Measurement of internal Strain/Stress Fields by Means of...

Measurement of internal Strain/Stress Fields by Means of Holography

by Carlos Castro Abadía

This thesis was submitted as part of the requirement for the MEng. degree in Engineering

School of Engineering, Univ. of Aberdeen

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Abstract

The goal of the project is the verification of the evaluation method published by

Dändliker. This includes the implementation of his Finite Difference Method as a

computer program as well as the measurement of the surface deformation of an object,

and the determination of the internal stresses and strains. The results of this evaluation

shall be compared with the analytically calculated stresses and strain of the chosen

object. The steps in more details are:

- Implementation of the Dändliker [6] algorithms on a computer. For the

implementation standard software systems may be used, e.g. MatLab.

- Measurement of the bending of a rectangular steel bar under a point load. The steel

bar shall be fixed on one end in the longitudinal direction and the load shall be

applied on the other end, fig. 1.

Fig. 1: Sketch of the set-up

The deformation of the surface shall be measured by Digital Holography in three

dimensions in a sufficient number of surface points.

- The internal stresses and strains shall be evaluated by the Finite Difference Method.

The data must be presented in a form to be compared to an analytical solution.

- The internal stresses and strains shall be evaluated by known analytical solutions.

The two results – measured and computed - for the internal stresses and strains shall be

compared. Differences have to be discussed.

Fsteel bar

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Contents

1- Introduction 8

1.1- Brief History of Holography 9

1.1.1- Principles of Holography 11

1.1.2- Two mains distributions in Holography 16

1.1.3- In-Plane/Out-of-Plane 16

1.1.4- Non-Destructive Testing 17

1.1.5- Temporal Coherence 18

1.1.6- Spatial Coherence 19

1.1.7- Diffraction 19

1.1.8- Digital Holography 20

1.2- Brief History of the Finite Element Method 22

1.2.1- Finite Elements in Engineering 22

1.2.2- Finite Elements in Mathematics 23

2- Generation Surface Deformation 26

2.1- Selection of the Metal Beam 26

2.2- Application of Finite Element Method 35

3- Dänliker Method 46

3.1- Dänliker Algorithms 46

3.2- Finite Difference 47

3.3- Elasticity Equations 50

3.4- Checking the Dänliker Results 51

4- Set Up in the Holography Lab 53

4.1- Initial Sketch 53

4.2- Improvement of the Sketch 55

4.3- Set Up 60

4.3.1- Utilized Instruments 61

4.3.2- Real Disposition of the Instruments 62

4.3.3- Procedure Explanation 63

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4.4- Problems Encountered 73

4.5- Results Obtained 74

4.6- Applicable Improvements 74

5- Discussion and Interpretation 75

6- Conclusions 76

7- Recommendations and Suggestions for Future Work 77

8- References 78

9- Appendices 79

9.1- Appendix I. Results of the Nodes in the Finite Element Analysis. 80

9.1.1- Superficial Face 81

9.1.2- Section at 1 mm below the top surface 83








9.2- Appendix II. Matlab’s Program and Results 107

9.3- Appendix III. Laser Safety 119

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List of tables and figures

Figure 1.1.2.1 Recording of transmission hologram 13

Figure 1.1.2.2 Scheme of volumetric hologram 14

Figure 1.1.2.3 Recording of reflection hologram 16

Figure 1.1.6.1 Michelson-interferometer 18

Figure 1.1.9.1 Recording a digital hologram 20

Figure 1.1.9.2 Reconstruction with reference wave 21

Figure 1.2.1 Sketch of the metal beam 24

Figure 2.1.1 First disposition of the metal beam 27

Figure 2.1.2 Second disposition of the metal beam 28

Figure 2.1.3 Third disposition of the metal beam 28

Figure 2.1.4 Sketch of the studied system 29

Figure 2.1.5 Plan of the metal beam 31

Figure 2.1.6 Diagram of the metal beam 32

Figure 2.2.1 Distribution of the forces 36

Figure 2.2.2 Types of elements 37

Figure 2.2.3 Mesh of the metal beam 39

Figure 2.2.4 Displacement in X direction 39

Figure 2.2.5 Displacement in Y direction 40

Figure 2.2.6 Displacement in Z direction 40

Figure 2.2.7 Strain in X direction 41

Figure 2.2.8 Strain in Y direction 42

Figure 2.2.9 Strain in Z direction 42

Figure 2.2.10 Stress in X direction 43

Figure 2.2.11 Stress in Y direction 43

Figure 2.2.12 Stress in Z direction 44

Figure 2.2.13 Von Misses Stress 44

Figure 2.2.14 Selected Nodes 45

Figure 3.2.1 Grid in the Finite Difference 48

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Figure 4.1.1 Initial Sketch 53

Figure 4.1.2 Paths lengths 54

Figure 4.2.1 Description of the Sensitivity Vector 55

Figure 4.2.2 Referent points 56

Figure 4.2.3 Generation of the ellipse 57

Figure 4.2.4 Components of the Sensitivity Vector 57

Figure 4.2.5 Components of the two illumination points 58

Figure 4.2.6 Situation of the points in the metal beam 59

Figure 4.2.7 Improved sketch 60

Figure 4.3.2.1 Detailed sketch 62

Figure 4.3.3.1 Regulation of the laser level 64

Figure 4.3.3.2 Regulation of the speck of the reference beam 64

Figure 4.3.3.3 Regulation of the reference beam level 65

Figure 4.3.3.4 Positive telescopic system 66

Figure 4.3.3.5 Negative telescopic system 67

Figure 4.3.3.6 Regulation of the Defocusing Lend 68

Figure 4.3.3.7 Regulation of the horizontally of the convex lend 68

Figure 4.3.3.8 Regulation of the vertically of the convex lend 69

Figure 4.3.3.9 Regulation of the last beam splitter for the illumination beam 71

Figure 4.3.3.10 Regulation of the last beam splitter for the reference beam 71

Figure 4.3.3.11 Real assembly in the holography lab 1. 72

Figure 4.3.3.12 Real assembly in the holography lab 2. 72

Table 9.1.1.1 Displacement of the Nodes at superficial face 80

Table 9.1.1.2 Strain of the Nodes at superficial face 81

Table 9.1.1.3 Stress of the Nodes at superficial face 82

Table 9.1.2.1 Displacement of the Nodes at 1mm 83

Table 9.1.2.2 Strain of the Nodes at 1mm 84

Table 9.1.2.3 Stress of the Nodes at 1 mm 85

Table 9.1.3.1 Displacement of the Nodes at 2 mm 86

Table 9.1.3.2 Strain of the Nodes at 2 mm 87



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Table 9.1.6.3 Stress of the Nodes at 5mm 97










Table 9.3.1 Classification of the lasers 123

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1- Introduction

Measurement of internal strain/stress by means of holography is a field really

interesting.

One of the most interesting points is the capacity of measure behaviors of the studied

object without damage the object. This means that it is possible to get important

information without change elastic, plastic or even thermal characteristics of the

material and the object.

It can be possible when it is applied the Non-Destructive Testing (NDT) by Digital

Holography (DH), which mean it is going to be explained afterwards.

This kind of studies are used in aircraft industry like is explained afterwards in the brief

history of the holography.

It is also important the role of the Finite Element Method (FEM) in this study. This

analysis allows check and compares the two results. However, the FEM is not an

exactly method but very approximated as well.

In this project is going to be explain how obtain the strains and stresses inside a metal

beam by Finite Element (a simulation technique very used nowadays by mechanical and

structural companies) by the software of finite elements Abaqus.

Then, the application of the Dänliker algorithms to extrapolate the deformation of the

surface of the metal beam inside it. This equations are really interesting when you only

can obtain the surface displacement, like in holography measurement.

And finally the set up to compare the results of FEM with Digital Holography and

Dänliker algorithms.

Once explained that, it is interesting to understand why these are the methods followed

in this project, doing a brief introduction about the holography (to comprehend Digital

Holography) and FEN; and taking a look of the evolution of this two fields.

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1.1- Brief History of Holography

Holography dates from 1947, when British (native of Hungary) scientist Dennis Gabor

developed the theory of holography while working to improve the resolution of an

electron microscope. Gabor coined the term hologram from the Greek words holos,

meaning "whole," and gramma, meaning "message". Further development in the field

was stymied during the next decade because light sources available at the time were not

truly "coherent" (monochromatic or one-color, from a single point, and of a single

wavelength).

