Part 1 Analysis Case

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Part I - Analysis Case

midas GTS 1

Table of Contents

1.0 Overview ............................................................................................................................... 1

2.1 Introduction to Static Analysis ............................................................................................... 3

2.2 Linear Static Analysis ............................................................................................................ 5

2.3 Nonlinear Static Analysis .................................................................................................... 16

2.3.1 Introduction .................................................................................................................. 16

2.3.2 Nonlinear Elastic Analysis ............................................................................................ 16

2.3.3 Elasto-plastic Analysis ............................................................... ................................... 17

3.1 Introduction to Seepage Analysis ........................................................................................ 28

3.2 Seepage Flow Rule ...................................................... ....................................................... 29

3.3 Governing equations ........................................................................................................... 30

3.4 Finite Element Formulation ................................................................................................. 33

3.5 Time Integration .................................................................................................................. 34

4.1 Introduction to Stress-Seepage Coupled Analysis .............................................................. 36

4.2 Effective Stress ................................................................................................................... 36

4.3 Governing Equation ............................................................................................................ 37

5.1 Introduction to Consolidation Analysis ................................................................................ 39

5.2 Governing Equation ............................................................................................................ 40

5.3 Formulation of Finite Element ............................................................................................. 41

5.4 Properties of Consolidation Element ................................................................................... 42

5.4.1 Degrees of freedom for plane strain element .......................................................... ... 44

5.4.2 Degrees of freedom for solid elements ........................................................... .............. 45

5.5 Precautionary Points about Consolidation Analysis ............................................................ 47



Table of Contents - Analysis Case

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5.5.1 Time and loading input ................................................................................................. 47

6.1 Introduction Construction Stage Analysis ............................................................................ 49

6.2 Addition and Removal of Elements ..................................................................................... 51

6.3 Adding and Subtracting of Load .......................................................................................... 53

6.4 Adding and Subtracting of Boundary Conditions ................................................................. 55

6.5 Load Distribution Factor ...................................................................................................... 55

6.6 Initialization of Displacements ............................................................................................. 56

6.7 Change in Material Properties .......................................................... ................................... 57

6.8 In-situ Stress ....................................................................................................................... 58

6.8.1 0 K Method .......................................................... ....................................................... 58

6.8.2 Self Weight Analysis Method ..................................................... ................................... 58

6.9 Restart Function.................................................................................................................. 59

6.10 Undrained Analysis ........................................................................................................... 59

7.1 Introduction to Dynamic Analysis ........................................................................................ 60

7.1.1 Eigenvector Analysis .................................................................................................... 60

7.1.2 Subspace Iteration Method .......................................................................................... 66

7.1.3 Lanczos Iteration Method ............................................................................................. 66

7.2 Consideration of Damping .................................................................................................. 66

7.2.1 Introduction .................................................................................................................. 66

7.2.2 Proportional damping ................................................................................................... 67

7.3 Response Spectrum Analysis ............................................................................................. 69

7.4 Time History Analysis .............................................................. ............................................ 74

7.4.1 Introduction .................................................................................................................. 74

7.4.2 Modal Superposition Method ..................................................... ................................... 75

7.4.3 Direct Integration Method ............................................................................................. 77




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7.5 Complex Response Analysis............................................................................................... 82

7.5.1 1-dimensional dynamic analysis (theoretical solution) .................................................. 82

7.5.2 2-dimensional dynamic analysis (finite element method) ............................................. 88

7.5.3 Transmitting boundary .................................................................................................. 92

8.1 Introduction Slope Stability Analysis ................................................................................... 93

8.2 Strength Reduction Method ................................................................................................ 94

8.2.1 Introduction .................................................................................................................. 94

8.2.2 Theoretical Background ............................................................................................... 95

8.2.3 Method of calculating Minimum Safety Factor .............................................................. 96

8.3 Stress analysis method based on the limit equilibrium method ........................................... 97

8.3.1 Introduction .................................................................................................................. 97

8.3.2 Evaluation of safety factor ............................................................................................ 98

8.3.3 Stress integration along the potential sliding surface ................................................... 99



Introduction

Chapter 1

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1.0 Overview

In general, geotechnical analysis is often compared with structural analysis for its

fundamental concept. In structural analysis, the focus is placed on the uncertainty of

applied loads such that various load cases are analyzed for a given structure, and the load

cases are combined to determine the maximum force effects on each member for its finaldesign. On the other hand, geotechnical analysis is focused on construction sequence and

uncertainty of ground materials. Geotechnical engineers primarily place emphasis on

analyzing the physical state and behavior of soils through numerical analysis. Therefore,

the analysis model should be able to reflect the actual ground composition and construction

conditions using solid elements. In addition, the elements must possess nonlinear and

anisotropic characteristics of soils to reflect the in-situ conditions considering the initial state

of stresses of the ground.

A geotechnical analysis software package must serve to well reflect the actual site

conditions in numerical modeling in order to determine the suitability of design and

construction conditions. As a total solution package for geotechnical engineering, midas

GTS is designed to provide all the necessary functions required for numerical analysis for

tunnel engineering and soil/rock mechanics. The analysis features included in are outlined

below:

A. Static Stress Analysis

1) Linear static stress analysis

2) Nonlinear static stress analysis (Nonlinear elastic & elasto-plastic

analysis)

B. Seepage Analysis1) Steady state analysis

2) Transient state analysis

C. Stress-Seepage Coupled Analysis

D. Consolidation analysis




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E. Construction stage analysis

1) Nonlinear analysis

2) Steady state analysis

3) Transient state analysis

4) Consolidation analysis

F. Dynamic analysis

1) Eigen value analysis

2) Response spectrum analysis

3) Time history analysis

4) Complex response analysis

G. Slope stability analysis

1) Strength reduction method

2) Limit state stress analys



Static Stress Analysis

Cha ter 2

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2.1 Introduction to Static Stress Analysis

Static stress analysis is theoretically used for analyzing structures, which are not subjected

to any kind of vibration. However, in practice, it is often assumed that static stress analysis

can be performed on a structure if the externally applied load has a frequency that is equal

to or less than one third of the natural frequency of the structure. Static stress analysis

problems can be divided into linear and nonlinear behavior according to the behavior type.

Linear behavior can be recognized as a special case of nonlinear behavior. The primary

sources of nonlinearity in static stress analysis are as follows:

Material nonlinear behavior

Geometric nonlinear behavior due to finite strain

Nonlinear behavior due to slip and spread of boundaries

In geotechnical analysis, material nonlinear behavior is applied primarily to represent soil

behavior, and to simulate the soil behavior in detail, structural material nonlinearity can be

imposed for the analysis. The material models that are used to consider material

nonlinearity can be divided as follows

Linear elastic model

Nonlinear elastic model

Elasto-plastic model

Visco-elastic model

Visco-elasto-plastic model

Among the above models, midas GTS provides the linear elastic, nonlinear elastic, and

elasto-plastic models. More detailed list of the models is shown in Part 3. Constitutive

Model in the manual.

The strain due to the body action may be divided into large strain and finite strain. Most

geotechnical problems relate to large strain, and the physical interactions are evaluated

based on the assumption that the strain of foundation is small enough. The geometric

nonlinear behavior, including finite strain, would heavily occur on large deformed structures



Part 1 – Analysis Case

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consisting of ductile materials such as mild steel and rubber.

However, the geometric nonlinear effects would be trivial for structural behavior exhibiting

large deformation composed of ductile materials such as mild steel and rubber. Therefore,

the aforementioned effects are not commonly considered in geotechnical problems.

The boundary nonlinear effect reflects the nonlinear behavior that occurs in the interfaces

between different materials. A variety of behaviors may occur in the interfaces betweendifferent materials such as shear slip, crack opening, and contact according to the types of

external loadings. To take into account these behaviors, it is common to describe the

behavior of interfaces with the special types of elements. The special type of elements and

behaviors are summarized as follows:

Interface, Contact Element : slip, cracking or opening, contact, bending

Elastic Link : tension only, compression only, hook, gap Spring

1 : tension only, compression only, hook, gap

The static stress analysis types provided by midas GTS can be divided as follows based on

the feature of behavior:

Linear static stress analysis

Nonlinear static stress analysis

Linear analysis is regarded as an independent analysis type in midas GTS because of its

convenience and immediacy. In linear static stress analysis, the material model is limited to

linear elastic models, and elastic link element and tension only, compression only, hook,

and gap behaviors of truss element are available. Therefore, linear static stress analysis is

utilized to observe the approximate behavior of a foundation, construction in the initial

condition of construction stage analysis, or to perform a tunnel lining analysis.When linear static stress analysis is performed on a foundation or structure that is assigned

with nonlinear material models, midas GTS internally converts the material models into

linear type and performs a linear static stress analysis.

1 The current spring element of midas GTS does not consider the nonlinear behavior.



Chapter 3 – Seepage Analysis

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2.2 Linear Static Stress analysis

Linear static stress analysis is used to determine the overall behavior of soil and rock

structures by assuming that all ground elements are composed of linear elastic materials.

Linear static stress analysis neglects the failure mode and idealizes the stress-strain

relationship in a straight line. Since finite element equations are expressed in a linear

elastic form based on Hooke’s Law, nonlinear elastic analysis and elasto -plastic analysis

can be also executed in the form of the linear elastic equations.

The inception of Nonlinear or elasto-plastic analysis by finite element method became

common in the practice of geotechnical engineering in the early 1990s. Initially, there were

technical difficulties in performing nonlinear analysis in practice. Relatively long analysis

time required for convergence for nonlinear or elasto-plastic analysis had to be overcome

as well. At this time, linear elastic analysis prevailed. However, due to the rapid

development of computer analysis speed and nonlinear and elasto-plastic analysis

algorithms, such analyses are now widespread and easily applicable to many geotechnical

projects. Linear analysis is still widely used in practice for efficiency in conducting

preliminary analysis and simple analysis involving materials with relatively less inherent

nonlinearity.

In most civil engineering applications, analysis problems can be condensed to “whether or

not a structure is safe under a given set of loads” and “what the amount of deformation is

prior to structural failure”. In order to find geotechnical deformation, a stress -strain

relationship is required. However, the stress-strain relationships of geo-materials are very

complex. A stress-strain relationship varies widely depending upon the material

composition, porosity, stress history and loading method.

Since stress-strain relationships exhibit very complex nonlinear material characteristics, it is

not practical to mathematically represent them in numerical models. In practice, the

material models are idealized to a simpler form to model stress-strain relationships. In the

process of idealizing the material models, the modulus of elasticity and Poisson’s ratio

alone cannot adequately represent ground behavior realistically. But, approximations can

be made to analyze the behavior at some specific stress states. In such a case, a clear

definition of the modulus of elasticity is required to idealize the material model.




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The key material parameters are the modulus of elasticity, Tangent modulus, and Secantmodulus. For a perfectly linear elastic material model, the last two moduli are identical. The

Secant modulus is often referred to as the deformation modulus of a nonlinear material in

the stress range we are interested in.

One of the equilibrium equations used for linear elastic analysis is expressed in Eq. (2.1.1).

Ku p (2.2.1)

where,

K : Stiffness matrix

u : Displacement vector

p : Load vector or unbalanced force vector

From Eq. (2.1.1), the displacement vector can be determined. This method is known as the

displacement method, which obtains the displacement vector from the first found analysis

results. The strain is calculated from the displacement vector through the compatibility

equation, and then the stress is determined from the strain through the constitutive

equation.

When materials undergo deformation, as shown in Fig. 2.2.1, an internal point e(x, y, z)

moves to a new coordinate B (x+u, y+v, z+w). Unless the element is rigid, the

displacement vector , ,u v wU continuously changes, and it can be represented by a

function of x, y, and z coordinates.




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x y

z , ,u v wU

, , x u y v z w B , , x y z A

F igur e 2.2.1 Displacement vector (u, v, w)

The strain is expressed as follows using the distributed displacements.

x

y

z

xy

yz

zx

u x

v

y

w

z

v u

x y

w v

y z

u w

z x

(2.2.2)




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Applying uniaxial stress in an elastic material results in strain in the same direction as thestress.

z z

x y z

E

(2.2.3)

where,

, , x y z : Strain components in the x, y, z direction

E : Modulus of elasticity

: Poisson’s ratio

If shear stress, zx , exists, the shear strain can be expressed as Eq (2.2.4)

zx zx

G

(2.2.4)

where, G : Shear modulus

Shear modulus can be derived from the modulus of elasticity and Poisson’s ratio.

2 1

E G

(2.2.5)

The volumetric strain of soil is expressed as follows:

( ) (1 2 ) x y z

x y z V

V E (2.2.6)




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where,

1 [ ( )]

1[ ( )]

1[ ( )]

x x y z

y y z x

z z x y

E

E

E

(2.2.7)

Thus, the bulk modulus, K, can be defined as follows:

[( ) / 3]

/ 3(1 2 )

x y z E K

V V

(2.2.8)

Due to the characteristics of the ground continuum, the use of the bulk modulus (K) and

shear modulus (G) may be a little controversial. But they are more conveniently used as

they can be expressed more concisely and clearly compared to E and . Figure 2.2.2

illustrates the physical significance of K and G.




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F igur e 2.2.2 Vari ous types of modul i




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When boundaries are constrained horizontally, the constrained modulus, M, can bedetermined by deformation in one dimension. Eq. (2.2.9) and (2.2.10) give the equations for

horizontal stresses and the constrained modulus for 0 x y .

