable and is not composed of a single grainy component. Fluid in pores is compressible and which is comprised of a single phase taking into account the air dissolved in it. The validity of this theory is due to a fluid saturation which is more than about 95%. For slightly less degrees of saturation, Biot’s theory is still considered valid provided that the air and liquid pressures are equal (Okusa, 1985). The principle of effective stress constitutes the most crucial aspect of the theory of poroelasticity as it provides a clear distinction between the role of pore water and that of soil skeleton to understand the coupled process. This way, it is now possible to distribute the total stresses applied to a porous medium between the two phases. For this reason, part of the applied stresses are transferred to the pore fluid generating pore pressure while the rest is transferred to the solid skeleton. The latter is called “the effective stress” which is the sole aspect that controls the deformation state in soil. Therefore, we now need to write the constitutive equations of classical mechanics in terms of effective stresses.
Also, in addition to the momentum balance laws leading to governing equilibrium equations, law of conservation of mass leading to continuity equation is now necessary for a complete depiction of the role of pore fluid flow in soil mechanics. The latter is governed by the Darcy’s law in terms of a fictitious flow velocity. de Boer (1996, 2000) summarizes some of these aspects in a more detailed fashion. Using this theory, some computational models which include plastic deformations and creep are developed by (Lewis and Schrefler, 1998; Brinkgreve and Vermeer, 2002).
2.2. 1D Theory of Classical Plasticity
A structure disfigures when it is exposed to external forces. The deformation is elastic if it is recoverable, which means deformation disappears rapidly when the external loads are taken back (unloading). In this case, it is considered that the relation between load and displacement as well as between stress and strain are linear. Most of the engineering materials exhibit a specific degree of linear-elasticity. The deformation of a material is plastic if it is irrecoverable. Fragile materials such as glass, concrete and rocks under tension have only elastic deformation before they fail. Oppositely, metals, rocks and concrete under high limited pressure can create significant plastic deformation before they fail, hence they display ductile material properties.
7
Theory of Plasticity is a subdivision of solid mechanics concerning the plastic deformation and the load carrying limits of materials and structures. It deals with the hypotheses of yielding of materials under complex stress paths and the calculation of stress-strain relationship after yielding.
In 1864, Tresca offered two studies which deal with the experimental surveys on plastic flow of metals. He assumed a yield condition for the continuum solutions. It is known as the ‘Tresca yield criterion’. Saint-Venant (1870) was the first to propose the basic equations of plasticity and to use them to solve quite a few practical problems.
Tresca criterion was used as a yield condition in their study. This is referred to as the
“maximum shear stress criterion” or single-shear criterion. Using that theory, Saint- Venant solved problems such as torsion of circular shafts, the plastic deformation of hollow circular cylinders under the internal pressure, and pure bending of rectangular prismatic bars. Saint-Venant named his new work as ‘Plastico-dynamics’. Ludwig Prandtl (1875, 1953) made extra developments in the history of plasticity. He studied on more complex two-dimensional problems of a semi-infinite body under constant pressure. Rapid advancements in plasticity began with Prandtl (1928) and the book of Nadai (1931). Some books relating the plasticity were published around 1950s by Sokolovsky (1946), Drucker (1950), Freudental (1950), Hill (1950), Nadai (1950), Prager and Hodge (1951) and Prager (1955). After the Tresca yield criterion, Huber (1904) and von Mises (1913) suggested the Huber-von Mises yield criterion. Main concepts contain the assumption of coaxiality of the main stress-strain rate tensors by de Saint-Venant (1870), plastic potential theory by von Mises (1928) and Melan (1938), maximum plastic work principle by Hill (1948), Drucker's stability postulates (Drucker, 1952, 1959), and kinematic hardening laws by Prager (1955) and Ziegler (1959). The classical plasticity is applied in mechanical engineering and metal forming where materials have the same amount of strength in both compression and tension. In 1961, Haythorthwaite (1961) presents the maximum deviatoric stress yield criterion and Yu (1961) proposes the twin-shear stress yield criterion.
Soils are composite materials made up of solid, air and water. They exhibit a various ranges of stiffness and strength depending upon their classification, load history, void ratio and density. Among many others at the time, soil plasticity is primarily examined by Roscoe et al. (1963), Schofield et al. (1968), Roscoe (1968) and Burland (1969).
The Cam Clay model is considered as one of the most successful theories building the
8
foundations of soil plasticity and establishing the “critical state theory” by Roscoe (1968), Schofield and Wroth (1968), Schofield et al. (1968), Atkinson and Bransby (1978), Atkinson (1978, 1981) and Wood (1990). They have overall constructed a very useful theoretical framework to explain the clay soil behavior which was later on adapted to other soils. Chen (1975), Salencon (1977) and Chen and Baladi (1985), Chen et al. (1985) and Zeng et al. (2002) are the other prominent works on soil plasticity. Now, plastic behavior of soil is still one of the most functional and investigated research subjects. There have surely been many many models developed in the last three decades alone based on theories of classical plasticity, generalized plasticity or bounding surface plasticity. However, as far as pioneering ones, especially for the monotonic behavior of soil, the Mohr-Coulomb (Mohr, 1900) and the Drucker- Prager (1952) criteria are still quite frequently used in soil mechanics.