This barrier was overcome in 1960 by Russian scientists N. Bassov and A. Prokhorov

and American scientist Charles Towns with the invention of the laser, whose pure,

intense light was ideal for making holograms. In that year the pulsed-ruby laser was

developed by Dr. T.H. Maimam. This laser system (unlike the continuous wave laser

normally used in holography) emits a very powerful burst of light that lasts only a few

nanoseconds (a billionth of a second). It effectively freezes movement and makes it

possible to produce holograms of high-speed events, such as a bullet in flight, and of

living subjects. The first hologram of a person was made in 1967, paving the way for a

specialized application of holography: pulsed holographic portraiture.

In 1962 Emmett Leith and Juris Upatnieks of the University of Michigan recognized

from their work in side-reading radar that holography could be used as a 3-D visual

medium. In 1962 they read Gabor's paper and "simply out of curiosity" decided to

duplicate Gabor's technique using the laser and an "off-axis" technique borrowed from

their work in the development of side-reading radar. The result was the first laser

transmission hologram of 3-D objects (a toy train and bird). These transmission

holograms produced images with clarity and realistic depth but required laser light to

view the holographic image. Their pioneering work led to standardization of the

equipment used to make holograms. Today, thousands of laboratories and studios

possess the necessary equipment: a continuous wave laser, optical devices (lens, mirrors

and beam splitters) for directing laser light, a film holder and an isolation table on which

exposures are made. Stability is absolutely essential because movement as small as a

quarter wavelength of light during exposures of a few minutes or even seconds can

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completely spoil a hologram. The basic off-axis technique that Leith and Upatnieks

developed is still the staple of holographic methodology.

Also in 1962 Dr. Yuri N. Denisyuk from Russia combined holography with 1908 Nobel

Laureate Gabriel Lippmann's work in natural color photography. Denisyuk's approach

produced a white-light reflection hologram which, for the first time, could be viewed in

light from an ordinary incandescent light bulb.

Another major advance in display holography occurred in 1968 when Dr. Stephen A.

Benton invented white-light transmission holography while researching holographic

television at Polaroid Research Laboratories. This type of hologram can be viewed in

ordinary white light creating a "rainbow" image from the seven colors which make up

white light. The depth and brilliance of the image and its rainbow spectrum soon

attracted artists who adapted this technique to their work and brought holography

further into public awareness. Benton's invention is particularly significant because it

made possible mass production of holograms using an embossing technique. These

holograms are "printed" by stamping the interference pattern onto plastic. The resulting

hologram can be duplicated millions of times for a few cents apiece. Consequently,

embossed holograms are now being used by the publishing, advertising, and banking

industries.

In 1972 Lloyd Cross developed the integral hologram by combining white-light

transmission holography with conventional cinematography to produce moving 3-

dimensional images. Sequential frames of 2-D motion-picture footage of a rotating

subject are recorded on holographic film. When viewed, the composite images are

synthesized by the human brain as a 3-D image.

In 70's Victor Komar and his colleagues at the All-Union Cinema and Photographic

Research Institute (NIFKI) in Russia, developed a prototype for a projected holographic

movie. Images were recorded with a pulsed holographic camera. The developed film

was projected onto a holographic screen that focused the dimensional image out to

several points in the audience.

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Holographic artists have greatly increased their technical knowledge of the discipline

and now contribute to the technology as well as the creative process. The art form has

become international, with major exhibitions being held throughout the world.

1.1.2- Principles of Holography

There are two physical phenomena as the principles of the holography: interference and

diffraction of light waves. Holograms are photographs of three dimensional impressions

on the surface of light waves. Therefore, in order to make a hologram you need to

photograph light waves. This presents something of a dilemma. As we all know, it can

be problematic to take a photograph of a quickly moving object. If you've ever had a

picture come back blurred from the film lab, you know all too well. When a person

moves too quickly in a photograph, their image blurs. Try to imagine the problems

associated with trying to photograph a photon. To start, a light wave moves at the speed

of light. That is about 300.000 kilometers per second. That is more than half way to the

moon in a second. Considerably faster than someone's hand waving. In fact, its so fast

that the very idea of even capturing it on film would appear impossible. What we need

is a way to stop the photon so it can be photographed. And this technique is called

interference.

Imagine yourself standing on a small bridge over of still water. Let’s further imagine

that you were to drop a pebble into the pond. As it hits the water it creates a circular

wave. This wave radiates outwards in an ever growing circular path. We've all done

this. Now, if you drop two pebbles in the water, you would create two circular waves,

each of which would grow in size and eventually cross the path of the other wave and

then continue on its individual expanding path. Where the two circular waves cross each

other, you might say that they interfere with each other. And the pattern that they make

is called an interference pattern. Not too difficult to envision. This is what interference

is. Two waves interfering with each other as they cross paths. No permanent impact is

left on either wave once it leaves the area of overlap. Each wave looks exactly the same

as it did before it crossed the other waves path. Well, maybe it is grown a little bit

bigger, but that's about it. So, what's the big deal about interference in that case?

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Here it is. As waves cross paths and interfere, the pattern they make is called a standing

wave. It is called a standing wave because it stands still. And since it stands still, it can

be photographed. This solves the problem of how we can photograph something moving

at the speed of light. So, to photograph interference pattern we should use special light

source. It is laser, which was first made to operate in 1960.

Laser light differs drastically from all other light sources, man-made or natural, in one

basic way which leads to several startling characteristics. Laser light can be coherent

light. Ideally, this means that the light being emitted by the laser is of the same

wavelength, and is in phase.

When two light waves pass through each other each wave acts like a bump to the other.

And the result is like rapids of light. The standing wave patterns are stationary even

though the light waves energy continues to move. When waves meet they perform

addition and subtraction. When two waves of equal size meet at their high points (called

crests), they add together to make a wave twice as high at that point. Conversely, where

two waves of equal size meet at their low points (call troughs) they add together to

become twice as low. And when one wave at its high point meets another wave at its

low point they subtract and cancel out. But it isn't really cancelled out in the sense of

being destroyed. Its more a case of there being no light at that spot. If you follow the

wave down its path just a drop further it will be meeting the other wave at a different

relationship and once again be visible. It is a situation of infinite possibilities. Just like

the patterns possible as the waves of two pebbles meet in a pond. At any point you may

notice that the standing wave pattern has produced a place where the waves have added

together to get higher or subtracted to become lower or even just gone flat. There are

few terms that are used to describe the possible encounters. If the waves add and get

higher it is called constructive interference. If the waves subtract or cancel altogether its

called destructive interference.

In holography, there are two basic waves that come together to create the interference

pattern. First and foremost is the wave that bounces off the object we are making a

hologram of. Since it bounces off the object, thereby taking its shape, it is called the

object wave. You can't have interference without something to interfere with. So a

second wave of light that has not bounced off an object is used to perform this function.

It is called the reference wave. When an object wave meets a reference wave creating a

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standing wave pattern of interference, it is photographed and called a hologram. Semi-

transparent mirror divides laser beam into two beams. The first beam which is called a

signal beam is directed by mirror, expanded by lens and it illuminates object. The

second beam, called a reference beam, is also directed by mirror, expanded by lens and

it falls directly onto photoplate. The photoplate registers an interference pattern between

the bearing beam and the light beam, reflected from the object. A transmission

hologram appears after an ordinary photo-chemical treatment (hologram of Leith-

Upatnieks). If such a hologram is exposed to a laser light beam, you may see a 3-d

image of the object. The transmission hologram does not reconstruct the image in

ordinary white light, and it is necessary to copy it to the reflection hologram.

Figure 1.1.2.1 Recording of transmission hologram [9]

If we write down the hologram in some volumetric medium, the received model of a

standing wave reproduces, for sure, not only amplitude and phase, but also spectral

distribution of the radiation, written down on it. This circumstance was necessary as a

basis of making of the three-dimensional (volumetric) holograms.

As a basis of operation of the volumetric holograms the Bragg's diffraction effect is

used: as a result of an interference of waves, propagating through thick emulsion, the

planes lighted by light of the greater intensity are formed. After processing of the

hologram on the lighted planes the layers of a blackening are formed. As a result of it

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the "Bragg planes" are created which have property to reflect light partially. I.e. in an

emulsion the three-dimensional interference figure is created.

Such thick-layer hologram provides effective reconstruction of an object wave if the

angle of incidence of a reference beam at recording and reconstruction remains

constant. The change of a wave length of light is not supposed also at reconstruction.

