Intro to bhj

8/13/2019 Intro to bhj

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CEL760 : FINITE ELEMENT METHOD INGEOTECHNICAL ENGINEERING

3 Credits (3-0-0) Coordinator : K.G. Sharma

Course Content

Introduction.

Steps in FEM.

Stress-deformation analysis: One-, Two- and

Three-dimensional formulations;

Solution algorithms;

Discretization;

Use of FEM2D Program and Commercial packages.

Analysis of foundations, dams, underground structures and

earth retaining structures.

Course Contents Contd.

Analysis of flow (seepage) through dams and

oun a ons.

Linear and non-linear analysis.

Insitu stresses.

Sequence construction and excavation.

Joint/interface elements.

. .

Evaluation of material parameters for linear and

non-linear analysis.

Recent developments.


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References

Desai, C.S. and Kundu T. (2001) Introductory Finite Element

Method. CRC Press.

Desai, C.S. and Abel, J.F. (1972) Introduction to Finite Element

Method. Van Nostrand Reinhold, New York.

Bathe, K.J. (1982) Finite Element Procedures in Engineering

Analysis. Prentice-Hall, Inc.

Zienkiewicz, O.C. and Taylor, R.L. (1989) The Finite Element

Method. Vols. 1 & 2, 4th Edition, McGraw-Hill Book Company.

Desai, C.S. and Christian, J.T. (1977) Numerical Methods in

Geotechnical Engineering. John Wiley & Sons.

References

Naylor, D.J. and Pande, G.N. (1981) Finite Elements in

Geotechnical Engineering. Pineridge Press.

Hinton, E. and Owen, D.R.J. (1977) Finite Element

Programming. Academic Press.

Evaluation

Minor Test I : 20%

Minor Test II : 20

Major Test : 40%

Assignments : 20%

Note: Students having less than 75% attendance will be

given one grade less than the grade scored by them.


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Introduction to

Finite Element Method

Dr. K. G. SharmaProfessor, Department of Civil Engineering,

Indian Institute of Technology Delhi

Many problems in engineering and applied science are

overned b differential or inte ral e uations.

Closed form (Analytical) Solutions (CFS)

Methods of Solution

The solutions to these equations would provide an exact,

closed-form solution to the particular problem being studied.

Gives the values of unknown quantity at any location in a body.

,simplified situations.

However, complexities in the geometry, properties and in theboundary conditions that are seen in most real-worldengineering problems usually means that an exact solutioncannot be obtained or obtained in a reasonable amount of time.


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Methods of Solution

Numerical Methods

Problems involving complex material properties& boundary conditions: Numerical Methods

Numerical Methods provide approximate butacceptable solutions.

o u on o a ne on y a a scre e num er opoints in the body.

Numerical Methods contd.

Process of selecting only a certain number of discrete pointsIn the body is termed as Discretization.

Divide the body into an equivalent system of smaller bodies/units. The assemblage of these units then represents the

original body.

We do not solve the problem for the entire body in one.

Instead, solutions are formulated for each unit and combinedto obtain the solution for the original body.

This approach is known as Going from Part to Whole.


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Discretization of Body

Numerical Methods Contd.

Analysis procedure is considerably simplified.

Amount of data to be handled depends upon the

number of small bodies.

Manual calculations for 1-D problems.

Computer required for large data: 2-D, 3-D problems.


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Methods of Solution contd.

Numerical Methods

Finite Element Method (FEM)Boundary Element Method (BEM)Coupled Finite Element Boundary Element Method(FEBEM)Distinct Element Method (DEM)Infinite Elements

Best known numerical method is Finite Difference Method.

The method has been adopted for use with computers.

.

FEM is essentially a product of electronic computer age.

Many of the features common to the previous numerical

methods.FEM possesses certain characteristics that take advantage

of the special facilities offered by computers.

FEM can be systematically programmed to accommodate

Such complex and difficult problems as non-homogeneous

materials, nonlinear stress-strain behaviour and complicated

boundary conditions.

It is difficult to accommodate these complexities in other

methods.


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The approach is similar to the extension of the familiar

concepts of analysis of framed structures as 1-D bodies to

problems involving 2-D and 3-D structures.

Physical or Intuitive approach to the learning and using

the method.

Or a rigorous mathematical interpretation of the method.