1 x y z

(2.2.9)

1

1 1 2 M E

(2.2.10)

The modulus obtained from field testing is one of the moduli above. This can be used in

practice after appropriately converting the modulus.

Boundary conditions for one dimensional consolidation behavior are identical to those for

the constrained modulus, which is closely related to the properties of one dimensionalconsolidation behavior in soft ground. Table (2.2.1) summarizes the relationship between

the one dimensional consolidation property parameter and constrained modulus.

The values shown in Table (2.2.2) are based on testing results obtained from uncracked

small specimens of various materials. Accordingly, reduced moduli of elasticity of rocks

should be used to consider the discontinuity of materials in a large scale. Figure (2.2.3)

shows a graph indicating the testing data of elastic modulus reduction ratio relative to theRock Quality Designation (RQD). RQD is defined as the cumulative length of core pieces

longer than 100 mm in a run divided by the total length of the core run expressed in a

percentage. Higher RQD values imply better the rock quality. An RQD of 100% does not

necessarily mean that the rock is perfectly continuous without defects. Weathered rocks

exhibit small RQD values.

As shown in Figure 2.2.3, the Modulus of Elasticity reduces from 100% to 20%, as the RQDreduces from 100% to 70%.

The 3D stress-strain relationship is expressed as follows:




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1/ / / 0 0 0

/ 1/ / 0 0 0

/ / 1/ 0 0 0

0 0 0 1/ 0 0

0 0 0 0 1/ 0

0 0 0 0 0 1/

x x

y y

z z

xy xy

yz yz

zx zx

E E E

E E E

E E E A

G

G

G

(2.2.11)

The above equation can be rewritten in an inverse matrix form:

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

2

1

0 0 0 0 02

10 0 0 0 0

2

x x

y y

z z

xy xy

yz yz

zx zx

A

(2.2.12)

where, 1 2 1

E A

That is,

σ = D ε (2.2.13)

3

x y z

x y z K

(2.2.14)

where, 3 1 2

E K




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The matrix D related to the compatibility matrix is as follows:

1 2 2

2 1 2

2 2 1

3

3

3

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

D D D

D D D

D D D

D

D

D

(2.2.15)

where,

1

2

3

4

3

2

3

D K G

D K G

D G

Table 2.2.1 Relationship between Consoli dation Parameters and Constrained Modulus

Consolidation Parameters Relationship with M

coefficient of volume change, vm 1

vm

M

coefficient of compressibility, va 01

v

ea

M

compression index, cc 0(1 )

0.435

vac

ec

M




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Table 2.2.2 Modul i of Elasticity and Poisson’s Ratios of r ocks & other materials

Material Elastic Modulus (tonf/m2) Poisson’s Ratio

amphibolite 9.4~12.1 106 0.28~0.30

anhydrite 6.8 106 0.30

siabase 8.7~11.7 106 0.27~0.30

siorite 7.5~10.8 106 0.26~0.29

solomite 11.0~12.1 106 0.30

sunite 14.9~18.3 106 0.26~0.28

deldspathic

gneiss8.3~11.9 106 0.15~0.20

gabbro 8.9~11.7 106 0.27~0.31

granite 7.3~8.6 106 0.23~0.27

ice 7.1 106 0.36

limestone 8.7~10.8 106 0.27~0.30marble 8.7~10.8 106 0.27~0.30

mica Schist 7.9~10.1 106 0.15~0.20

obsidian 6.5~8.0 106 0.12~0.18

oligoclasite 8.0~8.5 106 0.29

quartzite 8.2~9.7 106 0.12~0.15

rock salt 3.5 106 0.25

slate 7.9~11.2 106 0.15~0.20

aluminum 5.5~7.6 106 0.34~0.36

steel 20.0 106 0.28~0.29




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Rock Quality Designation (%)

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

?

?

?

M o d

u l u s R e d u c t i o n R a t i o ( E L / E M )

Results from DWORSHAK DAM, Deereet.al., 1967

Results after Coon and Merritt, 1970

ORANGE FISH TUNNEL ㅡ VERTICAL JACKING

TESTS, Oliver, 1977

ORANGE FISH TUNNEL ㅡ HORIZONTAL JACKING

TESTS

DRAKENSBERG TESTS

ELANDSBERG TESTS

OTHER DATA, 1978

F igur e 2.2.3 RQD vs. Reduction Ratio of Elastic Modulus ( / M L E E )




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2.3 Nonlinear Static stress analysis

2.3.1 Introduction

In geotechnical analysis, the nonlinear behavior is induced by the material nonlinearity of

foundation. The material nonlinear behavior is divided into nonlinear elastic, elasto-

plastic, and visco-elasto-plastic models according to the assumption on behavior. Each

model can be subdivided with respect to the characteristics of the foundation to analyze.

For small strain problems, nonlinear elastic model can be applied usefully. For large

strain problems which have something to do with the speed of applied load and seepage,

visco-elastoplastic analysis is appropriate. In majority of all other cases, elasto-plastic

analysis is used. In midas GTS, nonlinear elastic and elasto-plastic analyses are

available.

2.3.2 Nonlinear Elastic Analysis

A nonlinear elastic material changes its elastic characteristics depending upon the

analysis results. A hyperbolic model such as the Duncan-Chang model is a typical

example of a nonlinear material model which the stress and strain relationship is

described by a hyperbolic curve. The ground parameters are a function of confinement

stress and shear stress. The model is quite useful because it can be easily defined by

ground material properties obtained from triaxial tests or published literatures.

Hyperbolic curve

F igur e 2.3.1 Stress-Strain Cur ve of Duncan-Chang model




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2.3.3 Elasto-plastic Analysis

2.3.3.1 Introduction

The plastic strain is present after removing the applied load in the plastic behavior of

material unlike the elastic behavior. To represent the characteristic behavior, the total

strain can be calculated as follows:

e p ε ε ε (2.3.1)

where,

ε : Total Strain

eε : Elastic Strain

pε : Plastic Strain

Hook’s law defines the stress-strain relationship in elastic range, and from Eq. (2.3.1)

the stress can be expressed as the following:

e p σ Dε D ε ε (2.3.2)

where,

σ : Stress Vector

D : Material Stiffness Matrix

The stress at an arbitrary point in the structure subjected to loading is a criterion to

estimate the elasto-plastic stress state of that point. The criterion to evaluate the

elasticity and plasticity is defined with respect to the characteristics of the materials

such as steel and concrete, and which is called yield criterion. The yield criterion of

material is defined based on experimental data, and the stress state at which the

plastic flow is initiated can be expressed as a function in stress space below:




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, 0 f σ (2.3.3)

where,

f : yield function

: hardening parameter

If the value of the yield function f is less than zero, no plastic flow will occur. If f

is greater than zero, plastic flow will occur.

2.3.3.2 Plastic Flow Rule

The yielding point of a material would initiate the plastic flow, which induces the stress

redistribution for maintaining the equilibrium of material. The calculation of the plastic

flow is usually formulated by the incremental form to apply nonlinear analysis. The

representative variables for computing the plastic flow are the directions of

incremental stress and plastic strain. The direction of incremental stress can be written

as below:

ii

f

n

σ

(2.3.4)

where, n is the slope vector for the direction of the incremental stress perpendicular

to the yield surface, i is the total number of yield surfaces for the case that yield

function consists of multiple yield surfaces.

The incremental plastic strain is decomposed into two parts: magnitude and direction.

According to the Koiter’s law they are expressed below:

n n

i p i i i

i 1 i 1

g

ε mσ

(2.3.5)

where, i g is the plastic potential function which is considered as a function of the

stress and hardening parameter , and is obtained from the experiment test of

material.




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The plastic multiplier i is restricted by the Kuhn-Tucker conditions as follows:

0 f ;i

0 ; i 0 f (2.3.6)

These conditions show that no plastic flow will occur since i is always zero if the

yield function f is less than zero.

In Eq. (2.3.5), m is defined as a directional vector of incremental plastic strain. When

including the yield function f defines the incremental plastic strain, instead

of plastic potential function g , it is called the associated flow rule. When the yield and

plastic potential functions differ, / g σ is used as the relation, which is known as the

non-associated flow rule.

In general, the associated flow rule is applied to material models which have constant

deviatoric stress distributions along the hydrostatic axis in stress space such as von

Mises and Tresca models. The non-associated flow rule is used for material models

where the deviatoric stress distribution is not constant along the hydrostatic axis in

stress space such as Mohr-Coulomb and Drucker-Prager models. When the non-

associated flow rule is used for models with varying deviatoric stress distributions

along the hydrostatic axis, it can prevent the excessive volume expansion (dilatancyeffects) induced by inconsistency of directions of stress and strain. It is important to

note that slower convergence should be expected whenever a non-symmetric stiffness

matrix is required to solve the system of equations.

2.3.3.3 Linearized Consistency Condition

The internal state of elasto-plastic material is defined as the incremental constitutive

relation that is expressed by the large rate form. The plastic flow initiates when the

stress point reaches the yield criterion, and it is controlled by the parameter of plastic

state such as .

/ f σ




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The constitutive equation for the large strain of a non-associate plastic flow is given asbelow based on Eq. (2.3.2):

p( ) ( ) σ D ε ε D ε m (2.3.7)

When the stress point lies on the yield surface, the linearized consistency condition is

defined as the following first order expansion Taylor series.

T

T, 0

f f f h

σ σ n σ

σ

(2.3.8)

where, f

nσ

, and hardening modulus f

h

.

The / f

is calculated from the f

relationship based on the experiment. The/ depends on the material’s yield model, and it is calculated using the

relationships defined with respect to the yield models outlined in Section 1.2.

Substituting Eq. (2.3.7) into Eq. (2.3.8) and rearranging it for the plastic multiplier

give the following:

T

Th

n Dε

n Dm (2.3.9)

Substituting Eq. (2.3.9) into Eq. (2.3.7) gives the incremental form of stress-strain

relationship as:

T

epTh

Dmn Dσ D ε D ε

n Dm (2.3.10)

where, epD is the elastic-plastic tangent operator that means the tangential stiffness

matrix of material.




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Applying the non-associated flow rule, i.e., m n yields the non-symmetric matrixepD .

2.3.3.4 Integration of rate form

The integration methods for rate form of equations can be globally divided into the

explicit forward Euler algorithm with sub-increment and implicit backward Euler

algorithm.

In the explicit forward Euler algorithm, the direction of plastic flow is obtained at point

A, refer to Figure 2.3.2 & 2.3.3, where the stress rate intersects with the yield surface.

In the implicit backward Euler algorithm, it is calculated at point B, refer to Figure 2.3.4,

that is the final stress point.

The explicit forward Euler algorithm is relatively simple and does not require theiterative calculation at the integration points. However, the following shortcomings are

as follows:

conditionally stable

sub-increment is needed to obtain the admissible precision (Figure 2.3.3)

additional calibration is required for returning to the yield surface (Figure 2.3.3)

In addition, the consistent tangent stiffness matrix cannot be derived in the explicit

forward Euler algorithm. The implicit backward Euler algorithm produces the accurate

results within an acceptable range and is unconditionally stable, but it requires the

iterative calculation at the integration points. The method is more efficient than the

explicit forward Euler algorithm since the consistent tangent stiffness matrix can be

derived.




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(a) Location of the intersection point A

(b) Moving fr om A to C in the tangential direction and cali brating to the point D

F igure 2.3.2 Expl icit forward Euler algorithm

Yield surface

X

A

B

∆ σ e

Yield surface

X

A

B

C

D

r ∆ σ

e

( 1 - r ) ∆

σ e

-∆λ m A




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F igure 2.3.3 Sub-increment in the expli cit forward Euler algorithm

2.3.3.5 Explicit forward Euler method

In the explicit forward Euler method, the elastic stress increment is calculated based

on the assumption of elastic strain. Refer to point B in Figure 2.3.2(a).

e

e

B X

σ D ε

σ σ σ

(2.3.11)

In the next step, the incremental stress specifying the elastic limit is calculated. The

initial elastic stress increment is decomposed into the allowable stress increment

e1 r σ within elastic region, and the unallowable stress increment er σ outside

the yield function. The stress increment specifying the elastic limit is calculated from

the following equation. Refer to point A in Figure 2.3.2(a).

e

X

B

B X

1 , 0 f r f

r f f

σ σ

(2.3.12)

where the suffixes in Eq. (2.3.10) and Eq. (2.3.11) are given in Figure (2.3.2).

A

X

B B´ C

D

Yield surface

-∆λ m A A

-∆λ mBB

-∆λ mB´B´




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To apply the sub-increment method, the stress incremente

r σ outside the yieldfunction should be approximately subdivided into small k sub-increments of stress

(Figure 2.3.3). The number of sub-increments is directly related to the size of error, and

is calculated from Eq. (2.3.13).

effB effA effAINT 8 1k (2.3.13)

where, effA and effB are the effective stresses at the points A and B in Figure

2.3.2(a).

When the final stress point is not on the yield surface, the stress point should be

transferred to the point on the yield surface through the next iteration using a returning

algorithm. Refer to point D in Figure 2.3.3.

CC T

C C

D C C C

f

h

n Dm

σ σ Dm

(2.3.14)

2.3.3.6 Implicit backward Euler method

The explicit backward Euler method always requires the calculation of intersection

point since the next stress point is estimated using the direction vector perpendicular

to the yield surface on the intersection point A. However, the calculation of intersection

point is not required if the direction vector perpendicular to the yield surface were

reckoned on the point B.