Such selectivity of the volumetric transmission hologram allows to write down on a

plate up to several tens images, changing the angle of incidence of a basic beam

accordingly at recording and reconstruction.

The scheme of recording of the transmission volumetric holograms is same as the Leith-

Upatnieks's scheme for the two-dimensional holograms.

At reconstruction of the volumetric hologram, as against the flat transmission

holograms, only one image is forming due to reflection from the hologram of a

reconstructing beam only in one direction determined by Bragg's angle.

Figure 1.1.2.2 Scheme of volumetric hologram [9]

The recording procedure of reflective volumetric holograms is another. The idea of

making of such holograms belongs to Uri N. Denisyuk. Therefore holograms of this

type are known under the name of their founder.

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The reference and object light beams are formed with the help of a semi-transparent

mirror and directed by a mirror on a plate from two opposite sides. The object wave

shines a photographic plate on the part of an emulsion layer, reference wave - on the

part of a glass substrate. The Bragg's planes in such requirements of recording scheme

almost parallel planes of a photoplate. Thus, the thickness of a photolayer may be rather

small.

On the recording scheme the object wave is formed from the transmission hologram. I.e.

in the beginning usual transmission holograms are made by the technology, described

above, and then make in a copying mode the Denisyuk's holograms from these

holograms (which are termed the master holograms).

The basic property of the reflective holograms is an opportunity of reconstruction of the

image with the help of a source of white light, for example, incandescent bulb or the

Sun. The important property is the colour selectivity of such hologram. It means, that at

the image reconstruction by a white light, it will be restored in colour, that was used

during hologram recording. For example, if ruby laser (red light) was used for recording

procedure, the restored image of the object will be red.

According to property of colour selectivity it is possible to manufacture colour

hologram of the object, exactly reproducing its natural colour. For this purpose it is

necessary to mix three colours at the hologram recording: red, green and blue or to shine

consistently a photoplate by these colours. But, technology of the colour holograms

manufacturing is in an experimental stage and some more efforts and experiments will

be required. It is remarkable, that a lot of people, visitors of exhibitions of the

holograms, usually say, that they can see the colour volumetric images.

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Figure 1.1.2.3 Recording of reflection hologram [9]

1.1.3- Two mains distributions in holography

Distribution in-line: white light beam and beam reflected by the object are in the same

way or in the same line; it means that the angle between the two beams is 00.

Distribution off-axis: white light beam and beam reflected by the object arrive to the

recording plate with an angle between them.

1.1.4- In-plane/Out-of-plane

One of the most powerful fields in the interferometric optical techniques is the measure

of surface displacement. A point on the surface has a set of local Cartesian co-ordinate

associated with it. The Z axis of this coordinate system is normal to the surface and the

X and Y axes are tangents in the horizontal and vertical directions, respectively. So, “in-

plane” for the local co-ordinate system is the plane tangent to the surface (XY).

An object being viewed by an optical system is located within a global co-ordinate

system. The optical line of sight is usually aligned along the Z axis of the global co-

ordinates, and in-plane sensitivity of the optical instrument refers to the global X-Y

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plane. Thus, in order to determine local surface in-plane data, the shape of the object

and its orientation in the optical arrangement must be known, and the global in-plane

data must be transformed from one co-ordinate system to the other. Holographic

contouring, as described earlier, is the most expedient means of gathering data for the

co-ordinate transformations.

1.1.5- Non-Destructive Testing

Non-Destructive Testing (NDT) is used to test materials and components without

destruction. NDT methods are applied e.g. in aircraft industry, in power plants and in

automotive production.

Holography Non-Destructive Testing (HNDT) measures the deformation due to

mechanical or thermal loading of a specimen. Flaws inside the material are detected as

an inhomogeneity in the fringe pattern corresponding to the surface deformation.

HNDT can be used wherever the presence of a structural weakness results in a surface

deformation of the stressed component. The load can be realized by the application of a

mechanical force or by a change in pressure or temperature. Holographic NDT indicates

deformations down to the submicrometer range, so loading amplitudes far below any

damage threshold are sufficient to produce detectable fringe patterns [2].

In HNDT it is sufficient to have one fringe pattern of the surface under investigation.

Quantitative evaluation of the displacement vector field is usually not required. The

fringe pattern is evaluated qualitatively by the human observer or, more and more, by

fringe analysis computer codes.

In the field of holography is really important know several concepts and properties

about the light and the waves of a beam light. So, it is very important explain some

concepts like ‘Temporal Coherence’, ‘Spatial Coherence’ and ‘Diffraction’. These

concepts have to be well known for making after the experiment.

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1.1.6- Temporal Coherence

The Temporal Coherence is explained by the Michelson-interferometer (figure 1.1.6.1).

Figure 1.1.6.1 Michelson-interferometer

Light emitted by the light source S is split into two partial waves by the beam splitter

BS. The partial waves travel to the mirrors M1 respectively M2, and are reflected back

into the incident directions. After passing the beam splitter again they are superimposed

at a screen. Usually the superimposed partial waves are not exactly parallel, but are

interfering with a small angle. As a result a two-dimensional interference pattern

becomes visible.

The optical path length from BS to M1 and back to BS is s1, the optical path length

from BS to M2 and back to BS is s2. Experiments prove that interference can only

occur if the optical path different s1-s2 does not exceed a certain length L. If the

optical path difference exceeds this limit, the interference fringes vanish and just an

uniform brightness becomes visible on the screen. The qualitative explanation for this

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phenomenon is as follows: Interference fringes can only develop if the superimposed

waves have a well defined phase relation.

The phase difference between waves emitted by different sources varies randomly and

thus the waves do not interfere. The atoms of the light source emit wave trains with

finite length L. If the optical path difference exceeds this wave train length, the partial

waves belonging together do not overlap after passing the different ways and

interference is not possible [1].

1.1.7- Spatial Coherence

Spatial coherence describes the mutual correlation of different parts of the same

wavefront.

Spatial coherence describes also the ability for two points in space, x1 and x2, in the

extent of a wave to interfere, when averaged over time. More precisely, the spatial

coherence is the cross-correlation between two points in a wave for all times. If a wave

has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent.

The range of separation between the two points over which there is significant

interference is called the coherence area, Ac. This is the relevant type of coherence for

the Young’s double-slit interferometer [1].

1.1.8- Diffraction

To explain the phenomenon of Diffraction it is necessary to imagine a light wave which

hits an obstacle. This might be an opaque screen with some transparent holes, or vice

versa, a transparent medium with opaque structures. From geometrical optics it is

known that the shadow becomes visible on a screen behind the obstacle. By closer

examination, one finds that this is not strictly correct. If the dimensions of the obstacle

are in the range of the wavelength, the light distribution is not sharply bounded, but

forms a pattern of dark and bright regions.

Diffraction can be also explained with the Huygens ’ Principle:

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‘Every point of a wavefront can be considered as a source point for secondary spherical

waves. The wavefront at any other place is the coherent superposition of these

secondary waves.’ [1]

Once done a little review of holography it will be good to do a little review of digital

holography, because the holography that is going to be used is digital holography.

1.1.9- Digital Holography

The concept of digital hologram recording is illustrated in figure 1.1.9.1. A plane

reference wave and the wave reflected from the object are interfering at the surface of a

Charged Coupled Device (CCD). The resulting hologram is electronically recorded and

stored. The object is in general a three dimensional body with diffusely reflecting

surface, located at a distance d as well, but in the opposite direction from the CCD, see

figure 1.1.9.2 [1].

Figure 1.1.9.1 Recording a digital hologram

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Figure 1.1.9.2 Reconstruction with reference wave

There are several vantages and disadvantages between common holography and digital

holography, and these are the cause of using digital holography in this work [8]:

+ Digital holograms have strong anti-disturb property.

+ They are easy to be modified.

+ They are difficult to be imitated.

+ By using the pixel holograph and embossment, hologram films can be fabricated

by transforming digital holograms. This can make digital hologram anti-

counterfeiting identifiers.

+ Digital holography uses holographic printers, exposing the photosensitive

emulsion with computer generated images. This leads to the creation of

conventional holograms with digital content rather than real scenery.

+ 2D/ 3D graphics or digital photographs and movies can be printed which helps

in the holographic recording of real outdoor scenes, completely synthetic

objects, and objects in motion. This is impossible to achieve with optical

holography.