ENGINEERING MATHEMATICS

Finite differences

Variational

FAMILY TREE OF FINITE ELEMENT METHODS

Richardson 1910

Liebman 1918

Weighted

residuals

Trial Functions

Piecewisecontinuous trial

Functions

Rayleigh 1870

Ritz 1909

Courant 1943

Prager Synge 1947

Southwell 1940

Structural

analogue

substitution

Hrenikoff 1941

McHenry 1943

Newmark 1949

Gauss 1795

Galerkin 1915

Biezeno Koch 1923

Variational finite

differences

PRESENT DAY

FINITE ELEMENT METHOD

Direct continuum

elements

Argyris 1955

Turner et al. 1956

Varga 1962


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Methods of Solution contd. Empirical Methods

Observational Methods

CONTINUUM DISCONTINUUM

,

Isotropic Anisotropic

DISCRETIZATION

Going from Part to Whole

Discretization Scheme


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Discretization Scheme

PROTYPE-TRACKANALYSISDiscretization

Total No. of Elements = 971

Mesh Diagnostics

Model Track

Component Nodes Elements

Rail and 644 62

Total No. of Nodes = 5771

Prototype

Track

Sleepers

Ballast 592 72

Sub-ballast 660 81

Subgrade 3875 756


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Finite Element Mesh of Dam-Foundation System

70 m


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Analysis Using UDEC

Discretisation

3-D layout of the underground storage caverns from the reference of

Benardos and Kaliampakos, (2004). (ref 4 )


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PHASE 2 model of the oil storage cavern

Finite Element Method

In the FEM, a complex region defining a continuum is

discretized into simple geometric shapes called elements.

e proper es an e govern ng re a ons ps are assume

over these elements and expressed mathematically in terms

of unknown values at specific points in the elements called

nodes.

An assembly process is used to link the individual elements

.

conditions are considered, a set of linear or nonlinear

algebraic equations is usually obtained.

Solution of these equations gives the approximate behavior

of the continuum or system.


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Finite Element Method (cont.) The continuum has an infinite number of degrees-of-freedom

(DOF), while the discretized model has a finite number of DOF.

This is the origin of the name, finite element method.

The number of equations is usually rather large for most real-

world applications of the FEM, and requires the computational

power of the digital computer. The FEM has little practical value

if the digital computer were not available.

vances n an rea y ava a y o compu ers an so warehas brought the FEM within reach of engineers working in small

industries, and even students.

Finite Element Method (cont.)

Two features of the finite element method are worth noting.

The iecewise a roximation of the h sical field

(continuum) on finite elements provides good precision even

with simple approximating functions. Simply increasing the

number of elements can achieve increasing precision.

The locality of the approximation leads to sparse equation

systems for a discretized problem. This helps to ease the

solution of problems having very large numbers of nodal

unknowns. It is not uncommon today to solve systems

containing a million primary unknowns.


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Origins of the Finite Element Method It is difficult to document the exact origin of the FEM,

because the basic concepts have evolved over a period of

.

The term finite element was first coined by Clough in 1960.

In the early 1960s, engineers used the method for

approximate solution of problems in stress analysis, fluid flow,

heat transfer, and other areas.

The first book on the FEM by Zienkiewicz and Chung was

published in 1967.

In the late 1960s and early 1970s, the FEM was applied to a

wide variety of engineering problems.

Origins of the Finite Element Method (cont.)

The 1970s marked advances in mathematical treatments,

including the development of new elements, and convergence

studies.

Most commercial FEM software packages originated in the

1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s

(FENRIS, LARSTRAN 80, SESAM 80.)

The FEM is one of the most important developments in

computational methods to occur in the 20th century. In just a

few decades, the method has evolved from one with

applications in structural engineering to a widely utilized and

richly varied computational approach for many scientific and

technological areas.


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How can the FEM Help the Design Engineer? The FEM offers many important advantages to the design

engineer:

- ,

composed of several different materials and having

complex boundary conditions.

Applicable to steady-state, time dependent and

eigenvalue problems.

Applicable to linear and nonlinear problems.

One method can solve a wide variety of problems,

including problems in solid mechanics, fluid mechanics,

chemical reactions, electromagnetics, biomechanics, heat

transfer and acoustics, to name a few.

How can the FEM Help the Design

Engineer? (cont.)

-

at reasonable cost, and can be readily executed on

microcomputers, including workstations and PCs.

The FEM can be coupled to CAD programs to facilitate

solid modeling and mesh generation.

Many FEM software packages feature GUI interfaces,

auto-meshers, and sophisticated postprocessors and

graphics to speed the analysis and make pre and post-

processing more user-friendly.


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How can the FEM Help the Design

Organization?

Simulation using the FEM also offers important business

advantages to the design organization:

Reduced testing and redesign costs thereby shortening

the product development time.

Identify issues in designs before tooling is committed.

Refine components before dependencies to other

components prohibit changes.

Optimize performance before prototyping.