C B C σ σ Dm (2.3.15)




midas GTS 25

The iterative analysis is needed to find the solution since both left and right hand sideterms have the values of the unknown point C as shown in Eq. (2.3.15). Then, the

residual vector r which means the difference between the current stress and the

backward-Euler stress is introduced for the iterative analysis.

C B C r σ σ Dm (2.3.16)

F igure 2.3.4 Impl icit forward Euler algorithm

The residual vector r converges to zero when the final stress point is on the yield

surface. The new residual vector newr during the iterative analysis can be expressed

as the following using the first order expansion of Taylor series:

new old

mr r σ Dm D σ

σ

(2.3.17)

The residual vector newr equals to zero according to the converged final stress.

Substituting new0r into Eq. (2.3.7), and rearranging it for σ gives the Eq. (2.3.18).

Yield surface

X

B

∆ σ e

C




midas GTS 26

1

1

old old

mσ I D r Dm R r Dm

σ

(2.3.18)

The yield function can be written as the following using the first order expansion Taylor

series:

new old old 0

T f f

f f f h

σ

n σ

σ (2.3.19)

Therefore, substituting Eq. (2.3.19) into Eq. (2.3.18), yields the following equation:

1

old old

1

T f

h

n R r

nR Dm

(2.3.20)

2.3.3.7 Constitutive matrix

The way to construct the plastic constitutive equation is explained as follows. The

large stress rate is determined from the elastic part of the large strain rate vector. To

wit,

p σ D ε ε Dε Dm (2.3.21)

Since the current stress point is always on the yield surface, the consistency condition

should be satisfied. Substituting Eq. (2.3.21) into Eq. (2.3.9) yields the large stress

rate of as shown in Eq. (2.3.22):

h

T

ep

T

Dmn Dσ D ε D ε

n Dm (2.3.22)

The converged solution can be obtained faster due to the quadratic convergence

property of the Newton-Raphson method when the consistent tangent stiffness matrix

is used with the Newton-Raphson method in the explicit backward Euler method. To




midas GTS 27

consider this quadratic property, differentiating Eq (2.3.15) gives the following:

m mσ Dε Dm D σ D

σ

(2.3.23)

where, is the rate form of .

Eq (2.3.23) is rewritten as:

Aσ Dε Dm (2.3.24)

where,

mA I D

σ

,

mm m

Putting 1H A D , Eq (2.3.24) can be rewritten as

σ H ε m (2.3.25)

Rewriting Eq (2.3.25) with respect to the total strain based on the linearized

consistency condition gives the following:

Tep

Th

Hmn Hσ H ε C ε

n Hm (2.3.26)

Where, epD in Eq (2.3.22) is the tangent stiffness matrix, and epC in Eq (2.3.26) is the

consistent tangent stiffness matrix.



Seepage Analysis

Chapter 3

midas GTS 28

3.1 Introduction to Seepage Analysis

This chapter describes the techniques for developing and organizing the seepage function

in the analysis solver, the formulation procedure, and the relevant technologies. This

chapter will enable the user to become familiar with the programming methodology, solve

problems, and determine the acceptability of the analysis results.

The seepage analysis is globally divided into steady state and transient analysis.

Steady state analysis calculates a time-independent solution, whereas transient analysis

calculates a time dependent solution as a result of changing inflow/and or out-flow

conditions of the steady state are imposed.

The groundwater seepage may occur if water-permeable aquifer, head difference or flux

exists between region interfaces.

The seepage flow streams along the waterway connected to the empty space between soil

particles inside the foundation. The flow governed by Darcy’s law. According to this law, the

seepage quantity of water passing through the volume of soil in steady state is equal to the

permeability coefficient multiplied by hydraulic gradient and area. Darcy’s law is originallyfor saturated zones, but also applies for unsaturated zones.

Unsaturated zones includes from bond drying to the state close to saturation. As the degree

of saturation decreases less than 100%, formation of air bubbles as well as water within the

pore between soil particles. When the degree of saturation is significantly low, the water

drops stick to the gaps of soil particles in the form of a concave surface.

As the degree of saturation decreases, the pore pressure gradually develops a tensile state

under the influence of surface tension, and accordingly, the negative pore pressure is called

suction pressure. In most cases, the suction pressure increases as the degree of saturation

decreases.




midas GTS 29

The transient analysis is different from the steady state analysis in that the boundaryconditions vary with time and the volumetric water content is needed. When the

groundwater level rises or descends, the water content and porosity of unsaturated zone

are necessary due to the rising effects and decreasing velocities.

There is a great difference between the times required to reach the steady state of seepage

inside when the case that the water filling of reservoir begins in perfectly dry state of soil

dam is compared to that the water filling of reservoir begins in the state containing somewater content.

3.2 Seepage Flow Rule

midas GTS provides seepage analysis based on laminar flow using Darcy ’s law., Darcy’s

law is expressed by the following equation :

q ki (3.2.1)

where,

q : Seepage quantity of water

k : Permeability coefficient

i : Hydraulic gradient

Darcy’s law is originally derived for saturated soil, and is verified to be applicable to the flow

of unsaturated soil according to published studies. The only difference is that the

permeability coefficient of unsaturated soil is not a constant but a function, and indirectly

varies with the change of pore pressure. Darcy’s law can be expressed as,

v ki (3.2.2)

where, v is known as the Darcian velocity. When the water flows through permeable or

porous media, the actual average velocity can be calculated by dividing the Darcian

velocity into the porosity of soil.




midas GTS 30

3.3 Governing Equations

midas GTS uses the following governing differential equation for the seepage:

x y z

H H H k k k Q

x x y y z z t

(3.3.1)

where,

H : Total water head

xk : Permeability coefficient for the direction of x

yk : Permeability coefficient for the direction of y

z k : Permeability coefficient for the direction of z

Q : Flow quantity : Volumetric water content

t : Time

The above equation shows that the change of volumetric water content equals to the

difference between the inflow and outflow quantities over an large volume in an arbitrary

location during an arbitrary time. In brief, the flow rates for the directions of x, y, and z axes

plus external flow quantity are same as the rate change of volumetric water content.

The governing equation is the seepage equation for the transient flow. Since the flow

quantities of inflow and outflow over the large volume are constant regardless of the time in

the steady state flow, the governing equation of Eq. (3.3.2) is obtained by assigning the Eq

(3.31) equal to zero.

0 x y z

H H H k k k Q

x x y y z z

(3.3.2)

The change of volumetric water content depends on the change of stress state and the

characteristics of soil.




midas GTS 31

The saturated and unsaturated stress state conditions are expressed as two statevariables. Those are a p and a w p p , where is the total stress, a p pore air

pressure, and w p pore water pressure.

The seepage analysis in midas GTS consists of the conditions of constant total stress. The

loading and unloading on soil does not exist. The pore air pressure is constant under the

atmospheric pressure during unsteady flow state since unloading on the soil by itself does

not exist. a p is a constant, and does not affect the change of volumetric water

content. Thus, the change of volumetric water content is only a function of the change of

pore pressure since it depends on the change of stress state a w p p , and a p is

constant.

The change of volumetric water content is related to the change of pore pressure according

to the stress state and the characteristics of soil, and the relation can be expressed as:

w wm u (3.3.3)

where,

wm : Storage coefficient

Also, the total head is expressed as the summation of pressure head and potential head as

shown in Eq (3.3.4):

w

w

p H y

(3.3.4)

where,

H : Total head

w p : Pore pressure

w : Unit weight of water

y : Height




midas GTS 32

Eq (3.3.4) is rewritten as:

w w p H y (3.3.5)

Substituting Eq (3.3.5) into Eq (3.3.3) gives the following,

w w

m H y (3.3.6)

Substituting Eq. (3.3.6) into Eq (3.3.1) :

x y z w w

H y H H H k k k Q m

x x y y z z t

(3.3.7)

Since height is constant, the derivative of y is eliminated but the following differential

equation still remains:

x y z w w

H H H H k k k Q m

x x y y z z t

(3.3.8)




midas GTS 33

3.4 Finite Element Formulation

The finite element formulation from application of the weighted residual Galerkin approach

to derivation of the governing differential equation is as follows:

,T T

V V A

T

t dV dV q dA B CB H N N H N (3.4.1)

where,

B : Hydraulic gradient matrix

C : Permeability coefficient matrix of element

H : Nodal water head vector

N : Interpolation function vector

q : Unit water quantity of element edge

: w wm Storage term of transient seepage

,t

H :t

H Change of water head with respect to time

For two-dimensional analysis, the thickness of all elements in midas GTS is considered

constant. Accordingly, the finite element formulation is expressed as:

,T T T

t A A L

b dA b dA qb dL B CB H N N H N (3.4.2)

where, b is the element thickness. When the thickness is constant over the entire area,

the integration is calculated over the entire area which is reduced to the integration along

the length from a node to the other.




midas GTS 34

The finite element formulation is simplified as given below:

,t KH MH Q (3.4.3)

where,

T

Ab dA K B CB : Element characteristic matrix

T

Ab dA M N N : Mass matrix

T

Lqb dL Q N : Applied water quantity vector

Eq (3.4.3) is a general finite element formulation for transient seepage analysis. For the

steady state seepage analysis, the equation is simplified since water head is not a function

of time, and accordingly the ,t MH term is eliminated.

KH Q (3.4.4)

3.5 Time Integration

The finite element solution of transient seepage analysis is a function of time as shown in

,t H term of equation 3.4.1. The time integration with respect to time is performed using the

finite difference approach.

The finite element equation is written in the finite difference method as follows:

1 0 1 01 1t t t K M H Q Q M K H (3.5.1)

where,

t : Time increment,

: Ration between 0 and 1,

1H : Water head at the final time increment

0H : Water head at the initial time increment




midas GTS 35

1Q : Nodal flow quantity at the final time increment

0Q : Nodal flow quantity at the initial time increment

K : Element property matrix

M : Element mass matrix

midas GTS uses the Backward Difference method with = 1.0. Since = 1.0, equation

3.5.2 can be rewritten as:

1 1 0t t K M H Q MH (3.5.2)

To determine the water head at the final time increment, the initial water head must be

known. Generally, the initial conditions must be known before carrying out a transient state

analysis.



Stress-Seepage Staggered Analysis

Chapter 4

midas GTS 36

4.1 Introduction to Stress-Seepage Coupled Analysis

Groundwater seepage flow is caused by the difference in water heads at the boundaries in

the seepage zone or the flux boundary condition. As the groundwater flows, seepage force

is developed due to the friction between the water and soil particles. This process induces

displacements and stresses. Seepage force is expressed as an integration of the pore

water pressure.

midas GTS determines the seepage force using the pore water pressure obtained by

seepage analysis.

Pore water pressure is the product of the weight density of water and the pressure head

(total head - elevation head). In general, the seepage force is extensively used in areas

near outflow boundaries where the total head reduces drastically. Due to a low constraint

pressure, shear and tensile strengths are relatively low in these areas. As a result, when

considering seepage force to analyze the effective stress, the chances of failure is more

likely to occur. Stress-seepage Semi-coupled analysis is a very important field for

determining the stability of ground structures.

4.2 Effective Stress

Pore water pressure in ground affects the total stress. According to the Terzaghi’s theory,

the total pressure can be divided into effective stress and pore water pressure ( pw ).



Chapter 4 – Stress Seepage Coupled Analysis

midas GTS 37

Since water cannot resist any shear stress, the effective shear stress is equal to the totalshear stress. Thus, the total stress is expressed as:

xx xx w

yy yy w

zz zz w

xy xy

yz yz

zx zx

p

p

p

(4.2.1)

The pore water pressure can be divided into “steady state pore water pressure” steady p

and “excess pore water pressure” excess p .

w steady excess p p p (4.2.2)

The steady state pore water pressure is input data obtained from the groundwater analysis,

which is determined by the height of groundwater table. The excess pore water pressure

occurs during the stress calculation of a material, which exhibits the behavior of undrained

materials. Calculation of excess pore water pressure is described in more detail in the

subsequent chapter.

4.3 Governing Equation

The inverse elastic Hooke’s law is considered to derive Hooke’s law expressed in large

strains.




midas GTS 38

1 0 0 01 0 0 0

1 0 0 01

0 0 0 2 2 0 0

0 0 0 0 2 2 0

0 0 0 0 0 2 2

e

xx xx

e yy yy

e zz zz

e xy xy

e yz yz

e zx zx

E

(4.3.1)

where,

, E : Effective material properties – modulus of elasticity and Poisson’s ratio respectively.

However, since the derivative of the steady state component is zero,

w excess p p

(4.3.2)

By applying the terms in Eq. 4.2.1, the above Eq. (4.3.1) can be rewritten as follows:

1 0 0 0

1 0 0 0

1 0 0 01

0 0 0 2 2 0 0

0 0 0 0 2 2 0

0 0 0 0 0 2 2

e xx excess xx

e yy excess yy

e zz excess zz

e xy xy

e yz yz

e zx zx

p

p

p

E

(4.3.3)



Consolidation Analysis

Chapter 5

midas GTS 39

5.1 Introduction to Consolidation Analysis

Consolidation analysis is closely related to drained/undrained analysis where the

dissipation of excess pore water pressure is calculated. The excess pore water pressure

takes place when the water resists loadings.

The entrapped water, which has the ability to resist loading, slowly drains with time by

applying drainage boundary conditions. The excess pore water pressure thus dissipates,

which results in the deformation of the soil skeleton. The decrease in water pressure results

in the increase in the effective pressure, which exists in the soil skeleton.

Consolidation analysis addresses the reduction of excess pore water pressure resulting in

the increase in effective pressure with respect to time. During the process, the ground soilsettles in the direction of gravity.