+ Digital holograms can be multiplexed.

+ Another basic advantage is that the content for digital holograms can easily be

created by non-experts and the printing process is not very expensive compared

to conventional holograms.

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+ A series of digital photographs or a short movie of a real object is enough for

producing digital holograms.

− Digital holograms lack in the quality in terms of color appearance, resolution,

sharpness, etc. of conventional optical holograms.

− It is really difficult to do digital holography off-axis, because CCD-Camera

needs a very small angle between the two lights beams, so it is better to use in-

line. It means that the resolution increase while the angle between the two beams

decrease.

1.2- Brief history of the Finite Element Method

To understand the progress in Finite Elements it is worthwhile to briefly review the

history of the method, which was developed more or less in parallel in engineering and

mathematics.

1.2.1- Finite Elements in Engineering

The major development in engineering FEM was related to computers. In 1952 a great

effort was made at Boeing to analyze aircraft structures. A procedure was developed,

and appeared in the literature only by 1956 in [Turner et al., 1956]. The method was

essentially based on classical ideas of matrix structural mechanics cast in the framework

of digital computers. In 1961, it was recognized that the FEM is a special version of the

variational approach of Ritz [Ritz, 1909] and Garlerkin [Galerkin, 1915]. By 1964

several aircraft companies had developed their own finite element programs and in 1965

the ASKA program, written by the group under J.H. Argyris, appeared on the market.

The book by Zienkiewicz and Cheung [Zienkiewicz and Cheung, 1967] which appeared

in 1967 stimulated further development and played an important role. In the late 1960s

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the program NASTRAN written by MacNeal-Schwendler Corporation appeared on the

market. Since then, dozens of various commercial and research codes were written and

are available.

1.2.2- Finite Elements in Mathematics

If the FEM is understood as an approximate method for solving differential equations

utilizing a variational principle and piecewise polynomial approximation, then likely

G.W. Leibnitz (1646-1716) in 1696 was a user of FEM. At the same time L.Euler

(1707-1783) introduced the variational methods with the approximation approach being

essentially the main tool employed for the derivation of the Euler equation. Leibnitz, in

a letter to Johann Bernoulli (1667-1748) [Leibnitz, 1962], addressed the problem of the

brachistochrone, replacing the general descent curve with a piecewise linear curve with

one and two internal nodes, he determined the coordinates of nodal points to achieve the

brachistochrone.

In the nineteenth century the variational approaches were tools for proving the existence

of a solution of differential equations and various discretizations were also used as

mathematical tools. As an example the history of the variational method for solving the

Poisson problem. G.F.B. Riemann (1826-1866) in his 1851 thesis used the variational

principle of energy minimization, which he called the Dirichlet Principle, to prove the

existence of the solution to this problem, essentially assuming that the minimize exists.

L.T.W. Weierstrass (1815-1897) found an essential flaw in Riemann’s argument about

the existence of the minimizer. This error was corrected much later by Hilbert [Hilbert,

1901] using essentially the principles of functional analysis. The use of the variational

method as a tool for obtaining approximate solutions was first proposed by Ritz [Ritz,

1909], and Galerkin [Galerkin, 1915]. Courant touched on the subject of the variational

method in six papers with the last one in 1943 [Courant, 1943] where he explicitly

proposed a piecewise linear function for approximate solution of the Poisson problem

via minimization of the energy. For this reason, the linear element is sometimes called

the Courant element. Courant proposed the method in 1922 [Hurwitz and

Courant,1922], where he gave the proof of existence of the conformal mapping via

minimization of the Dirichlet integral, by employing a sequence of functions, which he

named the minimal sequence, which has the property that the potential energy decreases

24

and converges to the infimum. A long footnote is given on the construction of the

minimal sequence, where subdivision of the domain into triangles and the use of

piecewise linear functions is proposed.

There are many other instances of discovery of the FEM.

The history of FEM shows that it originated from different sources and today it is a very

mature method where engineering, mathematics, and computer science are interwoven.

On the other hand, the number and richness of the results are so large that it is hard for a

single person to comprehend it. This has led to the development of various specialties

concentrating only on a few particular aspects. For example, focusing on the

computational aspect leads one to emphasize only ‘recipes’ whilst neglecting ideas

behind them, while concentrating on mathematical aspects leads sometimes one to lose

sight of the purpose of FEM, which is the computational analysis of engineering

problems. Focusing on the computer-science aspects, such as the parallelization of FEM

computations, also leads to one sided view.

Once explained a brief summary of the history of the finite element method it is

important to know that the study of the metal beam can be approximated by a two-

dimensional analysis. It means that the deformation of the metal beam can be studied as

if it was a problem of a planar body.

This study will be done in a section of the bar parallel to the direction of the forces.

Figure 1.2.1 Sketch of the metal beam

25

However, in my case it is necessary to do a three-dimensional analysis because it is

essential knowing the behavior of several points in the surface perpendicular to the

direction of the applied force.

Knowing this it is important to do a brief summary of the finite elements for three-

dimensional analysis.

Thus, the fundamental ideas governing the development of element properties for three-

dimensional analysis are unchanged respect two-dimensional analysis; and some

families of elements for three-dimensional analysis are logical extensions of the families

of elements for two-dimensional analysis.

For example, the triangle and rectangle of two-dimensional analysis become the

tetrahedron and rectangular hexahedron respectively of the three-dimensional analysis.

With regard to three-dimensional finite-element analysis it is perhaps appropriate at the

very beginning to emphasize the effect that increasing the number of dimensions has on

the size of a problem (i. e. on the total number of degrees of freedom of the problem).

For example, imagine that in a one-dimensional analysis it is necessary to use n nodes

(at which u is the degree of freedom) to achieve a satisfactory level of accuracy of

solution, i.e. n unknowns are required. Then, for a similar level of accuracy in two-

dimensional analysis we would need to use around n2 nodes with u and v as degrees of

freedom at each node, giving a total of about 2n2 unknowns. Proceeding to three

dimensions we would use about n3 nodes with u, v and w as degrees of freedom for the

same sort of accuracy, giving a total of about 3n3 unknowns. If n=10, say, the necessary

degrees of freedom increase from 10 to 200 to 3000 as we proceed from one to two to

three dimensions. It is clear that the finite-element analysis of practical three-

dimensional solids will often result in the generation of very large numbers of equations

[3].

26

2- Generation Surface Deformation

2.1- Selection of the Metal Beam

To generate the deformed surface is necessary knowing which are going to be the

dimensions, the material and the properties of the bar.

All this things are explained in the text which is below. In this part, some possibilities

have been considered, like the disposition of the metal beam in the lab and in which

way it is going to be clamped, roughly deformation of the extreme of the bar and

selection of the dimensions of the metal beam.

First of all, it is necessary to decide the dimensions of the bar and the material.

Well, there are mainly two options in the field related with the material choice. One

choice is using Steel and another choice is Aluminum Alloy.

Like you can use both of these materials in applications of holography, I’ve chosen

Aluminum Alloy, because its Young’s Modulus is lower, so for the same force applied

the deformation will be bigger, and then the study will be more intuitive.

For deciding the dimensions of the bar I have calculated some the deformations with a

different sections of the bar, so at the end, I can decide more or less which has to be the

definitively section of the bar.

To calculate this deformation, first we have to talk about the disposition of the bar in the

experiment or in which way the bar will be hold.

At the beginning, I thought in only leaning the bar by the two extremes. This idea is

drawn in the figure 2.1.1.

27

Figure 2.1.1 First disposition of the metal beam

This disposition is really good to calculate the deformation in the middle, but the main

problem which did that I couldn’t choose this disposition was the holders, because of

the bar slips when I applied the force.

Then I thought of using the same disposition but using two clamps instead of using two

supports.

28

Figure 2.1.2 Second disposition of the metal beam

The problem of using this layout is that I needed a bar which a length twice to obtain the

deformation that I wish. For this reason, I have taken the last one disposition.

Figure 2.1.3 Third disposition of the metal beam

29

To demonstrate that we need a longer bar if we used the second disposition instead

using the third disposition, we need to use differential equations related with the field of

the elasticity [5].

So, the sketch of the system is:

Figure 2.1.4 Sketch of the studied system

Explication of the letters:

X: Horizontal Displacement. X=0 is at the beginning of the bar.

2L: Length of the bar.

F: Force applied in the middle of the bar.

E: Young’s Modulus

I: Inertia of the bar.