Discover design problems before litigation.

Theoretical Basis: Formulating Element

Equations

Several approaches can be used to transform the physical

formulation of a problem to its finite element discrete analogue.

If the physical formulation of the problem is described as a

differential equation, then the most popular solution method isthe Method of Weighted Residuals.

e p ys ca pro em can e ormu a e as e m n m za on

of a functional, then the Variational Formulation is usually used.


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Sources of Error in the FEMThe three main sources of error in a typical FEM solution are

discretization errors, formulation errors and numerical errors.

Discretization error results from transforming the physical

sys em con nuum n o a n e e emen mo e , an can e

related to modeling the boundary shape, the boundary

conditions, etc.

Discretization error due to poor geometry

representation.Discretization error effectively eliminated.

Sources of Error in the FEM (cont.) Formulation error results from the use of elements that don't precisely

describe the behavior of the physical problem.

Elements which are used to model physical problems for which they are not

suited are sometimes referred to as ill-conditioned or mathematically

unsuitable elements.

For example a particular finite element might be formulated on the

assumption that displacements vary in a linear manner over the domain.

Such an element will produce no formulation error when it is used to model a

linearly varying physical problem (linear varying displacement field in this

example), but would create a significant formulation error if it used torepresent a quadratic or cubic varying displacement field.


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Sources of Error in the FEM (cont.)

Numerical error occurs as a result of numerical

calculation procedures, and includes truncation

errors an roun o errors.

Numerical error is therefore a problem mainly

concerning the FEM vendors and developers.

The user can also contribute to the numerical

accurac for exam le b s ecif in a h sicalquantity, say Youngs modulus, E, to an

inadequate number of decimal places.

Advantages of the Finite Element Method

Can readily handle complex geometry:

The heart and power of the FEM.

Can handle complex analysis types:

ra on

Transients

Nonlinear

Heat transfer Fluids

Can handle complex loading:

- .

Element-based loading (pressure, thermal, inertial

forces).

Time or frequency dependent loading.

Can handle complex restraints:

Indeterminate structures can be analyzed.


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Advantages of the Finite Element Method (cont.)

Can handle bodies comprised of nonhomogeneous materials:

Every element in the model could be assigned a different set

of material properties.

Can handle bodies comprised of nonisotropic materials:

Orthotropic

Anisotropic

Special material effects are handled:

Temperature dependent properties.

Plasticity

reep Swelling

Special geometric effects can be modeled:

Large displacements.

Large rotations.

Contact (gap) condition.

Disadvantages of the Finite Element Method

A specific numerical result is obtained for a specific problem. A

general closed-form solution, which would permit one to

examine system response to changes in various parameters, is

.

The FEM is applied to an approximation of the mathematical

model of a system (the source of so-called inherited errors.)

Experience and judgment are needed in order to construct a

good finite element model.

A powerful computer and reliable FEM software are essential.

Input and output data may be large and tedious to prepare and

interpret.


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Disadvantages of the Finite Element Method (cont.)

Numerical problems:

Computers only carry a finite number of significant digits.

Round off and error accumulation.

Can help the situation by not attaching stiff (small) elements

to flexible (large) elements.

Susceptible to user-introduced modeling errors:

Poor choice of element types.

Distorted elements.

Geometry not adequately modeled.

Buckling

Large deflections and rotations.

Material nonlinearities .

Other nonlinearities.

Advantages of Finite Element Method

Nonhomogeneity

Material & Geometric Nonlinearities

as c, on near e as c, as o p as c, creep, v scop as c

Irregular Geometry

Any Boundary Conditions

Generality 1-D 2-D 3-D

Applicable to Wide Range of Problems

Structural En ineerin

Geotechnical Engineering

Water Resources Engineering

Mechanical Engineering

Nuclear Engineering

Heat Transfer


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Advantages of Finite Element Method-Contd

Applicable to Wide Range of Problems Biomedical Engineering

Electro-magnetism

X

Y

Z


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Discretization Examples

One-Dimensional

Frame Elements

Two-Dimensional

Triangular

Elements

Three-Dimensional

Brick Elements


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Field near a Magnet

Anatomy of Hip Joint

Largest weight

bearing jointComposed of rounded

head of the femur

46

joining the

acetabulum of pelvis

in a ball and socket

arrangement


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Composite Model

Basic Composite

Model With Elements

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Maximum Shear Stress Region

n arge ew o e e orme em an or ca oneShowing the Maximum Shear Stress Region (Path Aa)

48


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Natural Knee Joint


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Idealization of Natural Femur

Fixation of Endoprosthesis


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Idealization with Endoprosthesis