2.3. Analysis of Nonlinear Soil Response
Ever since the soil response under external loadings have been investigated through numerical methods, the nonlinear analyses have become the major concept in evaluating the actual response of soils. The main reason for that is the fact that many civil engineering structures experience failure under such huge loads as a result of permanent deformations or the levels pore water pressure in the soil rises up leading to soil liquefaction. Therefore, it is of utmost importance to model actual soil response using powerful numerical methods. Many of the models given above are also implemented in computational codes; naming a few, Zienkiewicz and Humpheson (1977), Zienkiewicz and Pande (1977), Vermeer and de Borst (1984), Desai (1984), Zienkiewicz and Mroz (1984), de Boer (1988) and many others in the 90s and 2000s all the way up to now. Since more than three decades of time, important developments have been obtained in numerous constitutive models, which efficiently characterize the behavior of soils subjected to various stress paths. It is, however, not feasible to summarize a rather wide range of literature on numerical analyses of soils in this section due to a very large number of works done on this topic. Thus, considering the decades of research studies completed and the variety of literature ranging from 1-D simplified approaches to 3-D fully nonlinear analyses of the dynamic response of soils and soil-structure systems, it is more preferable to mention mostly the key textbooks and self-explanatory research books in this section as opposed to the endless number
9
of technical papers. These are: ‘The Finite Element Method’ by Zienkiewicz and Taylor (2000), ‘Nonlinear Finite Element Analysis of Solids and Structures vol. 1 and 2’ by Crisfield (1991, 1997), ‘Nonlinear Finite Element Analysis of Solids and Structures’ by Borst et al. (2012), ‘Finite Element Method: Linear Static and Dynamic Finite Element Analysis’ by Hughes (2000), ‘Computational Inelasticity’ by Simo et al. (2000), ‘Concepts and Applications of Finite Element Analysis’ by Cook et al.
(2002), ‘Dynamics of Structures’ by Clough et al. (2003) and ‘Dynamics of Structures:
Theory and Applications to Earthquake Engineering’ by Chopra (2011) which cover significant studies of nonlinear modeling of dynamic behavior.
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3. 1-D LINEAR DYNAMIC ANALYSES OF COUPLED FLOW AND DEFORMATION
3.1. Introduction
In nature, soils are composed of solid particles and of pore fluid, which fills in the pores between those particles. Motion of solid skeleton and flow of pore fluid together result in the response of soils under external loads. In the solution of geotechnical engineering problems, the actual behavior of soils, as observed in nature, is represented by the analysis of coupled flow and deformation. Thus, related analyses performed to understand such behavior should consider this coupled process.
In this chapter, analytical and numerical solutions to the governing equations of coupled flow and deformation are summarized briefly. While the analytical solutions derived by the thesis supervisor, Dr. Ulker, are used herein, the numerical solutions are derived by the classical “ Finite Element Method (FEM)”. Numerical solutions are obtained by using the Newmark- Implicit Time Integration Method to solve time- dependentequations In conclusion, the results are verified by comparing analytical and numerical solutions and are further interpreted by performing a set of parametric studies.
3.2. Mathematical Formulations
In a saturated soil subjected to external stress, pore water pressure increases causing water drainage through the pores. In consequence of the drainage process, volume change occurs in the soil. In highly permeable soils, depending on the rate of loading, pore pressure rises and resulting volume changes occur almost rapidly. As long as there is flow of pore water in the soil, the dissipation of pore pressure takes place under constant loading which is called ‘consolidation’. As a result of consolidation, soil settles and permanent displacements occur. This process of consolidation was developed by Karl Terzaghi in 1925 for the first time, which is called the One- Dimensional Theory of Consolidation. The consolidation process is dependent upon
12
the permeability of the soil and hence, is a function of rate of flow through the soil.
Terzaghi’s ‘effective stress principle’ written in terms of the law of conservation of mass and law of momentum balance along with D’Arcy’s law for the flow of pore fluid form the basis for understanding consolidation.
Biot (1941) and later Biot, (1955, 1962) who extended the theory of consolidation by including the inertial terms associated with the motion of fields, generalized the theory as the Theory of Coupled Flow and Deformation, (mentioned above) essentially describing the dynamic consolidation for 3-D case and for the majority of porous materials.
The related mathematical formulation is based on three basic physical laws:
1. Constitutive Law, that represents the collaborative effects of the intermolecular forces when the substance is deformed. According to this, stress–strain relationships are obtained in the material.
2. Law of Conservation of Momentum, is an important consequence of the Newton's second law of motion and states that the change in momentum in a controlled volume of a solid medium remains constant until the medium/object is forced by an external effect. By means of this, equilibrium equations (of both solid and liquid phases) are developed.
3. Law of Conservation of Mass, states that the change in mass of a bodyis zero during a dynamic process and leads to thecontinuity equation or also called the mass balance equation.