So, if I call:

W: vertical displacement.

W’: angle turned and also means the first derivate.

M: moment in the restriction.

Q: charge or force applied in this point.

30

In this way, we know different things related with the displacement in the clamps.

W=0

W’=0

M≠0

Q≠0

And the differential equation is:

′′ (2.1)

So, the equation is integrated twice we can obtain the vertical displacement depending

in the length of the bar:

′ 1 (2.2)

1 2 (2.3)

Now, if is applied the boundary conditions explained some lines ago, it is possible to

obtain the definitively equation.

0 (2.4)

2 (2.5)

′ 0 0 1 1 0 (2.6)

0 0 2 2 0 (2.7)

And the final equation is:

(2.8)

31

So, if we have to obtain the same maximal deformation, the length of the bar in the

second disposition has to be twice the third disposition, because of in the second

disposition the X will be in the middle of the bar and in the third disposition the X will

be in the extreme of the bar.

Now, I know how to calculate the deformation approximately it is possible to choose

the final section of the bar.

Here it is shown the different possibilities of the dimensions of the bar.

I think that it will be good obtaining a high displacement, because it is more visual that

if the displacement was small.

So, I’ve done the calculations with different thickness, high and length.

So, the dimensions of the bar will be:

a= length

b= thickness

c= height

Figure 2.1.5 Plan of the metal beam

The material chosen is Aluminum Alloy 2014-T6.

For this material de Young Modulus is 69GPa = 69000Pa/mm2 = 69 MPa.

Now, for calculating the inertia and to put in a right way the axis is necessary to show it

in a sketch:

32

Figure 2.1.6 Diagram of the metal beam

So, the inertia will be respected axis X:

(2.9)

In this way, applying the equation of the deformation and the different parameters

written before it is obtained:

(2.10)

(2.11)

(2.12)

33

I’ve done the calculations for different forces and dimensions which are:

• F=100N >>>>>>>> Deformation at the end of the bar = 0,3865 mm

a=150mm

b=15mm

c=15mm


a=150mm

b=15mm

c=15mm


a=150mm

b=8mm

c=12mm


a=150mm

b=12mm

c=8mm


a=150mm

b=12mm

c=8mm

• F=250N >>>>>>>> Deformation at the end of the bar = 0,96618mm

a=150mm

b=15mm

c=15mm

• F=250N >>>>>>>> Deformation at the end of the bar = 2,3588mm

a=150mm

b=12mm

c=12mm

34

Once I’ve done these calculations, I think that it will be good to take a bar with

thickness between 8mm and 15 mm, and with a height of 8mm and 15mm.

By other hand, I think that I will only need 150mm to do the experiment but I will also

need more length to hold this bar; so I think that it is a good idea take a bar of 300 mm.

Note: The dimensions of the bar are not really important in this project because the

main point is measuring its displacement.

Material: Aluminum Alloy 2014-T6

Length = a = 300mm

Area of the bar could be (b*c)= 12x12, 10x12, 8x10, 10x8, etc.

At the end, it was available in the workshop a bar made of Aluminum Alloy 2014-T6

with the next dimensions:

Length= 300mm

Section of the bar=12x12mm

So, in this way to appreciate in a right way the vertical displacement I am taking the

next dimensions:

200

12

12

150

And using the equation of the vertical displacement at the extreme of the bar it is

possible to approximate the maximum displacement at:

3,3548

35

2.2- Application of Finite Element Method

Once decided the exactly dimensions it is time to assembly the bar in the lab.

Like it is going to be explained in the point of Digital Holography, I need to study the

perpendicular face to the table of the lab of the metal beam; so it is necessary to change

a little the way of applying the force.

So the procedure it is going to be the next:

The direction of the force is going to be parallel to the workbench, it means that the

force will have a horizontally direction.

Nevertheless, it is important applying one of the main properties of the holography

which is the non-destructive testing. It is important to take into account this because it

guarantees that the metal beam doesn’t suffer any kind of permanent deformation.

One explained all these things; it is really important explaining the most important

points of the procedure in the generation of the surface.

So, the software used in this part is Abaqus 6.8.

In this program the coordinate system has changed some components respect the initial

components described upper. So, the next figure shows how it is placed the bar in the

space and the direction of the applied forces.

36

Figure 2.2.1 Distribution of the forces

The reason why I have chosen two concentrated forces is because it simulates the

behavior of the clamp which does the force to the bar.

This is going to be explained in detail in the next chapters, especially in which it is

explained the assembly in the lab.

As I mentioned before about non-destructive testing I will do a maximum deformation

in the extreme of the metal beam about 50 μm.

So, knowing the deformation at the extreme is easy to obtain the necessary force

applied.

(2.13)

0,05 (2.14)

, 1,2161 (2.15)

37

As I am going to apply the force in two concentrated points, the force in each node will

be:

0,608 (2.16)

Once decided the magnitude of the force it is important and necessary knowing which

kind of finite element is going to be applied in the bar.

As I have explained in the introduction of Finite Element Method there are several types

of finite element, depending which analysis you are doing.

In this case, the software Abacus has mainly six types of element, which are for each

kind of analysis.

So, the next picture shows these elements.

Figure 2.2.2 Types of elements [4]

38

In my analysis I have chosen the Hexahedra because it is the most appropriate to realize

a analysis of an object which sections are rectangular.

The rectangular hexahedral element is obviously the three-dimensional element

equivalent of the basic rectangular element in two-dimensions. The element has a total

of 24 degrees of freedom (the nodal freedom are u, v and, w at eight corner nodes) and

the co-ordinate origin is taken for convenience to be at the centroid of the body.

In assuming a displacement field for the element it is natural to begin by considering

expressions for each of u, v, and w as products of linear functions in x, in y, and in z, i.e.

as expressions of the form:

(2.17)

where , , etc. are constants. Multiplying this out gives expressions of the form [3]:

. (2.18)

Once shown this, it is clearer how Abaqus works with this element; so if the number of

element is very huge the computer will have problems to solve the job.

Once selected the type of the element is time to choose the size of the element.

I would like to put the element’s size as small as possible, but if the size is so small the

software has problems to simulate. So, the final and minimum size chosen is 1 mm.

Thus, the metal beam looks like this:

39

Figure 2.2.3 Mesh of the metal beam

At the end, the whole metal beam has been meshed with a total of 35280 elements.

Once meshed the metal beam and submitted and run the job; everything is ready to

obtain the different results.

So, the displacement of every node of the bar in the X, Y and Z directions will be like

the next figures show.

Figure 2.2.4 Displacement in X direction

40

Figure 2.2.5 Displacement in Y direction

Figure 2.2.6 Displacement in Z direction

It is important to remark that the maximum displacement correspond at the extreme of

the bar, and it is 4,995 μm, very close to the first approximation calculated roughly at

the end of this chapter.

41

This small difference can be due to errors of the finite element and the software.

Nevertheless, the numerical approximation is an approximation, it means that it is not

an exactly calculation.

By other hand, the strains in the mains directions are:

Figure 2.2.7 Strain in X direction

42

Figure 2.2.8 Strain in Y direction

Figure 2.2.9 Strain in Z direction

43

And finally the stresses in the mains direction and the Von Misses stress are:

Figure 2.2.10 Stress in X direction

Figure 2.2.11 Stress in Y direction

44

Figure 2.2.12 Stress in Z direction

Figure 2.2.13 Von Misses Stress

All these images represent the whole metal beam. However it is interesting knowing the

displacement in some particular nodes.

45

I have chosen 36 nodes close to the extreme of the bar. These nodes are shown in the

next figure:

Figure 2.2.14 Selected Nodes

Then I have obtained the stress and strains of the nodes just below them.

I have done eight sections with 1mm between two consecutives sections. So, the last

section has reached eight mm below the first surface.

The different magnitudes of the displacement, strains and stresses and the coordinates of

the nodes are shown in the Annex I.

46

3- Dänliker Method

3.1- Dänliker Algorithms [6]

It is well known that holographic interferometry allows determining the displacement

field on the surface of solid objects under mechanical load.

It is the purpose of this paper to show that the strain and stress field can be extrapolated

below the surface from the measurable displacements of the surface. This will be

verified, at least for isotropic elastic materials, following Hooke’s law.

To calculate the strains and stresses inside the metal beam, it is necessary first knowing

the way in which the information will be obtain and in which way it will be processed.

So, the differential deformation of the material are completely described by the vector

gradient

, (3.1)

of the displacement field u(x).