Problems & Loads

Problem Types One Dimensional Problems

Two Dimensional Problems Plane Stress

Plane Strain

Axisymmetric

Three Dimensional Problems

Loads Point loads

Pressure loading

o y orces

Due to insitu stresses

Due to Temperature


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BODY FORCE

Xb dV

Xc dVVolume

element dV Body force: distributed force per

unit volume (e.g., weight, inertia,

etc)

zSurface (S)

Volume (V)

uv

w

x

a

=

c

b

a

X

X

X

X

NOTE: If the body is accelerating, then the

inertia force

&&

&&

&&

u

xy

may be considered as part of X

=w&&

vu

u~

&&= XX

Xc dVVolumeelement dV

pz

Traction: Distributed force

per unit surface area

SURFACE TRACTION

z

ST

Volume (V)

uv

wXa dV

Xb dV

pypx

=z

y

x

pp

p

T S

xy


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Analysis Types

Analysis: Stress Deformation

Linear elastic

Elasto Plastic

Elasto Viscoplastic

Simulation of

Sequential

Construction Excavation

Joint / Interfaces

Confined flow

Unconfined flow

Finite elements Infinite elements

Joint / Interface elements

Line / Bar elements

The width of the concrete face is 0.3 m at the top and 1 m at the bottom of the

dam. Upstream slope of the dam is of 1V : 1.4H. Downstream slope is of 1V :

1.5H.


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Discretization

Themeshfortheanalysisofthedamconsistsof 7500elements

Sequential loading

Theloadingisdonein23layerswitheachlayerbeing5mhigh.


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Excavation sequence for the power house cavern

Categories of problem in Geotechnical

Engineering

FEM is applicable to a wide range of boundary

.

Boundary Value Problem: A solution is sought in

the region of the body, while on the boundaries ofthe region, the values of unknowns are prescribed.

Initial Value Problems: Initial values of the unknowns

are also prescribed.


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Steady or Equilibrium

Static stress-deformation analyses for foundations,

, , ,

Steady state fluid flow

Eigen value

Natural frequencies of foundations and structures

Categories of problem in Geotechnical

Engineering

Transient or Dynamic

ress e orma on e av our o oun a ons, s opes,

banks, tunnels, and other structures, under time

dependent forces

Viscoelastic analysis

Consolidation

Transient fluid flow

Wave propagation


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Governing Equations

Static and Steady State

nergy r nc p es, eren a qua ons

Transient or Dynamic

Energy Principles, Differential Equations

Initial Value Problems

Time Marchin Schemes Forward difference method (Euler): Explicit

Backward difference: Implicit

Central difference: Implicit

Time Step, t

Modules

Pre Processor: Data, Mesh

Analysis FEM2, , , , , ,

MIDAS, UDEC, 3DEC

Post Processor

Mesh

Deformed mesh / shape Deformation vector / contour plots

Stress vector / contour plots

Flow vectors

Analysis Module

Solid Elements: Finite Infinite

Joint Elements: Zero thickness Thin

Bar Elements


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PROTYPE-TRACK ANALYSIS

Displacement Contours

Deformed Shape

Intact D-F Case in CC

Dam-Foundation in CC Displacement Vectors for D-F

Minor Principal Stress ContoursMajor Principal Stress Contours


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Flow vectors at 4th Stage with grout zone in the presence of

water table

Flow net for cavern with oil with grout zone in the presence

of water table


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Approaches

Displacement Method

Equilibrium Method

Mixed Method

Displacement Formulation

Primary & Secondary Unknowns

Problem Primary Secondary

Stress Deformation Static Dis lacements Strains Stresses

Dynamic foundations, dams,

embankments, Slopes, pavements

Accelerations,

velocities

Seepage, flow Fluid potentials Velocities, Discharge,

Quantity of flow

Coupled consolidation,

Liquefaction

Displacements,

Pore pressure

Strains, Stresses,

Quantity of flow

Non-homogeneity

Complex Boundaries

Material Non-linearity

Geometric Non- Linearity


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Commercial Software

ABAQUS

ax s , ax s

Phase2, Midas GTS

SOFiSTiK, CESAR-LCPC

Examine

2D

, Examine

3D

31 July 201273/189

FLAC, FLAC3D

UDEC, 3DEC

Books

Desai & Kundu: Introductory Finite Element Method

Desai & Abel: Introduction to Finite Element Method

Bathe: Finite Element Procedures in Engineering Analysis

Zienkiewicz: Finite Element Method

Rao SS: Finite Element Method in Engineering

Krishnamoorthy: Finite Element Analysis


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Documents

Intro to bhj