The increasing deformation with respect to time induces settlement foundation of structure.

The differential settlement of the foundation has a great influence on the structural stability.

In consolidation analysis, ground failure is less likely to occur since the ground soil

parameters strengthen over time. Undrained analysis generally precedes consolidationanalysis. If failure does not occur in the undrained analysis, failure does not occur in

consolidation analysis.




midas GTS 40

5.2 Governing Equation

Until now solid and fluid analyses have been considered individually or by superposition.

However, Biot has formalized the theory of coupled solid-fluid interaction applied to soil

mechanics. Ground is treated as porous elastic solid, and the pore fluid coupled with the

solid is based on the conditions of compressibility and continuity.

The governing equation of Biot is given in Eq. (5.2.1).

2 2 2

2 2 2

w w w w x y z

w

K p p p p pk k k

x y z t t

(5.2.1)

where,

K : Bulk modulus of ground, p : Average total stress

A two dimensional state is assumed to simplify the equation. When there is no element load

in an equilibrium condition, the rate of change in effective stress is increased as Eq. (5.2.2)

by the rate of change in pore water pressure.

0

0

xy x w

xy y w

p x y x

p

x y y

(5.2.2)

The constitutive rules for solid Eq. (5.2.1) and fluid Eq. (5.2.2) have been defined

separately before. If we now assume full saturation and incompressibility, the discharge of

water from the ground elements reduces the volume of the elements. Accordingly,

y x qq d u v

x y dt x y

(5.2.3)



Chapter 5 – Consolidation Analysis

midas GTS 41

The third differential equation from Eq. (5.2.1) is given as Eq. (5.2.4).

2 2

2 20

y x w w

w w

k k p p d u v

x y dt x y

(5.2.4)

5.3 Formulation of Finite Element

In general, displacement methods eliminate the following parameters and by

them with u and v terms. Accordingly the final coupled variables are u , v and pw .

These are expressed as Eq. (5.3.1) in accordance with general finite element

discretization.

w w w

u

v

p

Nu

Nv

N p

(5.3.1)

where,

u, v, pw : x & y direction displacements and pore water pressures

,w

N N : shape functions for displacement and pore water pressure

Even though a higher order shape function is commonly used for the displacement, a linear

shape function is preferred for the pore water pressure.

Arranging the governing equation for equilibrium and continuity into a finite element

formulation using the Galerkin method, the coupled equation as Eq. (5.10) is obtained.

w

T

w w

d

dt

Ku Cp p

uC K p 0

(5.3.2)

where, C is a quadrilateral connection matrix, which is composed by the terms of the




midas GTS 42

following type:

j

i

N N dxdy

x

(5.3.3)

The above finite element equation needs to be integrated again over time. This is executed

by the following linear interpolation by the finite difference method.

1 1 0 0

2

1 1 0 0

1 1

1

w w

T T

w w w wt t

Ku Cp Ku Cp p

C u K p C u K p (5.3.4)

where, the loading, p is assumed to be unchanged with time.

If the fully implicit method

1 is used, the above Eq. (5.3.4) can be expressed in the

recurrence relationship as Eq. (5.3.5).

1 0

1 0

T T

w w wt

K C u u0 0 p

C K p pC 0 0 (5.3.5)

5.4 Properties of Consolidation Element

The important variables commonly used in a finite element solution for consolidation

analysis are displacement and pore water pressure. As incremental displacements are

related to nodal displacements through an interpolation function, incremental pore

water pressures are also related to nodal pore water pressures through an

interpolation function in a similar way.

nodal

w w w p N p (5.4.1)

From the above relationship equation, the right side represents the incremental pore

water pressure, and the left side represents the interpolation function of each pore

water pressure and the nodal incremental pore water pressure. midas GTS assumes

that the interpolation function of pore water pressure is identical to that of displacement.




midas GTS 43

In consolidation analysis, elements retain the degrees of freedom for nodaldisplacements together with the degrees of freedom for pore water pressures

additionally. Consolidation analysis in midas GTS assumes that all elements retain

the degrees of freedom for pore water pressures unless two boundary conditions (non-

consolidation condition and draining condition) are defined. If elements are not

expected to become subjected to consolidation, for instance, embankment materials,

then we need to assign them a non-consolidation condition, as shown in the Figure

5.4.1, so that they act as general structural elements.

If draining conditions are modeled, then the consolidation elements must be subjected

to drainage conditions at the boundaries. Once consolidation analysis is carried out

with such boundary conditions, the excess pore pressures must become “0” in regions

where non-consolidation and draining conditions are assigned.

F igur e 5.4.1 Boundary condition of consoli dation element

Elements that can be assigned as consolidation elements presently are only plane

strain elements (3-node, 6-node, 4-node and 8-node) and solid elements (4-node, 10-

node, 6-node, 15-node, 8-node and 20-node). Detail description about the degrees of

freedom for displacements and pore water pressures per element and the integration

methods are outlined below.




midas GTS 44

5.4.1 Degrees of freedom for plane strain element

If elements are of higher order, excess pore water pressure is first linearly interpolated to

obtain stress analysis results. The nodes representing the excess pore water pressure

act as a dummy variable, see Figure 5.4.2. In consolidation analysis, every node for

excess pore water pressure is located at corners, and a linear function is used to

interpolate all of the elements. This is known to be appropriate for the distribution of

excess pore water pressure.

Triangular elements

Structural DOF

Pore Pressure DOF

Structural DOF

Dummy node for porepressure output which is

linearly interpolated

1 Integration Point

Pore Pressure DOF

3 Integration Point

F igure 5.4.2 DOF for a lower order 3-node and a higher order 6-node tr iangul ar elements




midas GTS 45

Quadrilateral elements

Structural DOF

Pore Pressure DOF

Structural DOF

Pore Pressure DOF

Dummy node for pore

pressure output which

is linearly interpolated

4 Integration Point

4 Integration Point

F igure 5.4.3 DOF f or a lower order 4-node and a higher order 8-node quadri lateral elements

5.4.2 Degrees of freedom for solid elements

Hexahedral elements

Structural DOF

Pore Pressure DOF

Structural DOF

Pore Pressure DOF

Dummy node for pore

pressure output which is


27 Integration Point

8 Integration Point

F igure 5.4.4 DOF for a lower order 8-node and a higher order 20-node hexahedral elements




midas GTS 46

Pentahedral elements

Structural DOF

Pore Pressure DOF

Structural DOF

Dummy node for pore

pressure output which is


Pore Pressure DOF

9 Integration Point

2 Integration Point

F igure 5.4.5 DOF for a lower order 6-node and a higher order 15-node Pentahedral elements

Tetrahedral elements

Structural DOF

Pore Pressure DOF

Structural DOF

Dummy node for porepressure output which islinearly interpolated

1 Integration Point

Pore Pressure DOF

4 Integration Point

F igure 5.4.6 DOF for a lower order 4-node and a higher order 10-node tetrahedral elements




midas GTS 47

5.5 Precautionary Points about Consolidation

Analysis

5.5.1 Time and loading input

For consolidation analysis, time and loading distribution is carried out by entering thevalues as shown in Figure 5.5.1:

(a) Initial load (b) Additional load (c) Long-term load placement

F igure 5.5.1 I nput dialog for rimes and load factors

1.0

1.0

loading

time 1.0

1.0

loading

time10.0

...

:

=1.0

1

loading

time160.0

82 4 ...

(a) Initial load (b) Additional load (c) Long-term load placement

F igure 5.5.2 Time and load graphs




midas GTS 48

Consolidation analysis using material models such as Cam-Clay or Modified Cam-Clay,

an initial analysis must be performed in order to take into account initial loads, i.e.

dead loads. Because the analysis is intended to define the initial stress, generally

within the linear range, a load is applied immediately as one “time step” as shown in

the Figure 5.5.1“(a) Initial Load.”

When additional loads are loaded as in construction staged consolidation, the total

additional load is loaded at appropriate intervals over the total time as shown in the

above Figure 5.5.1 “(b) Additional loads”. Figure 5.5.1 (b) shows that additional loads

are loaded at 10 equal intervals over a total of 10 days. The sum of the total load

factors must equal to 1.0. If additional loads or the analysis time are inappropriate,

convergence difficulties may arise in consolidation analysis. In such cases, the time

step needs to be appropriately reduced and the load factor needs to be set smaller

accordingly in order to improve the convergence.

Once the embankment operation is completed, analysis for a long-term load

placement is carried out during which only time elapses. Figure 5.5.1 (c) shows that

only time passes while the final embankment is placed for a long time. The load factor

0.0 means that no additional external load is applied. Due to the characteristics of

consolidation analysis, stress variations in the initial stages are rather drastic. Figure

5.5.1 (c) shows that the initial time steps start with small time intervals and gradually

increase to larger intervals to achieve a better convergence and faster analysis speed

simultaneously.



Construction Stage Analysis

Chapter 6

midas GTS 49

6.1 Introduction Construction Stage Analysis

Numerical analyses in many geotechnical applications with construction phases can be

easily simulated in the construction stage analysis. Geotechnical analysis usually involves

material nonlinear analysis in which the material nonlinear properties can be obtained from

the initial condition of the ground. The initial condition, which is the condition prior to any

construction events, refers to the in-situ condition or initial ground condition.

Once the initial ground stress is obtained, excavation loads can be obtained. In addition, the

shear strength of a nonlinear material can be calculated by applying material properties

such as Mohr-Coulomb. Therefore, the construction stage analysis entails the entire

construction process from the initial ground condition. Since the actual sequence of a

construction site is very complex and variable, the analysis is simplified to focus on the

major construction events in sequence.

For example, the construction stages for a tunnel are shown below:

1st Stage : Initial ground stress

2nd

Stage : First face excavation

3rd Stage : Install reinforcement to first face + 2nd face excavation

4th Stage : Install reinforcement to second face + 3

rd face excavation

5th Stage : Install reinforcement to third face + 4

th face excavation

……… (Continue) …………




midas GTS 50

midas GTS considers the following items in construction stage analysis:

1. Construction stage definition

Inactivate /Activate/ Deactivate Elements and Loads

Modify boundary condition

Reflect change in material properties

Load distribution ratio

Sequential groundwater level

Drained/undrained analysis

Displacement initialization

In-situ analysis

Restart

2. Seepage Analysis

Sequential Steady State Seepage Analysis.

Sequential Transient Seepage Analysis

3. Stress-Seepage Semi-Coupled Analysis

Effective stress analysis using pore water pressure results obtained from seepage

analysis.

4. Consolidation Analysis

Consolidation analysis according to embankment

When “Defining Construction Stages”, all elements, loads, and boundaries can be classified

as the following for each relevant stage:

A. Inactive

B. Activated

C. Deactivated

When additional elements are activated in a particular construction stage, these elements

are not affected by the previous construction stages New construction stages are created

per construction sequence to reflect any structural changes and updates in the analysis.

This means that inactive elements activated in a relevant construction stage will have zero

stress regardless of the loadings to which the structure is subjected.



Chapter 6 – Construction Stage Analysis

midas GTS 51

midas GTS does not create and analyze independent analytical models for each

construction stage. After completing the analysis at each stage where the changes in

structure and loadings are reflected appropriately, the analysis results from the previous

stage are accumulated and analyzed. The concept of an accumulated model is used.

Therefore, any change in structure or loadings in a stage influences the subsequent

construction stages. For example, once a load is applied to the structure in a construction

stage, the load remains active for the subsequent construction stages.

6.2 Activation and Deactivation of Elements

Inactive elements that are activated in a construction stage are assigned with zero initial

stress at which point their self weights are applied to the existing structure. These activated

will become part of the total structure in the subsequent stages.

All ground elements are subject to stress before deactivation. If a load is applied around thedeactivated elements, proper stress redistribution for the remaining activated elements must

take place so that the new free faces do not become subjected to any stress. This can be

explained by using an example:

Let us assume that Object A will become removed from Object B as shown in Figure 6.2.1.

A

B

AO

BO

T AB

F

ㅡ T BA F

(a)

(b)

(c) F igure 6.2.1 Concept of El ement Deactivation & Equil ibri um Forces




midas GTS 52

The stresses in Object A and Object B prior to removal of Object A are AO and BO

respectively. All external forces have been reflected in these stresses. Since these two

objects are in equilibrium, the load, AB F , must be applied to Object B due to Object A to

maintain the stress, BO . Similarly, the load, BA F , must be applied to Object A. Therefore,

any excavation load applied to a boundary depends on the stress state and the self weight

of the elements being excavated. This is expressed in Eq. 6.2.1.

A A

T T

BA AO A AV V

F dV dV B N (6.2.1)

where,

B : Strain-Displacement Matrix,

AV : Volume of excavation,

N : Element shape function,

: Ground weight density

The global stiffness for the current construction stage is calculated by assembling the

stiffness parameters from each activated element as shown in Eq. (6.2.2). All elements used

in every construction stage are stored in the database, and the element information in the

current stage is defined by adding all elements retrieved as activated from initial to current

stage and by subtracting all elements deactivated.

1

nT

i i i

i

K L k L (6.2.2)

where,

K : Global stiffness of current construction stage

iL : Matrix defining the location of element stiffness

ik : Element stiffness activated in the current construction stage

n :Total number of elements activated in the current construction

stage




midas GTS 53

6.3 Activation and Deactivation of Loads

The construction stage analysis consists of the phased analysis with incremental form, and

the incremental load of the current construction stage is needed for the analysis. The

incremental load currentF of the current stage is the difference between total load currentF

of the current stage and total load previousF of the previous stage.

current current previous F F F (6.3.1)

All the loadings used in every construction stage are stored in a database, and the total load

currentF of the current stage is defined by adding all loadings retrieved as activated from

initial to current stage and by subtracting all deactivated loads.