It is assumed that μik is completely known in a plane (x, y, z) where z is constant.

So the linear extrapolation in a plane which normal vector is parallel to axis z is possible

through

, , , , , , , (3.2)

if the derivates μik,z = δμik/δz to the plane are known. However, μik,z cannot be directly

calculated, since only the in-plane derivatives μik,α (α=1,2), are accessible from the

given distribution μik in the plane (x, y, z) where z is constant.

The additional relations to determine the nine components of μik,z are given by the

mathematical compatibility conditions

, , (3.3)

47

where i = x, y, z and α = x, y.

The remaining three components of μik,z are found to be explicitly

, , , , , , (3.4)

, , , , , , (3.5)

, , , , , , , (3.6)

As I said in the introduction of the report, the holography system applied is Non-

Destructive System, so the deformation is almost inappreciable and consequently the

curved surface is depreciable and then it is not necessary to rotate the base in which are

placed all the points.

3.2- Finite Difference

Once explained how obtain the derivates of each displacement it is necessary knowing

how obtain the information.

So, the information will be done with the coordinates of the point and the displacement:

Point A: PA= (225; 5; 0) Displacement PA: U = 0.00181923 mm V = -0.000000403692 mm W = 0.0437715 mm To apply this numbers to the equations explained before it is necessary use Finite Difference. This method is a numerical way of doing differentiation. In the Finite Difference there are three possible ways of calculating this differentiation: forward, backward and central. Before explaining each of them it is important remark that the points are going to be operates like they were points of a gird. In which the horizontal distance (ΔX) and vertical distance (ΔY) between points are h. In the next figure is shown the grid with the distributions of the points that will be used for the application of Dänliker algorithms.

48

Figure 3.2.1 Grid in the Finite Difference Forward In this case are used two consecutive points, but it is calculated the first one.

(3.7) Backward In this case are used two consecutive points too; but it is also calculated the first one.

(3.8) Central

In this case are used two non consecutive points, and it is calculated the point between

them.

(3.9)

49

And the method followed to apply the Dänliker algorithms is Central.

Once known how calculated the derivates explained in the Dänliker algorithm is time to

show, with the point B for instance, how all this derivates will be for this point [11]:

0 0 0

(3.10)

This matrix [μik] has to be calculated for the points A, C, D and E because all the terms

in these matrixes are necessaries to construct the matrix [μik,z] for point B.

,

,

,

0 0 ,

(3.11)

where

, ,

(3.11)

, ,

(3.12)

, , (3.13)

Note: In the matrix notation these components means the same x=1, y=2 and z=3.

50

And now, applying the equation 3.2 is possible to obtain the matrix [μik] and [μik,z] in

each of the sections wanted in z direction.

3.3- Elasticity Equations

In this moment, well known these two matrixes are possible to apply the elasticity

equations to obtain the different strains and stresses.

Strain tensor:

(3.14)

where

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

Stress tensor:

(3.21)

51

where [σ] is symmetric and

11

1

0 0 00 0 00 0 0

0 0 00 0 00 0 0

0 0

0 0

0 0

(3.22)

The Matlab’s program and the results for one point are in appendix II.

3.4- Checking the Dänliker Results.

Now, it is important knowing if the extrapolation is well done.

To know this, it is necessary calculating the matrix of first derivates in a section

(knowing the displacements in this section) and then compares the new principle

components of the strain tensor (ε11’ and ε22’) with the extrapolated components in this

section (ε111 and ε122). Here, the high in the extrapolation that I have taken is h = 1mm,

so it is the section 1.

The method followed to obtain these two components is similar to the method done

before.

Now, it is only necessary four points (A, B, C and D) to calculate the matrix [μik] for the

point B which is in the middle of these points.

It is important to know that all the points and their displacements have been taken in the

section situated at 1 mm below the top surface.

This new program and its results are also in the appendix II.

52

Now, to compare these two results is necessary to calculate the relative error.

(9.1)

Thus, for ε11 and ε22 the error will be respectively:

1 . . .

100 41,3% (9.2)

1 . . .

100 36,41% (9.3)

The big errors are due to the thin difference between the numbers of the matrix E1for

the first algorithm and E0 for the second algorithm (see in appendix II) are really near.

However, some cases which are near but changing the magnitude orders. It does that the

error will be so huge.

It is also important remark that an increment of 1 mm between points it is really big for

using Finite Difference; because Finite Difference is an approximation of the integral

for single points.

All these aspects do that the results were a bit different, but if the increment is smaller

the new results will be nearer.

53

4- Set up in the Holography Lab

4.1- Initial Sketch

Before going to the lab is necessary to know how to put the different devices in the table

of the lab.

The most important things that I must have in my mind are the security and the sketch

of the table of the lab to execute the experiment.

So, the security rules followed in the lab are explained in the Annex II.

The first sketch of the experiment will be like the picture shown in the next figures.

In this first picture it is possible to identify the different components that I will use in

the experiment.

Figure 4.1.1 Initial Sketch

In this second picture, it is to identify the lengths that are necessary to control in the

experiment.

54

It is due to the Temporal Coherence; so the sums of the different ways for the reference

wave and the illumination wave have to be more or less the same.

It means that the whole distance ran for the reference wave has to be the same that the

distance ran for the illumination wave.

1 2 3 1 2 3

Where 1,2,3 is the distance rans in each piece of the way of the

reference wave, and 1,2,3 is the distance ran in each piece of the way of

the illumination wave.

Figure 4.1.2 Paths lengths

55

4.2- Improvement of the Sketch

As the objective of the experiment is measuring the displacement of the bar, it is

necessary take into account several important aspects.

First of all, it is necessary to remember which directions affect in the movement of the

object in holography.

So, if it is drew an Illumination Point (IP) and an Observation Point (OP), both two are

the focus of an ellipse. So, if you move the object through this ellipse there is no

changes in the image taken by holography.

However, if you move the object in a perpendicular direction of the observed point of

the object, you will observe the maximum variation. This movement or this direction is

called “Sensitivity Vector”. This explication is represented in the next figure.

Figure 4.2.1 Description of the Sensitivity Vector

56

So, knowing this now, it is possible to improve the sketch of the experiment and apply

this to the practical. This means that it is necessary to apply the force to the beam in the

horizontal direction (axis X), so we will obtain the displacement in the direction of the

observation direction.

As, I will see the Sensitivity Vector with a certain angle, I need to use two illuminations

points to see the two components of this vector.

So, I will illuminate the metal beam from the right and the left of the observe point, it

means from both sides of the CCD Camera.

The theoretical explanation is the next:

When you want an illumination point and an observation point both respected a

determinate point or studied point or observed point; there is a way or field trough the

observed point can be removed and nothing suffer any kind of change. It is due to the

definition of the ellipse.

′ ′ 0 (4.1)

This is represented in the next two figures; the letter with accent means that is the

displacement vector after the movement.

Figure 4.2.2 Referent points

57

Figure 4.2.3 Generation of the ellipse

Now, if it is taken only one illumination point it is really easy to see the two

components reflected in the observed point. One of them is the component “in-plane”

which is tangent the surface in the observed point, and the other one is the component

“Out-of-plane” which is perpendicular to the tangent in the observed point.

Figure 4.2.4 Components of the Sensitivity Vector

58

So, if we put two illumination points for the same observed point it will be obtained the

next four vectors; two of them are the components “Out-of-plane” (one for each

illumination point) which have the same value and the same direction and orientation.

Otherwise, the components “in-plane” will have the same value and direction but

opposite orientation.

So, at the end it will be obtained only one vector in the direction perpendicular to the

observed point.

Figure 4.2.5 Components of the two illumination points

So, this is the reason why is necessary to use two illumination points; because it is

fundamental to obtain the component “Out-of-plane” to obtain the final displacement by

digital holography.

59

Once explained this, the surface of the metal beam that will be studied is going to

change.

If we take as reference plane a plane parallel situated around 24 cm of the top of the

table, nevertheless the camera is going to record in a horizontal and parallel line from

the surface of the table. So the surface of the metal beam that is going to be observed is

perpendicular to the table. This is shown in the next figure.

Figure 4.2.6 Situation of the points in the metal beam

Now, it is necessary to vary the initial sketch because a new illumination point has to be

introduced.

So, in the next figure is shown the final sketch in the lab.