In developing a generalized mathematical formulation for a two-phase soil, the following assumptions need to be made:
1. The water and the gas phases within the porous medium are considered as a single compressible fluid.
2. The effects of gas diffusing through water and movement of water vapor are ignored.
With reference to the above-mentioned laws, total stress can be decomposed into the summation of effective stress and pore water pressure as follows;
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ij ij ijp
(3.1)
whereij' is effective stress, ij is total stress, is Kronecker delta and ij pis pore fluid pressure. Strain is defined as; ij
, ,
1( )
ij 2 ui j uj i
(3.2)
where ui j, and uj i, are the derivatives of the solid displacement with respect to spatial coordinates. The effective stress-strain relation can be written as;
( 0)
ij Dijkl kl kl
(3.3)
whereDijklis the tangent coefficient matrix and is the initial strain. According to kl0 previously stated laws, by considering the inertial forces associated with both phases (solid skeleton and pore fluid), Fully Dynamic (FD) formulation is summarized as;
,
ij j gi ui fw
(3.4)
,
f i f i
i f i f i i
i
w g
p g u w
n k
(3.5)
, ,
n
i i i i
f
u w p
K
(3.6)
where is total density of medium and is density of the fluid.f n is porosity, Kf is bulk modulus of the pore fluid,gis the gravitational acceleration andkis the permeability coefficient of medium. In addition, u is solid part’s displacement, w is the relative fluid displacement and pis the pore fluid pressure.
By ignoring the inertial forces associated with pore fluid, the equations turn into the Partly Dynamic (PD) formulation as a simplified form as follows;
,
ij j gi ui
(3.7)
,
f i
i f i f i
i
p g u g w
k
(3.8)
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, ,
n
i i i i
f
u w p
K
(3.9)
If all the acceleration terms are neglected, the formulation becomes the most simplified one named as the Quasi Static (QS) form and is written as;
, 0
ij j gi
(3.10),
f i
i f i i
i
p g g w
k
(3.11)
, ,
n
i i i i
f
u w p
K
(3.12)
3.3. Analytical Solutions: 1-D Simplified Responses under Constant Loadings Considering the 1-D response of soil, analytical solutions for 1-D response, which are developed for a column of porous medium under constant loading, are needed. In this section simplified mathematical formulations of 1-D case developed by Ulker (2009) are presented.
For all simplified forms, equations are obtained in accordance with laws and formulations mentioned above. For the 1-D case, by combining (3.1), (3.2) and (3.3), the equation becomes;
D u p
z
(3.13)
After ommitting the static gravity terms, substituting (3.7) into (3.13) gives;
2 2
u p
D u
z z
(3.14)
Combining (3.8), (3.9) results in;
n
f f
u k p k p
t z z g z gu K t
(3.15)
Equations (3.14) and (3.15) are final forms of the PD formulation of 1-D case. As mentioned before, if we neglect all inertial terms, equations become the QS form of 1- D case as;
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2
2 0
u p
D z z
(3.16)
n
f f
u k p p
t z z g z K t
(3.17)
3.3.1. 1-D quasi-static (QS) response of constant step loading
In this section, QS solution of the porous soil column under q0 constant step loading is presented (see Figure 3.1). By taking pore water to be incompressible, and using ɛ as the first derivative of displacement in depth, Equation (3.16) can be rewritten as follows;
p 0 D z z
(3.18)
1 p p
D zdz D
(3.19)Figure 3.1: A saturated porous soil column under constant step loading.
If we substitude (3.17) and (3.18) into (3.19) equation becomes;
0
f
p k p
t D z g z
(3.20)
2 2
v f
k p p
m g z t
(3.21)
t q0
q q q 0 (p0, D
dudz q)H
(u0, dpdz0, w0) z
q
q q
16 where mv 1D and (1 )
(1 )(1 2 )
D E
, which is the elastic constrained modulus of soil and E is the Young’s modulus. The coefficient v
v f
C k
m g
is called the coefficient of consolidation.
Equation (3.21) can be solved by using the “method of separation of variables”, and the final form of the solution to the 1-D consolidation equation as presented in Ulker (2009) yields;
2 2H 0
0
2 2cos( )
( , ) sin
2H
v n
C t
n
n n z
p z t q e
n
(3.22)
3.3.2. 1-D partially dynamic (PD) and quasi-static (QS) responses of constant harmonic loading
When loading is in harmonic form applied on the surface of saturated soil column, such as q e0 i t , where is the angular frequency and q is the amplitude, we get a 0 problem as shown in Figure 3.2 below.
Figure 3.2: A saturated porous soil column under constant harmonic loading.
QS response of the governing equations for solid part’s displacement and pore fluid pressure is obtained by Ulker (2009) by omitting inertial terms as follows;
2 H H
2
q q
( , ) q 1 1
1
z A z
A
i t A
e e
u z t z e
AD e D
(3.23)
qq e0 i t
H
(u0, dpdz0, w0)
z (p0, D