The total loads of the current stage are divided into surface force and body force as the

following:

current current current

surface body F F F (6.3.2)

where, surface force includes the loadings related to nodes such as nodal force or

distributed force, and body force includes the loadings related to elements such as self-

weight. While the surface force is assembled by using the information about the degrees of

freedom of the nodes where the loads are imposed, the body force is assembled by using

the matrix iL in Eq. (6.2.2) since it is related to adding and subtracting of elements.

The sub-incremental load step is defined as the incremental load currentF in Eq. (6.3.1)

multiplied by load factor of the current construction stage. It can be expressed for an

arbitrary load step i as

current previous current current

i i F F F (6.3.3)

The linear algebraic equation of static stress analysis calculates the response of structure




midas GTS 54

for an arbitrary load step i is given as

ubf

i i iF K d (6.3.4)

where iK is the global stiffness of i th load step, and id is the nodal displacement of i th

load step. Theubf

iF is the unbalance force of the i th step, and is computed by subtracting

the total internal force of previous step from the total current load.

ubf current internal

1i i i F F F (6.3.5)

In Eq. (6.3.5),current

iF is given in Eq. (6.3.3), andinternal

1iF is the total internal force of the

previous step which is computed from the element stresses.

internal T T

1 1 1

dn n

T

i j j j j jV j j

V

F L f L B σ (6.3.6)

where,

jf : Element member force

jL : Matrix defining the location of loading in an element

jB : Matrix of slope for displacement-strain relation

jσ : Element stress

V : Volume

n : Number of elements activated in the current construction stage

For 1i ,internal

0F is the total internal force of the final load step of the previous

construction stage. The element stress is updated after calculating each load step.




midas GTS 55

6.4 Activation and Deactivation of Boundary

Conditions

Activated boundary conditions are stored in a database similar to elements and loads. In

case of “f ree nodes”, they must be constrained to keep the mesh consistent. If all elements

connected to a node are deactivated, then the degrees of freedom connected to this node

will automatically be removed from the system of equations. This will improve the

computational efficiency by eliminating the free nodes by use of the information of nodes

included in the activated elements.

6.5 Load Distribution Factor

In order to simplify the construction stage analysis, load distribution factors can be

considered. This is a numerical technique which enables the user to reflect the changingeffects of an element by applying “distribution factors”. Distribution factors are sequentially

applied to represent a number of construction stages, as shown in Figure 6.5.1. The

concept is generally used when construction stages are condensed in a 3 dimensional

analysis, or when a 3 dimensional model is simplified to a 2 dimensional model.

For example, if a sequential transfer of stresses takes place over 3 steps, 40%, 30% and

30%, from an excavation stage. In midas GTS, the user can define the excavation stageand activate the load distribution factor option. The user can then simply enter 0.4, 0.3 and

0.3 for After Current Stage 0, 1 and 2 respectively.




midas GTS 56

F igure 6.5.1 Load Distri bution Factor Dialog Box

If we assign a 100% load distribution factor to deactivating elements, all the internal

stresses in the deactivating elements are redistributed to the remaining elements of the

structure causing stress changes in the remaining elements. Conversely, if we assign 0%

load distribution factor to the deactivating elements, no redistribution of the stresses in thedeactivating elements takes place. In such a case, the stresses in the remaining elements

in the structure remain unchanged.

The load distribution factor controls the amount of stress transfer from the deactivated

elements to the remaining elements. This function becomes very useful when the user has

to consider partial or staged stress transfers even after completely removing elements.

For tunnel analysis, the stresses in the excavated elements are not transferred at once.

Instead, the stresses are gradually redistributed to the rock bolts and shotcrete over a

number of stages. In such a case, the load distribution factor is used to reflect partial

transfer of stresses.




midas GTS 57

6.6 Initialization of Displacements

During the construction stage analysis, it may be necessary to initialize displacements to

zero at the initial stage. For instance, displacements must be initialized to zero at the end of

the initial stress calculation in the first construction stage.

Since the displacements can be initialized at any construction stage in midas GTS, the

initial stage conditions and intermediate stages can be obtained.

6.7 Change Element Attribute

Change Element Attribute is a function in midas GTS that is used to modify material

attributes in a construction stage analysis in order to reflect any improvements, ground

disturbance, and material hardening behavior over time. For instance, changing the

structural lining attributes, such as hardening of lining concrete and lining thickness, may be

modified during a particular stage.

A precautionary note, although linear elastic materials can be easily implemented, care

should be used when dealing with nonlinear material models. For instance, nonlinear

material models such as foundations, it is the case to fill up with different material after

excavation in construction stage analysis, and to change the material properties only

without adding or removing the element. It is physically not exact behavior to change the

material properties because the stress state from the previous step will be reflected in

subsequent stages of the analysis. Therefore, the material properties for a nonlinear

material model should be changed in the new stage after “deactivating” the elements. When

new elements are created, midas GTS initialize the stresses, strains, and internal state

variables of the new elements to zero.




midas GTS 58

6.8 In-situ Stress

midas GTS provides the following two methods for computing in-situ stresses.

6.8.1 0 K Method

The K0 method defines the earth pressure by calculating horizontal stresses from vertical

stresses using a non-dimensional quantity 0 K in ( 0 /h v K ).

The vertical stress v is obtained first by performing a self-weight analysis, and then

the horizontal stress is modified using the relation 0h v K . In this case, the shear

stresses on the vertical and horizontal planes of the soil element are unchanged.

If the ground surface is horizontal, this method presents no problem. If the surface,

however, is not horizontal, the stress state determined by this method under self weight

will not be in static equilibrium. midasGTS always performs iterations using the K0

modified stresses as a starting point, to reach equilibrium. That means that calculated

stresses at the end of the K0-construction stage do not necessary obey the K0-condition.

6.8.2 Self Weight Analysis Method

The initial stress state is determined by the stress state obtained by self weight analysis.

If the ground surface is horizontal, this method is identical to that of the K 0 method,

0 /(1 ) K . If it is not horizontal, the results will differ from those of the K 0 method

and shear stress will exist because horizontal strains exist.

Therefore, this method is more suited for a sloped ground surface. This method does not

allow us to specify a K 0 value greater than 1. If large values of K 0 are unavoidable, thenthe K 0 method is used and a null stage is used subsequently to satisfy an equilibrium

condition.




midas GTS 59

6.9 Restart Function

Construction stage analysis may be interrupted during analysis midway, or the program may

terminate due to system instability. In unforeseeable circumstances, GTS provides a restart

function “Save for restart” enables users to continue the analysis from where it has left off

and select the restart stage in the analysis controls option. Refer to Online help for details.

This option is only available for Construction Stages. This is a useful feature that will

significantly reduce the total analysis duration against unexpected errors during the analysis.Note: restarts can be defined per stage. The restart stages can be initialized in the

analysis control panel.

6.10 Undrained Analysis

In midasGTS, undrained analysis can be applied to the selected elements and associated

construction stages. There are two handles to activate the undrained option in midasGTS.

First, the material properties for the undrained analysis are defined in the material model ’s

drainage parameters. Second, they are activated under the construction stage. If the

drainage parameter option “undrained” is assigned, and not activated as “undrained” in the

construction stage, the material would assume drained behavior. Conversely, if the

drainage parameter option “drained” has been assigned, and activated as “undrained” in the

construction stage, the material would also behave as drained. For activating the undrained

option, the drainage parameter "undrained" and "undrained" option in the construction stage

must both be selected."

It is possible to conduct drained and undrained analysis in turn with respect to construction

stages. When undrained analysis was conducted in the previous stage, and drained

analysis is conducted in the current stage, it can be observed that the pore pressure

decreases due to the dissipation of pore pressure. When drained analysis was conducted in

the previous stage, and undrained analysis is conducted with applied loading in the current

stage, it can also be observed that the pore pressure changes due to increasing of pore

pressure.



Dynamic Analysis

Chapter 7

midas GTS 60

7.1 Introduction to Dynamic Analysis

7.1.1 Eigenvector Analysis

In midas GTS, Mode shapes and natural periods of an undamped free vibration are

obtained from the characteristic equation below:

n n n K M (7.1.1)

where,

K : Stiffness matrix

M : Mass matrix

n : n-th mode eigenvalue

n : n-th mode eigenvector (mode shape)

Eigenvalue analysis is also referred to as “free vibration analysis” and is used to analyze

the dynamic characteristics of structures.

The dynamic characteristics obtained by an Eigenvalue analysis include vibration modes

(mode shapes), natural periods of vibration (natural frequencies) and modal participation

factors. They are determined by the mass and stiffness of a structure.

Vibration modes take the form of natural shapes in which a structure freely vibrates or

deforms. The first mode shape or natural vibration shape is identified by a shape that

can be deformed with the least energy or force. The shapes formed with increase in

energy define the subsequent higher modes. Figure (7.1.1) shows the vibration modes

of a cantilever beam, arranged in the order of their energy requirements for deflected

shapes, starting from the shape formed by the least energy.

A natural period of vibration is the time required to complete one full cycle of the free

vibration motion in the corresponding natural mode.



Chapter 7 – Dynamic Analysis

midas GTS 61

(a) M ode shape

sec sec sec

1 1.87510407

1 1.78702 secT

2 4.69409113

2 0.28515 secT

3 7.85475744

3 0.10184 secT

(b) Natural peri od

F igure 7.1.1 Mode shapes and corr esponding natural periods of a pr ismatic canti lever beam

The methods to calculate the natural period in a single degree of freedom system

(SDOF) assumes that the load and damping terms equal to zero. The equation of




midas GTS 62

motion in SDOF yields to the free vibration equation of a linear second order differential

equation as given in Eq. (7.1.2).

( )

0

mu cu ku p t

mu ku

(7.1.2)

Since A is the displacement due to vibration, we simply assume cosu A t , where,

A is a constant related to the initial displacement. Therefore, Eq. (7.1.2) can be written

as:

2( ) cos 0m k A t (7.1.3)

In order to satisfy Eq. 7.3, the value in the parenthesis must be zero, which leads to Eq.

(7.1.4)

2 1, , ,

2

k k f T

m m f

(7.1.4)

where,

2 : Eigenvalue

: Rotational natural frequency

f : Natural frequency

T : Natural period

The modal participation factor is expressed as a contribution ratio of the corresponding

mode to the total modes and is written as:

2i im

m

i im

M M

(7.1.5)

where,

m : Modal participation factor,

m : Mode number,

D i A l i




midas GTS 63

i M : Mass at location i,

im : m-th mode shape at location i

In most seismic design codes, it is stipulated that the sum of the effective modal masses

included in an analysis should be greater than 90% of the total mass. This will ensure

that

the critical modes that affect the results are included in the design.

2

2

im i

m

im i

M M

M

(7.1.6)

where,

m M : Effective modal mass

If certain degrees of freedom of a given mass are constrained, the mass will be included

in the total mass but excluded from the effective modal mass due to the restraints on the

corresponding mode vectors. Accordingly, if the user attempts to compare the effective

modal mass with the total mass, the degrees of freedom pertaining to the mass

components must not be constrained.

In order to analyze the dynamic behavior of a structure accurately, the analysis must

closely reflect the mass and stiffness, which are important factors for determining the

eigenvalues. In most cases, finite element models can readily estimate the stiffness

components of structural members. In the case of mass, however, the user is required to

pay particular attention for obtaining an accurate estimate. The masses pertaining to the

self-weights of structural components are relatively small compared to the total mass. It

is quite important that an eigenvalue analysis accounts for all the mass components in astructure.

Mass components are generally specified as 3 translational masses and 3 rotational

mass moments of inertia consistent with 6 degrees of freedom per node. The rotational

mass moments of inertia pertaining to the rotational mass inertia do not directly affect the

A l i C




midas GTS 64

dynamic response of a structure. Only translational ground accelerations are typically

applied in a seismic design. However, when the structure is of an irregular shape, where

the mass center does not coincide with the stiffness center, the rotational mass moments

of inertia indirectly affect the dynamic response by changing the mode shapes.

Mass components are calculated by the following equations: (See Table 7.1.1)

Translational mass: dm (7.1.7)

Rotational mass moment of inertia: 2r d

where, r : distance of the total mass center to the center of an large mass.

The units for mass and rotational mass moment of inertia are defined by the unit of

weights divided by the gravitational acceleration, W(T2/L) and the unit of massesmultiplied by the square of a length unit, W(T2/L)L2 respectively. Here, W, T and L

represent the weight, time and length units respectively. In the case of an MKS or

English unit system, the mass is determined by the weight divided by the gravitational

acceleration. The masses in an SI unit system directly use the weights in the MKS units,

whereas the stiffness or loads in the MKS units are multiplied by the gravitational

acceleration for the SI unit system.

Ch t 7 D i A l i




midas GTS 65

Table 7.1.1 Calcu lations for Mass data

: mass per uni t area : mass center

Shape Translation massRotational mass moment

of inertia

Rectangular shape

M bd

3 3

12 12m

bd db I

2 2

12

M b d

Triangular shape

M area of

triangle m x y

I I I

Circular shape

2

4

d M

4

32m

d I

General shape

M dA m x y I I I

Part 1 Analysis Case




midas GTS 66

Line shape

L mass per unit

lengh

L M L

3

12m L

L I

Eccentric mass

Eccentric mass : m

M = m

Rotational mass moment

of inertia about its mass

center : o I

2

m o I I mr

midas GTS uses lumped masses in analyses for efficiency. Mass data can be entered in

the main menu through Model > Load > Nodal Mass.

midas GTS adopts the subspace iteration method for the solution of an eigenvalue

analysis, which is suitable for the analyses of large structures.