60

Figure 4.2.7 Improved sketch

4.3- Set Up

Now, that it is known which diagram is going to be followed in the lab it is necessary to

name and explain the different devices that will be used, the real disposition of these in

the lab and which is the procedure follow to reach the final assembly.

61

4.3.1- Utilized Instruments:

• Laser: This is a He-Ne Laser which power is 20 mW.

• Beam Splitter: This device is used to separate the wave in two beams. The

Beam Splitter used is made of Fused Silica which refractive index (R) is 1,47.

So, the reflection of the beam in the BS is around 3%. It means that one beam

will have around 3% of the power of the initial beam, and the other beam will

have around 96% of the power of the initial beam. The other 1% is due to some

leaks in the BS.

• Mirror: Used to divert the beam.

• Concave Lend or Defocusing Lend: This kind of lend increase the size of the

speck of the beam. In my case I have used a concave lend -25; it means that the

focus is 25 mm behind it.

• Convex Lend: This lend is used to maintain the diameter of the speck constant.

The lends used in this set up are convex lend 500; it means that this lend has to

be placed 500 mm from the focus of the concave lend.

• CCD Camera: Charge-coupled device Camera is used to record or take a

picture of the hologram.

• Iris Aperture: This guarantees only the pass of one speck of the beam, blocking

the reflected speck.

• Nevertheless some devices like sticks, rails, platform for the rails, screws,

supports, etc. have been used to allow the right set up.

62

4.3.2- Real Disposition of the Instruments

Figure 4.3.2.1 Detailed sketch

In this sketch are represented all the optical instruments used in the set up and in which

beam are working (it is putted in brackets), which are:

1. Laser

2. Beam Splitter

3. Iris Aperture (Reference Beam)

4. Mirror (Reference Beam)

5. Filter for the brightness (Reference Beam)

6. Concave Lend (Reference Beam)

7. Convex Lend (Reference Beam)

8. Mirror(Reference Beam)

9. Iris Aperture (Illumination Beam)

10. Mirror (Illumination Beam 2)


63

12. Concave Lend (Illumination Beam 2)


14. Iris Aperture (Illumination Beam 2)

15. Concave Lend (Illumination Beam 1)


17. Iris Aperture (Illumination Beam 1)

18. Studied Object (Metal Beam)

19. Beam Splitter

20. CCD-Camera

Note: The mirror (10) goes up and down. When it is up, the Illumination Beam 2 is

working, while it is down the Illumination Beam 1 is working. It means that only one of

the two Illumination Beams works with the Reference Beam.

4.3.3- Procedure Explanation

First of all it is really important saying the order of the different kinds of devices is

going to be assembly. So, next to the number of step is written the way that belong this

step.

Step 1 (Calibration of the laser)

Put the way of the laser (1) speck parallel with the workbench. This is realized with a

stick in the rail of the laser, measuring in this way the high of the speck in several

lengths respected the laser.

64

Figure 4.3.3.1 Regulation of the laser level

Step 2 (Reference Beam)

Put the Beam Splitter (2)(BS) behind the laser (1) to divide the wave.

To regulate the BS is necessary to use an Iris Aperture (3) for knowing where is going

the laser speck. This device is also necessary to filter one speck, because the beam

splitter gives two specks due to the reflexion inside it.

Figure 4.3.3.2 Regulation of the speck of the reference beam

65


Putt mirror (4) behind the Iris Aperture (3), already in the right place, for knowing that

the wave is reflected back trough the Iris Aperture too. Doing this you can guarantee

that this tram of the way is parallel to table too.

Now, it is necessary to regulate the other hand of the mirror. To do this is necessary to

use a rule to guarantee the same high along this tram.

Figure 4.3.3.3 Regulation of the reference beam level


Put the last mirror (8) in this way to drive the reference beam to the last beam splitter.

This mirror has to turn the wave 90 degrees.

66


Calibrate optical lends used in the reference beam to assembly a telecopy system.

The first lend that is going to be collocated is a Defocusing Lend or Concave Lend. This

lend rises the speck of the laser proportionally with the distance respected this lend.

To put the speck in the middle of the lend it is necessary to put a paper behind the lend

and see if the speck is in the middle of the lend.

This is so roughly, so later on it will be accurate approximated with the mirrors.

Once it is done, it is collocated the Concave Lend.

To put the second lend in the right place, it is necessary to know which kind of

telescopic system we are assembling. So, there are two possibilities.

First possibility

With a Focusing Lend or Positive Focusing Lend and a Convex Lend.

In this system the focus of the Focusing Lend is behind it, so if the big Convex Lend is

for X mm, it has to be collocated X mm from the focus.

Figure 4.3.3.4 Positive telescopic system

67

Second possibility

With a Defocusing Lend or Negative Focusing Lend and a Convex Lend.

In this system the focus is in front of the Defocusing Lend, so the Convex Lend will be

collocated X mm from this focus.

This distance will be lower than in the first possibility.

Figure 4.3.3.5 Negative telescopic system

In my case, I am using a Defocusing Lend (6) of 25 mm and a Convex Lend (7) of 500

mm. So, the Convex Lend (7) has to be 475 mm from the centre of the Defocusing Lend

(6).

The way to regulate and collocated in the right place both lends is the next.

First of all is necessary to centre the speck of the laser in the centre of the Defocusing

Lend (6). To do this it possible to use a dark paper and put it behind the lend and see

that the speck is roughly in the middle of it.

68

Figure 4.3.3.6 Regulation of the Defocusing Lend

Now, it is necessary guarantee that the expanded speck fits into the Convex Lend (7).

This is also made through a dark paper (like brown paper) putting it in front of the lend

and regulating the big speck like is shown in the next figures:

Figure 4.3.3.7 Regulation of the horizontally of the convex lend

69

Figure 4.3.3.8 Regulation of the vertically of the convex lend

Step 6 (Illumination Beam 1)

Put again an Iris Aperture (9) behind the Beam Splitter (2). Then I put a mirror (16) to

know where has to be directed the beam. But now, I only use this mirror to guarantee

that the beam is come back to the Iris Aperture (9) in a right way.

Once I have done this, I must turn the mirror (16) and point with the metal beam. To do

this correctly, it is necessary use a rule to guarantee the same high along the last tram of

the way.

The procedure is identically to the step 3.

70

Step 7 (Illumination Beam 2)

Put a mirror (10) behind the Iris Aperture (9) to deviate the beam 90 degrees. Then I

have to drive the way equal that in the step 6. Finally, the two specks of the two

Illumination Beams have to be in the same place in the metal beam (18).

Step 8 (Illumination Beam 1 and 2)

Put the different lends in this two ways (Concave Lend 12 and 15).

In this case, I am going to use only a Defocusing Lend with out Convex Lend; for this

reason I only have to be into account putting the Defocusing Lend respect the mirror in

an enough length to allow that the expanded speck fits in the mirror.

The procedure is similar to the step 5, without the Convex Lend.

Step 9 (three both beams)

Now I have assembled the three ways; it is necessary to put the last Beam Splitter (19)

in a right place.

To do it, I need to put a mirror instead putting the metal beam (18). This does that the

mirror reflects the Illumination Wave perfectly into the Beam Splitter (19), then the

Beam Splitter superpose the two beams and you can see if they are accurate superposed.

To do this correctly, it is necessary putting an Iris Aperture (17 and 14 for Illumination

Beam 1 and 2 respectively) before the beam arrives to the metal beam. So in this way,

you can approximate the size of the speck of Illumination Wave to the size of the speck

of Reference Wave.

This comparison can be carried on with a piece of paper. For example, if you first block

one wave you only see the speck of one wave, so you could mark in the paper with a

pen where exactly this speck is. Later, you can block the other beam and do the same

with the other.

So, you can compare the two big specks and know exactly where putting the Beam

Splitter (19).

71

And finally, the CCD-Camera (20) is placed really closed to the Beam Splitter (19). The

only thing that has to be realize carefully is guarantee that sensor of the CCD is in the

middle of the laser speck.

Figure 4.3.3.9 Regulation of the last beam splitter for the illumination beam

Figure 4.3.3.10 Regulation of the last beam splitter for the reference beam

72

It is important to highlight the utilization of rails in the ways of the two illumination

ways. It is important because of the path length. So first of all, the way of the Reference

Beam has been assembly and after the other two ways. These have been assembly with

a path length roughly equal than the path length of the Reference Beam, and after have

been accurate these two distances through the graph rail.