7.2 Consideration of Damping

7.2.1 Introduction

Structural damping in a dynamic analysis can be largely classified into the following:

A. Viscous damping (Voigt model and Maxwell model)

B. Hysteretic damping

C. Friction damping

1. Internal friction damping (Material damping)

2. External friction damping

3. Sliding friction damping

D. Radiation damping

E. Modal damping

Chapter 7 Dynamic Analysis




midas GTS 67

1. Proportional damping

2. Mass proportional type

3. Stiffness proportional type

4. Rayleigh type

5. Caughey type

F. Non-proportional damping

1. Energy proportional type

Among the many different ways of expressing the damping phenomena above, modal

damping is most frequently used in the numerical analyses of structures. The values for

modal damping are determined for each modal natural frequency of a vibration system.

The modal damping can be classified into proportional and non-proportional damping.

midas GTS provides proportional damping, which includes mass proportional, stiffness

proportional and Rayleigh type damping.

In midas GTS, mass-proportional, stiffness-proportional, and Rayleigh types of damping

are available.

7.2.2 Proportional damping

The mass-proportional damping represents the external viscous damping caused by air

resistance, and is based on the assumption that damping matrix is proportional to mass.

The stiffness-proportional damping might overestimate the damping of high order modes

since it is assumed that damping is proportional to stiffness due to the difficulty in

considering the vibration energy release directly.

Rayleigh damping is from modifying the damping value of high order modes in stiffness-

proportional damping, and is expressed by combining the mass and stiffness types of

proportional damping.

The general type of proportional damping matrix C is defined by Caughey as follows:





midas GTS 68

N 11 j

j

j 0C M{ a ( M K ) }

(7.2.1)

where,

, j N : Degrees of freedom of nodes (Mode number)

In equation (7.2.1), 1 M K can be obtained from the equation of free vibration of an

undamped system below:

M{ y } K{ y } 0 (7.2.2)

iax{ y } { u }e (7.2.3)

Substituting into the equation (7.2.2), it becomes:

2( M K ){ u } { 0 } (7.2.4)

From the equation (7.2.4), 1 2 M K is obtained. Here, 2 exists in as many numbers

as the number of modes. Considering all the modes, it is expressed as 2

s .

Substituting 1 M K obtained from the equations (7.2.2)~(7.2.3) into the equation (7.2.1),

and multiplyingT

s{ u } on the left hand side of the equation and s{ u } on the right hand

side of the equation, the equation (7.2.5) becomes:

N 1 N 1T 2 j T 2 j

s s s j s s s j s s

j 0 j 0

{ u } C{ u } C a { u } M{ u } a M

(7.2.5)

Also, damping coefficient for Mode s, s h can be expressed as:

s s s sC 2 h M (7.2.6)

s h in equations (7.2.5) and (7.2.6) is:




Chapter 7 Dynamic Analysis

midas GTS 69

2 j s s j s

s s s

3 2N 301 s 2 s N 1

s

C 1h a

2 M 2

1 a( a a a ), s 1 N

2

(7.2.7)

The damping coefficients for N number of natural modes are then determined.

Damping constants and matrices for the mass proportional type and stiffness proportionaltype are as follows:

0 s 0

s

ah , C a M

2 : Mass proportional type (7.2.8)

1 s s 1

ah , C a M

2

: Stiffness proportional type (7.2.9)

In the case of the Rayleigh type:

0 s 1 s 0 1

s

1 ah ( a ), C a M a K

2

(7.2.10)

where,

1 2 1 2 2 1

0 2 22 1

2 ( h h )

a ( )

2 2 1 11 2 2

2 1

2( h h )a

( )

7.3 Response Spectrum Analysis

The dynamic equilibrium equation for a structure subjected to a ground motion used in a

response spectrum analysis can be expressed as follows:

( ) ( ) ( ) ( ) g M u t C u t K u t M w t (7.3.1)




Part 1 Analysis Case

midas GTS 70

where, M : Mass matrix,

C : Damping matrix,

K : Stiffness matrix,

g w : Ground acceleration

and, u(t), ( )u t , ( )u t are relative displacement, velocity and acceleration respectively.

Response spectrum analysis assumes the response of a multi-degree-of-freedom (MDOF)

system as a combination of multiple single-degree-of-freedom (SDOF) systems. A

response spectrum defines the peak values of responses corresponding to and varying

with natural periods (or frequencies) of vibration that have been obtained through a

numerical integration process. Displacements, velocities and accelerations form the basis

of a spectrum. Response spectrum analyses are generally carried out for seismic designs

using the design spectra defined in design standards.

To predict the peak design response values, the maximum response for each mode is

obtained first and then combined by an appropriate method. For seismic analysis, the

displacement and inertial force corresponding to a particular degree of freedom for the m-th

mode are expressed as follows:

xm m xm dmd S , xm m xm am x F S W (7.3.2)

where,

m : m-th modal participation factor

m : m-th modal vector at location x

S dm : Normalized spectral displacement for m-th mode period

S am : Normalized spectral acceleration for m-th mode period

W x : Mass at location x

In a given mode, the spectral value corresponding to the calculated natural period is

located from the spectral data through linear interpolation. It is therefore recommended that

spectral data at closer increments of natural periods be provided at the locations of




Chapter 7 y a c a ys s

midas GTS 71

curvature changes (refer to Figure 1.2.2). The range of natural periods for spectral data

must be sufficiently extended to include the maximum and minimum natural periods

obtained from the Eigenvalue analysis.

midas GTS can easily generate the spectral data for seismic design by inputting Dynamic

coefficient, Foundation factor, Zoning factor, Importance factor, Ductility factor (or Response

modification factor or Seismic response factor), etc.

Response spectrum analyses are allowed in any direction on the Global X-Y plane and in

the vertical Global Z direction. The user may choose an appropriate method of modal

combination for analysis results such as the Complete Quadratic Combination (CQC)

method or the Square Root of the Sum of the Squares (SRSS) method. 1

The following describes the methods of modal combinations:

SRSS(Square Root of the Sum of the Squares)

1 22 2 2

max 1 2 n R R R R (7.3.3)

ABS(ABsolute Sum)

max 1 2 n R R R R (7.3.4)

CQC(Complete Quadratic Combination)

1 2

max

1 1

N N

i ij j

i j

R R R

(7.3.5)

1 The user may reinstate the signs lost during the modal combination process and apply them to the

response spectrum analysis results. For details, refer to “ Analysis>Analysis Case > Response Spectrum Analysis

Control” in the Online Manual.




y

midas GTS 72

where,

2 3 2

2 2 2

8 (1 )

(1 ) 4 (1 )ij

r r

r r r

,

j

i

r

,

max R : Peak response,

i R : Peak response of i-th mode,

r : Natural frequency of i-th mode to j-th mode,

: Damping ratio

In Eq. 7.34, when i = j, then ρij = 1 regardless of the damping ratio. If the damping ratio (ξ)

becomes zero (0), both CQC and SRSS methods produce identical results.

The ABS method produces the largest combination values among the three methods. TheSRSS method has been widely used in the past, but it tends to overestimate or

underestimate the combination results in the cases where the values of natural frequencies

are close to one another . As a result, the use of the CQC method is increasing recently as

it accounts for probabilistic inter-relations between the modes.

If the displacements of each mode are compared for a structure having 3 DOF with a

damping ratio of 0.05, the results from the applications of SRSS and CQC are as follows:

Natural frequencies,

1 0.46 , 2 0.52 , 3 1.42

Maximum modal displacements : D ij (displacement components of i-th degree of freedom

for j-th mode)

0.036 0.012 0.019

0.012 0.014 0.005

0.049 0.002 0.017

ij D




p y y

midas GTS 73

If SRSS is applied to compute the modal combination for each degree of freedom,

1 2

2 2 2

max 1 2 3 0.042 0.046 0.052 R R R R

If CQC is applied,

12 21 0.3985

13 31 0.0061

23 32 0.0080

1 22 2 2

max 1 2 3 12 1 2 13 1 3 23 2 32 2 2

0.046 0.041 0.053

R R R R R R R R R R

Comparing the two sets of displacements for each degree of freedom, we note that the

SRSS method underestimates the magnitude for the first degree of freedom but

overestimates the value for the second degree of freedom relative to those obtained by

CQC. Thus, the SRSS method should be used with care when natural frequencies are

close to one another.




midas GTS 74

1nT nT 1T 2T 3T 4T 5T 6T 7T

6T 7T xT

1S

2S

3S 4S

5S

6S

7S

1nS nS

6S

7S

F igure 7.3.1 Response spectrum curve and li near interpolation of spectral data

7.4 Time History Analysis

7.4.1 Introduction

The dynamic equilibrium equation for time history analysis is written as:

[ ] ( ) [ ] ( ) [ ] ( ) ( ) M u t C u t K u t p t (7.4.1)

where, [ ] M : Mass matrix

[ ]C : Damping matrix

[ ] K : Stiffness matrix

( ) p t : Dynamic load

and, ( )u t , ( )u t and ( )u t are displacement, velocity and acceleration respectively.




midas GTS 75

Time history analysis seeks out a solution for the dynamic equilibrium equation when a

structure is subjected to dynamic loads. It calculates a series of structural responses

(displacements, member forces, etc.) within a given period of time based on the dynamic

characteristics of the structure under the applied loads. midas GTS uses the Modal

Superposition Method and Direct Integration Method for time history analysis.

The following describes the general concept of Modal Superposition Method and data

input precautions.

7.4.2 Modal Superposition Method

The displacement of a structure is obtained from a linear superposition of modal

displacements, which maintain orthogonal characteristics to one another. This method

premises on the basis of that the damping matrix is composed of a linear combination ofthe mass and stiffness matrices as presented below.

C M K (7.4.2)

( ) ( ) ( ) ( )T T T T M q t C q t K q t F t (7.4.3)

( ) ( ) ( ) ( )i i i i i i im q t c q t k q t P t ( 1,2,3, , )i m (7.4.4)

( ) ( )m

i i

i j

u t q t

(7.4.5)




midas GTS 76

( )

0

(0) (0)( ) (0)cos sin

1( ) sin ( )

i i

i i

t i i i ii i Di Di

Di

t t

i Di

i Di

q qq t e q t t

P e t d m

(7.4.6)

where,2

1 Di i i ,

, : Rayleigh coefficient,

i : Damping ratio for i-th mode,

i : Natural frequency for i-th mode,

i : i-th mode shape,

( )iq t : Solution for i-th mode SDF equation

When a time history analysis is carried out, the displacement of a structure is determined

by summing up the product of each mode shape and the solution for the corresponding

modal equation as expressed in Eq. (7.4.5). Its accuracy depends on the number of

modes used.

This modal superposition method is very effective and, as a result, widely used in linear

dynamic analyses for large structures. However, this method cannot be applied to

nonlinear dynamic analyses or to the cases where damping devices are included such

that the damping matrix cannot be assumed as a linear combination of the mass and

stiffness matrices.

The following outlines some precautions required during data entry when using the

modal superposition method:

Total analysis time (or Iteration number): the time to analysis or the total

number of iteration

Time Step: Time step can affect the accuracy of analysis results significantly. The

increment must be closely related to the periods of higher modes of the structure and the

period of the applied force. The time step directly influences the integral in Eq. (7.4.6),

and as such specifying an improper time step may lead to inaccurate results. In general,




midas GTS 77

one-tenth of the highest modal period under consideration is a reasonable value for the

time step. In addition, the time step should be smaller than that of the applied load.

10

pT t (7.4.7)

where, pT : the highest modal period being considered

Modal damping ratios (or Rayleigh coefficients): Values for determining theenergy dissipation (damping) properties of a structure, which relate to either

the total structure or individual modes

Dynamic loads: Dynamic loads are directly applied to the nodes or foundation

of a structure, and are expressed as a function of time. The change of loadings

must be well represented in the forcing function. A loading at an unspecified

time is linearly interpolated.

7.4.3 Direct Integration Method

The method of direct integration solves the dynamic equilibrium method by integrating it

over time steps. Without changing the form of the equilibrium equation, various methods

can be used to determine the solution.

midas GTS uses the Newmark method for the method of direct integration. The following

are the basic assumptions and integration process:

1 t t t t t t u u u u t

(7.4.8)

212

t

t t t t t t u u u t u u t

(7.4.9)

By solving above equations, t t u is determined from Eq. 7.4.9 and t t u is

determined from Eq. (7.4.8). Thus, the equilibrium can be expressed in terms of




midas GTS 78

displacement, velocity and acceleration of the current and previous step.

, , ,

, , ,

t t t t t t t

t t t t t t t

u f u u u u

u f u u u u

(7.4.10)

By applying the Eq. (7.4.11) to the dynamic equilibrium equation, Eq. (7.4.12),

equilibrium can be defined in terms of displacement, velocity and acceleration of the

previous stage and displacement of the current stage. The displacement of the current

stage can be obtained from Eq. (7.4.13). After determining the displacement of the

current stage, the current acceleration and velocity can be found as shown in Eq.

(7.4.14). Damping is determined using the ratio of strength and mass as shown in Eq

(7.4.15).

t t t t t t t t M u C u K u p

(7.4.11)

0 1 0 2 3

1 4 5

t t t t t t t

t t t

K a M a C u p M a u a u a u

C a u a u a u

(7.4.12)

ˆ ˆt t

t t K u p

(7.4.13)

where, 0 1ˆ K K a M a C

0 2 3 1 4 5ˆ

t t t t t t t t t t p p M a u a u a u C a u a u a u

0 2 3 6 7,t t t t t t t t t t t t t u a u u a u a u u u a u a u (7.4.14)

where, 0 1 2 32

1 1 1, , , 1

2a a a a

t t t

4 5 6 71, 2 , 1 ,2

t a a a t a t




midas GTS 79

a, : Newmark Integration Variable (Stable if 0.5 , 0.25 )

t : Integration time step

C a K b M (7.4.15)

where, a, b : Coefficients of strength and mass for damping

For analysis involving the nonlinearity of stiffness and damping, the direct integration

method should be used in most of these cases. The analysis time is proportional to the

total number of time steps since the direct integration method is performed with respect

to all time steps. The data and noteworthy points are as follows:

Total analysis time (Iteration number) : the time to analysis or the total

number of iteration

Time step : Time step can affect the accuracy of analysis results significantly.

The increment must be closely related to the periods of higher modes of the

structure and the period of the applied force. The time step directly influences

the integral in Eq. (7.4.12), and as such specifying an improper time step may

lead to inaccurate results. In general, one-tenth of the highest modal period

under consideration is a reasonable value for the time step. In addition, the

time step should be smaller than that of the applied load.

10

pT

t (7.4.16)

where, pT : the highest modal period being considered

The time step should not be unnecessarily small since the analysis time increases in

proportion to the number of time steps. increases

Defining the damping using stiffness and mass : the damping is defined as




midas GTS 80

the proportional expression of stiffness and mass

Time integration method : Integration parameters for Newmark method is

inputted. While constant acceleration provides the stable solution that

converges in every condition, linear acceleration might provide the unstable

solution that does not converge. If possible, it is reasonable to use the

integration parameters for constant acceleration.

Dynamic loads : Dynamic loads are directly applied to the nodes or

foundation of a structure, and are expressed as a function of time. The change

of loadings must be well represented in the forcing function. A loading at an

unspecified time is linearly interpolated.

The following describes the fundamentals to help the user understand better.

Figure (7.4.1) shows an idealized system to illustrate the motion of a SDOF structural

system. The equilibrium equation of motion subjected to forces exerting on a SDOF

system is as follows:

( ) ( ) ( ) ( ) I D E f t f t f t f t (7.4.17)

I f (t) is an inertia force, which represents a resistance to the change in velocity of a

structure. The inertia force acts in a direction opposite to the acceleration, and its

magnitude is ( )mu t .

E f (t) is an elastic force by which the structure restores its configuration to the original

state when the structure undergoes a deformation. This force acts in the opposite

direction to the displacement, and its magnitude is )(t ku .

D f (t) is a damping force, which is a fictitious internal force dissipating kinetic energy

and thereby decreasing the amplitude of a motion. The damping force may occur in a

form of internal friction. It acts in the opposite direction to the velocity, and its magnitude

is ( )cu t .




midas GTS 81

(a) I deali zed model (b) State of equi li bri um

F igur e 7.4.1 Moti on of a SDOF System

The above forces are now summarized as:

( )

( )

( )

I

D

E

f mu t

f cu t

f ku t

(7.4.18)

where,

m : Mass

c : Damping coefficient

k : Elastic coefficient

From the force equilibrium shown in Figure 7.4 (b), we can obtain the equation for

motion for a SDOF structural system.

( )mu cu ku f t (7.4.19)

Eq. 7.4.19 becomes the equation of damped free vibration if f(t)=0, and it becomes the

equation of undamped free vibration if the condition of c=0 is additionally imposed on thedamped free vibration. If f(t) is assigned as a seismic loading (or displacements,

velocities, accelerations, etc.) with varying time, the equation then represents a forced

vibration analysis problem. The solution can be determined by using either the Modal

Superposition Method or Direction Integration Method.

(elastic force)

(damping force)

(external force)(inertia force)




midas GTS 82

7.5 Complex Response Analysis

7.5.1 1-dimensional dynamic analysis (theoretical solution)

midas GTS uses free field analysis to calculate the response of foundation during

earthquake before structure is constructed. The free field analysis is mainly used in

predicting the vibration of earth’s surface in calculating the dynamic stress and strain for

liquefaction assessment, and determining the earthquake load that will cause theinstability of foundation or earth structure.

The free field analysis is used to compute the response of foundation due to the vertical

radio wave of shear waves passing through the linear visco-elastic zone. The foundation

to analyze is composed of a number of stratum which are infinite to horizontal direction

and the semi-infinite bottom layers as shown in Figure 7.5.1. It is assumed that each

stratum is homogeneous and isotropic. The vibration is induced by the shear wave thatis transmitted through and reflected by the foundation in vertical direction, and the

displacement is produced only in horizontal direction. Therefore, the wave equation of

Eq. (7.5.3) should be satisfied in every stratum.

2 2 3

2 2 22

u u uG G

t x x t

(7.5.1)

where u is the horizontal displacement in time t , and , G , and are mass

density, shear elastic modulus, and hysteretic damping ratio, respectively.

Expressing Eq. (7.5.1) as harmonic function of Eq. (7.5.2), and transforming Eq. (7.5.1)

into frequency domain give the governing equation of Eq. (7.5.3). The stress-

displacement

relationship is given in Eq. (7.5.4)

( , ) ( , ) i t u x t u x e (7.5.2)

2* 2

2( , ) ( , ) 0G u x u x

x

(7.5.3)




midas GTS 83

*

( , ) ( , ) x G u x x

(7.5.4)

where ( , ) x is shear stress, and complex shear modulus is * (1 2 )G G i .

1

. . .

m

m+1

. . .

u1

x1

u2

x2

um

xm

um+1

xm+1

um+2

xm+2

uN

xN

N

Particle motion

Incident wave

Reflected wave

h1

hm

hm+1

hN=∞

1 1 1 G

m m mG

1 1 1 m m mG

N N N G

LayerNo.

CoordinateSystem

PropagationDirection

Properties Thickness

Figure 7.5.1 1-dimensional free field analysis model

The free field foundation is generally expressed as shown in Figure 7.5.1 to calculate the

solution of one-dimensional wave equation. When the number of layer is given below

from the surface, and the response of m-th layer from the surface is expressed as mu ,

the response is expressed as a function of m x which is the depth from the top of m-th

layer as shown in Eq. (7.5.5)-(7.5.6).

* *

( , ) ( ) ( )m m m mik x ik x

m m m mu x A e B e

(7.5.5)




midas GTS 84

* ** *( , ) ( ( ) ( )m m m mik x ik x

m m m m m m

x ik G A e B e

(7.5.6)

where,* */m smk V , * * /

sm m mV G , and m A and m B are the layer response factors,

where m A is the elastic wave component that is transmitted upward, m B is the elastic

wave component that is transmitted downward. The compatibility condition of Eq. (7.5.7)

and force equilibrium should be satisfied in the boundary of adjacent layers

1 1

1 1

( ) ( 0) 1,2,...,( 1)

( ) ( 0)

m m m m m

m m m m m

u x h u xm N

x h x

(7.5.7)

Substituting Eq. (7.5.5) and (7.5.6) into Eq. (7.5.7) gives the relationship between the

factors as shown in Eq. (7.5.8).

* *

* *

1 1

* *

1 1 * *

1 1

, 1, 2,..., ( 1)

m m m m

m m m m

ik h ik h

m m m m

ik h ik hm mm m m m

m m

A B A e B e

m N k G A B A e B e

k G

(7.5.8)

Rearranging to derive the relationship between the response factors of adjacent soil

layers, the recurrence equation of Eq. (7.5.9) is obtained as

* *

* *

* *

1

* *

1

1 1(1 ) (1 )

2 2

1 1(1 ) (1 )

2 2

m m m m

m m m m

ik h ik h

m m m m m

ik h ik h

m m m m m

A A e B e

B A e B e

(7.5.9)

where,

mh : thickness of m th layer

*

m : ratio of dynamic stiffness between the adjacent layers.

* **

* *

1 1

m mm

m m

k G

k G

(7.5.10)




midas GTS 85

Since the shear stress is always zero, it is clear that 1 1 A B from Eq. (7.5.6). Therefore,

the response factor of m th layer can be calculated by applying the recurrence equation

of Eq. (7.5.9) subsequently from the first to m th layers.

1

1

( ) ( ) ( )

( ) ( ) ( )

m m

m m

A a A

B b B

(7.5.11)

where 1 1 1a b .

The transfer function between the boundary of layer i and j , ( )ij H is expressed

as

( ) ( ) ( )( )

( ) ( ) ( )

i i iij

j j j

u a b H

u a b

(7.5.12)

When the transfer function ( )ij H and the response ( ) ju for the boundary of layer

j are given, the response ( )iu for the boundary of layer i is obtained as Eq.

(7.5.13).

( ) ( ) ( )i ij ju H u (7.5.13)

When the frequency domain analysis is used, the effect of nonlinear behavior of

foundation is generally considered by the equivalent linearization technique. The reason

for using this equivalent linearization technique is that the frequency domain analysis

basically assumes the foundation as the linear behavior.

Since the constant stiffness (shear modulus) and damping are applied in frequencydomain analysis, the change of stiffness due to the actual shear strain is not considered

as shown in Figure 7.5.2. Therefore, the nonlinear behavior of foundation is considered

by iteratively conducting the linear analysis that includes the change of stiffness and

damping ratio of foundation according to the shear strain which is obtained from the




midas GTS 86

previous step, as shown in Figure 7.5.3. To determine the stiffness and damping, the

effective shear strain that is given as constant value less than one(e.g., 0.65) multiplied

by the maximum shear strain obtained from the previous step as shown in Eq. (7.5.5) is

used. The reason for using the effective shear strain is that the maximum shear strain

induces the more strain energy than the actual behavior does, as shown in Figure 7.5.2.

The method to consider the nonlinearity of foundation by using the equivalent

linearization technique is summarized in Table 7.5.1.

max max { ( )}t oct t (7.5.14)

2 2 2 2 21( ) ( ) ( ) { ( ) ( )} 6 ( ) 6 ( )

3oct x z x z yz xz t t t t t t t (7.5.15)

F igur e 7.5.2 Di ff erence between maximum shear strai n and effective strain




midas GTS 87

F igur e 7.5.3 The process to converge nonl inear shear modul us and damping factor by the equivalent

li neari zation technique

Table 7.5.1 The process for f ree field analysis using the equivalent li nearization techni que

Determine the initial values of the shear modulus( G ) and damping ratio( h )

of each layer. In general, the values where the strain is very small are used.

Calculate the maximum shear strain max of each layer from the free field

analysis using the initial values.

Calculate the effective shear strain eff of each layer.

maxeff R

where R is the ratio between the effective shear strain and maximum shear

strain. The 0.65 is used, or the following M is used according to the

magnitude of earthquake(Seed & Sun, 1992).

1

10

M R

Determine the shear modulus G and h according to the max/ eff G G

curve and eff h curve of each layer using the effective shear strain eff .

Repeat the step 2 to 4 until G and h converge. In general, it is considered

to be converged that the relative error is less than 5%, and the iteration is less

than 5.




midas GTS 88

7.5.2 2-dimensional dynamic analysis (finite element method)

7.5.2.1 Overview

The biggest difference between soil-structure interaction problem and general

structural dynamics problem is the radiation damping induced by the infiniteness of

foundation. While the general damping property decreases the structural movement

by material friction, the radiation damping decreases the structural kinetic energy by

releasing wave energy to the infinite region of foundation.

The radiation damping is considered by introducing a damping term in the equation of

motion, and the magnitude is dependent on the shape of wave that is propagated

outside. Since the wave shape can be modeled easily in frequency domain, a

frequency domain analysis would be efficient. In general, the soil material is basically

inhomogeneous, and the mechanical behavior of the soil is highly nonlinear.

To accurately describe soil-structure interaction problems, it is necessary to consider

the important characteristics of radiation damping and nonlinearity. Therefore, in a

frequency domain analysis in which the radiation damping can be easily modeled is

used, and the material nonlinearity is analyzed by the equivalent linearization method.

The analysis applies the standard frequency domain analysis that uses the FFT(Fast

Fourier Transform). In addition, the responses from foundation and structure can be

calculated in one operation since the equation of motion builds the integrated soil-

structure system with input motion of free field response. The interpolation method of

transfer function for single degree of freedom system is used to reduce the number of

frequencies for calculating the solution of the equation of motion. This method is used

to obtain the interpolated value for a transfer function based on the solution of two

frequencies in a row. Therefore, selecting the basic frequencies is very important, and

it is needed to define the cut-off frequencies high enough to guarantee the validity for

foundation analysis.




midas GTS 89

7.5.2.2 Equation of Motion

From the result of free field analysis as mentioned above, the analysis is performed by

inputting the earthquake response of bedrock into the equation of motion of Eq. (7.5.5).

[ ]{ } [ ]{ } { } { } { } { } M u K u m y V F T (7.5.5)

where

{ }u :Relative displacement vector of each node on the bottom of the

model,

M :Mass matrix of the two-dimensional soil-structure system with a unit

thickness,

[ ] K :Complex stiffness matrix including the damping effect for the two-

dimensional soil-structure system with a unit thickness.

{ }m :Mass vector for the direction of input motion,

y :Input motion acting on the bottom of the model.

{ }V :Force that is generated in the viscous boundary of three-dimensional

direction, and is given as Eq. (7.5.6).

1{ } [ ]({ } { }) f V C u u

L (7.5.6)

where L is the thickness of plane system, [ ]C is the diagonal matrix which includes

the characteristic of free field, and { } f u is velocity vector of free field.

{ } F is characterized as the load acting on the vertical face of soil-structure system,

and is given as Eq. (7.5.7)

{ } [ ]{ } f F G u (7.5.7)

where [ ]G is the complex stiffness matrix of free field, { } f u is the displacement




midas GTS 90

vector of free field. { }T is the force by the horizontal transfer of wave energy which is

expressed as the transmitting boundary condition.

{ } ([ ] [ ])({ } { }) f T R L u u (7.5.8)

where [ ] R and [ ] E are the stiffness matrixes according to the frequency of

boundary conditions developed by Lysmer, and Drake & Wass.

Assuming the input motion ( ) y t in Eq. (7.5.5) as the summation of finite number of

harmonic motions as shown in Eq. (7.5.9), the frequency domain analysis is

performed.

/2

0

( ) Re s

N i t

s

s

y t Y e

(7.5.9)

where, N is the number of points to represent the input motions. Therefore, the

response of soil-structure system is given as Eq. (7.5.10)

/2

0

{ } Re { } s

N i t

s

s

u U e

(7.5.10a)

/2

0

{ } Re { } s

N

i t f f s

s

u U e

(7.5.10b)

Substituting Eq. (7.5.10) and Eq. (7.5.6) into Eq. (7.5.5), and rearranging it give the Eq.

(7.5.11).

2([ ] [ ] [ ] [ ] [ ]){ }

{ } ([ ] [ ] [ ] [ ]){ }

s s s s s

s s s s f s

i K R L C M U

Li

m Y G R L C U L

(7.5.11)

In Eq. (7.5.11), { } f sU is expressed as Eq. (7.5.12), and { } f s A is the transfer




midas GTS 91

function with respect to the input motion of the bottom.

{ } { } f s f s sU A Y (7.5.12)

Substituting Eq. (7.5.12) into Eq. (7.5.11), and rearranging it give the Eq. (7.5.13).

[ ] s K is the stiffness matrix according to the frequency of the system, and { } s P is the

load vector for the input motion of 1 s

Y .

[ ] { } { } s s s s K U P Y (7.5.13a)

2[ ] [ ] [ ] [ ] [ ] [ ] s s s s s

i K K R L C M

L

(7.5.13b)

{ } ([ ] [ ] [ ] [ ]){ } { } s s s s f s

i P G R L C A m

L

(7.5.13c)

Using Eq. (7.5.14) which is obtained from dividing the both side terms of Eq. (7.5.13)by sY , [ ] s K and { } s P are the functions of frequency , and { } s A is also a

function of frequency.

[ ] { } { } s s s K A P (7.5.14)

Obtaining { } s A from Eq. (7.5.14), { }u of Eq. (7.5.15) is given as Eq. (7.5.10a).

{ } Re { } si t

s su A Y e (7.5.15)

Also, the response in time domain analysis can be obtained by inverse transformation

of Eq. (7.5.11) using FFT method.




midas GTS 92

7.5.3 Transmitting boundary

It is difficult to accurately describe the actual infinite ground soil from a 2-dimensional

model for soil-structure analysis. Therefore, it is necessary to assess the boundaries of a

model in appropriate positions, and to settle the boundary similarly to the actual site

conditions.

The boundary conditions in soil modeling are divided into element boundary, viscous

boundary, and transmitting boundary condition. The element boundary condition is

subdivided into free ends which include the force from earthquake response load in the

boundary for the free field, the fixed ends include the displacement, and the rotating end

represents the horizontal and vertical rotating ends. The element boundary condition

considers the effect of earthquake wave for the free field adequately, but it cannot

consider the effect of reflected waves that are reflected from a footing slab. Also, these

effects increase as the location of boundary becomes closer to the footing slab.

To overcome the shortcomings of element boundary condition, Kuhlemeyer, Ang,

Newmark, et al. developed the viscous boundary condition that can absorb the material

wave with a constant angle in the boundary. However, the viscous boundary condition

also needs to be placed at a certain distance away from a footing slab since it cannot

accurately consider the effects of complicated surface wave.

Although viscous boundary conditions have certain limitations, transmitting boundaryconditions can consider various effects from material waves and surface waves. The soil

layers parallel to horizontal direction are represented by a spring function of frequency

and damper. Since it is generally assumed that the property of each soil layer, in the

horizontal direction, is homogeneous, satisfactory results can be obtained for cases

when the boundary conditions are attached to the structure itself. However, it is effective

to place a certain distance between footing slab and boundary to accurately consider the

change of characteristic according to the strain of horizontal direction.

Chapter 8



Slope Stability Analysis

midas GTS 93

8.1 Introduction Slope Stability Analysis

Slope stability analyses for both embankments and excavation are subjects most frequently

encountered in geotechnical engineering. A slope naturally is subjected to potential energy

due to its self weight. If pore water pressure, external forces, seismic loads and wave

forces are exerted on a slope, slope instability may arise. When the shear stress due to selfweight and external load are greater than the shear strength of soil, either slip or failure

may occur. Determining such shear stress and soil strength can be summed up as slope

stability analysis.

Slope stability has been conventionally analyzed by performing displacement and stability

analyses separately (Duncan, 1984; Huang, 1983; Brunsden, 1984). However, actual slope

failure is caused by large displacements in local areas and develops slowly. It is known thatdisplacement and failure are not distinguished, but take place in the process of progressive

failure (Chowdhury, 1978; Griffiths, 1993). Therefore, it is important to formulate a slope

stability analysis method which can track the failure process from initial deformation to

ultimate failure.

step #1 geo static

potential energy (W)

(self weight + gravity)

step #2 external force action

W

step #3 shear stress

within the slope

Step #4 reacting on the shear

strength of the slope(S)

s

S F

T

W

T

pore water pressure, applied load,earthquake load, wave load, etc.

due to external loadshear force occurs

F igure 8.1.1 Slope Failu re Diagram




midas GTS 94

The following methods are currently suggested for slope stability analysis:

Limit equilibrium theory by Mass procedure / Slice method

Limit theory

Finite element method by elastoplastic theory

In order to monitor the stability of slopes on site, it is important to determine not only the

minimum factor of safety, but also the failure behavior. For a progressive slope failure, the

locations and the control values for monitoring must be determined in advance and

managed appropriately. The finite element method provides suitable analysis results and

also used to determine the progressive failure behavior of a slope (Anderson, 1987;

Duncan, 1996).

The on-going development in computational technology, finite element methods have been

widely used for analyzing structural ground behavior for displacements, stresses, pore

water stresses, etc. The finite element method recently has extended its applications from

simple behavioral analysis for ground structures to evaluating slope failures in stability

problems as well.

midas GTS uses the “Strength Reduction method”, which is developed for solving slope

stability problems, formulated using finite element techniques.

8.2 Strength Reduction Method

8.2.1 Introduction

The finite element method is a precise numerical analysis method which satisfies the

force equilibrium, compatibility condition, constitutive equation and boundary condition at

each point of a slope. It simulates the actual slope failure mechanism and determines

both the minimum factor of safety and the failure behavior. It can also reflect real in-situ

conditions better than most methods. Moreover, it can determine the failure process

without assuming any failure planes in advance. (Griffith 1999; Matsui, 1990).

Chapter 8 – Slope Stability Analysis



midas GTS 95

There are two types of methods used in finite element method to analyze slope stability

– Strength Reduction method and Indirect method (Pasternack, S.C. and Gao, S, 1988).

The Strength Reduction method is a direct method, which gradually reduces the shear

strength ( , )c of sloped ground materials until failure takes place. Failure is assumed

to occur when the analysis does not converge. Under this condition, the maximum

strength reduction factor where analysis fails to converge becomes the factor of safety.

The Indirect method determines the factor of safety by applying the calculated stresses

combined with the conventional Limit theory method.

8.2.2 Theoretical Background

The Strength Reduction method uses a finite element technique first proposed by

Zienkiewicz (1975). We now focus on a Gauss point, A, of an element in a sloped ground

structure to calculate the factor of safety of a slope as shown in Figure (8.2.1) The stress

state at this point is represented in a Mohr circle. In order to represent the sliding surface,

the shear stress at the point is divided by a factor of safety, F, so that the Mohr circle for

the stress state of the fictitious sliding surface becomes tangent to the failure criterion.

That is, the stress state of the point is corrected to the failure state. An increase in the

number of points results in a global slope failure. As soon as a finite element solution

diverges, the analysis stops and the limit value, F, becomes the minimum factor of safety

for the slope. This method requires stability in numerical analysis, but returns consistent

results and evaluates the actual failure behavior.




midas GTS 96

A

' '

Mohr ’s circle

for A

Modified

Line

Modified

Line

F igur e 8.2.1 Strength Reduction Method

8.2.3 Method of calculating Minimum Safety Factor

To determine the minimum factor of stability of slopes, the modulus of elasticity (E) and

Poisson’s ratio ( ) are assumed to be constant. The cohesion ( c ) and friction angle ( )

are simultaneously reduced, and the factor of safety, s F is determined at the diverging

point. The factor of safety for slope failure is determined on the basis of shear failure asfollows:

s

f

F

(8.2.1)

where,

: Shear strength of slope material

This is computed based on the Mohr-Coulomb criterion.

tannc (8.2.2)

The shear stress at the sliding surface, f is expressed in Eq. 8.2.3.




midas GTS 97

tan f f n f c (8.2.3)

where,

SRF f

cc : Coefficient of shear strength

1 tantan

SRF f

: Coefficient of shear strength

SRF : Strength reduction factor

In order to determine the SRF accurately, it is necessary to trace the resulting values of s F

causing the slope to fail. The incremental parameter s F is increased in very small steps

even though it may extend the analysis time duration. Otherwise, calculating the minimum

factor of safety may face difficulties.

8.3 Stress Analysis Method Based on the Limit

Equilibrium Method

8.3.1 Introduction

Slope stability analysis is globally divided into two methods: simplified method and

numerical analysis method. The limit equilibrium which belongs to the simplified methods,

is one way to approach slope stability problems and is most widely used in practice.

However, this method cannot consider the stress hysteresis effects during the formation

of slope and the variation of stress for foundations due to the groundwater. Although the

finite element techniques considers the formation process of slope and the property of

foundation, the method requires high cost and long analysis time, and the slope stability

data for evaluation purposes is relatively insufficient. Recently, a numerous researches

are conducted to concentrate on the good points of the simplified method and numerical

analysis method. Among these, the slope stability analysis method using the finite

element method which has been proposed by Kim(1998) in KAIST is applied for slope

stability analysis.




midas GTS 98

In this method, stress analysis is first performed on the slope using the finite element

method. Based on the stress analysis results, the factor of safety for potential sliding

surface is calculated, and the critical section is determined using the minimum safety

factor.

8.3.2 Evaluation of safety factor

The safety factor in the finite element method is defined as,

f S

s

mS

d

F

d

(8.3.1)

m

is shear stress, and f

is shear strength according to Mohr-Coulomb failurecriteria.

tan

1sin 2 cos 2

2

f n

m y x xy

c

(8.3.2)




midas GTS 99

where, the stress, n , normal to sliding surface is given as:

2 2sin cos sin2n x y xy (8.3.3)

c :Viscosity,

:Internal friction angle of material

:Angle between horizontal and sliding surfaces.

x and y :Normal stresses directions x and y respectively

xy :Shear stress.

xy

y

x

xy

sliding surface

F igure 8.3.1 Stress components of sli ding surface

To calculate the safety factor, stress integration is required with respect to the potential

sliding surface as shown in Eq. (8.3.1). Since the stress integration of continuous stress

field with respect to node is more reliable than that of Gaussian integration point, the

continuous stress field in every node is calculated using the global stress smoothing

method(Hinton & Compbell, 1974) in midas GTS.

8.3.3 Stress Integration Along the Potential Sliding Surface

The integration of shear stress along the potential sliding surface is required to evaluate

the factor of safety of the assumed potential sliding surface as shown in Eq. (8.3.1). This

process is performed by integrating over all elements along the sliding surface.




midas GTS 100

The shear stress of an arbitrary point within an element is expressed using the stresses

of nodal points that are calculated by the global stress smoothing methods:

nodenode

1

i i

i

σ N σ (8.3.4)

where,

i N : Shape function at node i

node

i : Nodal stress at node i

:Stress of an arbitrary point within the element.

The stress integration over the potential sliding surface in two-dimensional global

coordinate system is given as the following by transforming into the integral in one-

dimensional local coordinate system.

intn

2 1

1 11

,2 2

n

i in

i

L L x y d d W

T T (8.3.5)

where,

:Coordinate variable in local coordinate system

iW : Integral constant in integral point i

T :Transformation matrix transforming the stress from local coordinate

system into global coordinate system

L :Element length

:Shear stress m or shear strength f on potential sliding surface




The global safety factor for the potential sliding surface is given as

2

1

2

1

nel n

n1

nel n

n1

f

i s

m

i

d

F

d

(8.3.6)

where,

nel :Number of elements intersecting along potential sliding surface

1n :Starting point of potential sliding surface within an element

2n :Ending point of potential sliding surface within an element

Part 1 Analysis Case

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Transcript of Part 1 Analysis Case