So, the real assembly in the lab is like is shown in the next two figures:

Figure 4.3.3.11 Real assembly in the holography lab 1.

Figure 4.3.3.12 Real assembly in the holography lab 2.

73

4.4- Problems Encountered

One of the main problems was the huge difference brightness between the Reference

Beam and the two Illuminations Beams.

So, knowing this problem, it was looked for some possible sources of the problem,

which were:

Polarization in different direction: To solve this situation it was placed a Polarization

Rotator in front of the first Beam Splitter (2); so managing this device it was observed

that the brightness of the three beams were changing.

And finally, the Reference Beam and the Illumination Beam were polarized at 90

degrees, so it was impossible to achieve any kind of interferometry.

To correct this problem I turned the casing of the laser till the polarizing was different

of 90 degrees and the brightness of the Reference Beam and Illumination Beam were

similar.

Nevertheless, another problem related with the brightness arose. It was that the light

reflected in the metal beam was extremely low, so it was necessary putting a filter (5) in

the way of the Reference Beam to reduce its brightness and approximate it to the

brightness of the Reflected Beam.

To measure and regulate it in an exactly way, I used Laser Power Meter that told me

what was the exactly power (in mW) of each wave.

Once solve this little problem it was time to record some hologram, but it resulted

impossible.

First source of the problem that was though was the path length of the different beams

were wrong and the difference was upper than a few centimeters. So to check it, the

laser was changed by an Nd-YAG Laser which coherence length is around 30 metres (I

had to change the filters because this laser was much more powerful (around 100 mw)

and it was impossible to watch nothing in the screen of the computer.)

The result was unsatisfactory because it was impossible to watch some kind of

interference of the beams on the screen of the laptop.

74

4.5- Results Obtained

Once resolved the problem of the polarization and the brightness seemed that it was

going to be possible to take a hologram.

It was unsatisfactory and the results were the same.

To check if it was everything perfect, the metal beam was changed by a mirror that

reflected the Illumination Beam into Beam Splitter (19).

Now, the results were different. In the hologram there were the typical fringes but not

well defined and some tiny diagonal fringes.

These tiny diagonal fringes show that there is some kind of interferometry, so it means

that the two waves are arriving almost well.

However, the other fringes which were due to Reference Beam were really big and

really dirty.

So, it seems that the beams arrive with a lot of noise.

Another aspect that it is important to result is that when you knocked very weakly the

table of the lab the diagonal fringes disappeared what show that if something perturbs

the environment of the lab the interferometry between the two beams is over.

4.6- Applicable Improvements

Now, knowing the problems for taking a hologram, there is something that can be done

to improve it.

One of these things is clean perfectly each component and check very quietly that all of

them are right.

And the second thing that will be good to do it is changing whole components one by

one and realized which one is the focus of the problem.

75

5- Discussion and Interpretation

If it is compared the obtained results of the point B by Finite Element Analysis and by

Dänliker Algorithms is easy to guess that these results are bit different.

These results can be observed in the appendices I and II respectively. In the appendix I

the point with its strains and stresses is shady by orange color.

The reason because both results are a bit different is due to the size of the mesh (1 mm)

is so huge; the program (Abaqus) could not process a smaller mesh.

Like Finite Difference is an approximation of the derivate’s meaning and it is a limit

going to infinite; it is usual that an increment of 1 mm was so long.

Another important reason is the approximation of the turned angle by the metal beam to

zero [10]. It can variety a little the final result, but it does not have as importance as the

last justification.

Nevertheless it is important to remark that the stresses obtained by FEM and Dänliker

algorithms are really different because the strains are bit different, and this makes that

the difference for the stress will be huger than the difference for the strain.

And finally it is important to say, that if you compare a result of Finite Element Method

with one of Finite Difference both of them are approximations of the result; so is

common that they were not the same.

Other aspects about Dänliker algorithms have been discussed at the end of that chapter.

The result of the holography set up have been explained in the chapter four previously.

76

6- Conclusions

The global result of the project is successful.

The reason of this is because a lot of different aspects like FEM, Dänliker algorithms

and Digital Holography have been mixed to obtain some result and compare them. With

this comparison we can know if the Dänliker algorithms mixed with Digital Holography

are enough good for trusting them.

Nevertheless, this project is a brief study of something that can have a big impact in the

industry because it is one way for obtaining the behavior inside an object without

damaging it.

However, the objective of this project is not to analyze the different results if not only

compare them. It means, comparing the results coming from Finite Element Method and

the results coming from Digital Holography and Dänliker algorithms; and see that they

are more or less the same.

So, of my point of view, this is a field that it is necessary to study deeper because it is a

way to achieve good tests for some jobs that need a high level of security like can be

aircraft industry or something similar to it.

77

7- Recommendations and Suggestions for Future Work

For future work is important taking into account approximate deeper the Dänliker

algorithms with increments as short as possible.

Nevertheless, it will be important to get a mesh in FEM much thinner, like 0,1 mm. The

result of this will be more exactly and then the comparison between FEM and Dänliker

will be better.

With respect to the holography set up, the task will be to check component by

component and know which one is wrong and finally clean all of them.

Later on, the mission will be recording a hologram and if it is successful, measuring the

displacement of the metal beam with special software.

This part can be the hardest, because it can take an interval of time undefined. So, it is

important doing this improvement step by step and always noting down all different

aspects.

78

8- References

Books:

1. W. Jueptner, U. Schnars, 2005, “Digital Holography”, Springer

2. Edgar Conley, Jean Robillard, 2000, “Industrial Applications for Optical Data

Processing and Holography”, CRC Press

3. D. J. Wave, 1984, “Matrix and finite element displacement analysis of

structures”, Oxford University Press

4. B. J. Bell, M. J. Smith, 1980, “Theory of Structures”, George Godwin

5. Gross Hauger, Scknell Schröder, 2007, “Technische Mechanik 2, Elastostatik”,

Springer

6. R. Dänliker, 1979, “Extrapolation of Strain and Stress From Holographically

Measured Surface Displacement”, Journal of Applied Mechanics, Vol. 46, No. 3

7. Association of University Radiation Protection Officers, 2006, “Guidance on the

safe use of lasers in education and research”, AURPo Guidance Note No. 7

Web site:

8. http://www.hologramsuppliers.com/digital-hologram.html

9. http://www.holography.ru

10. http://nacc.upc.es/nacc-libro/node46.html

11. http://www.eng.uc.edu/~gbeaucag/Classes/Processing/Chapter2html/Chapter2.ht

ml

79

9- Appendices

9.1- Appendix I. Results of the Nodes in the Finite Element

analysis.

In this appendix is included the table of the displacements of each selected node in the

Finite Element analysis. There are nine tables, one for the superficial surface and one

per each section calculated.

80

9.1.1- Superficial Face

X Y Z U1 U2 U3 225,002 8 0 1,82E‐03 ‐8,07E‐07 4,38E‐02 226,002 8 0 1,82E‐03 ‐7,67E‐07 4,41E‐02 227,002 8 0 1,82E‐03 ‐7,27E‐07 4,44E‐02 228,002 8 0 1,82E‐03 ‐6,87E‐07 4,47E‐02 229,002 8 0 1,82E‐03 ‐6,47E‐07 4,50E‐02 230,002 8 0 1,82E‐03 ‐6,07E‐07 4,53E‐02 225,002 7 0 1,82E‐03 ‐4,04E‐07 4,38E‐02 226,002 7 0 1,82E‐03 ‐3,84E‐07 4,41E‐02 227,002 7 0 1,82E‐03 ‐3,64E‐07 4,44E‐02 228,002 7 0 1,82E‐03 ‐3,44E‐07 4,47E‐02 229,002 7 0 1,82E‐03 ‐3,24E‐07 4,50E‐02 230,002 7 0 1,82E‐03 ‐3,04E‐07 4,53E‐02 225,002 6 0 1,82E‐03 ‐5,12E‐12 4,38E‐02 226,002 6 0 1,82E‐03 ‐5,14E‐12 4,41E‐02 227,002 6 0 1,82E‐03 ‐5,17E‐12 4,44E‐02 228,002 6 0 1,82E‐03 ‐5,20E‐12 4,47E‐02 229,002 6 0 1,82E‐03 ‐5,23E‐12 4,50E‐02 230,002 6 0 1,82E‐03 ‐5,26E‐12 4,53E‐02 225,002 5 0 1,82E‐03 4,04E‐07 4,38E‐02 226,002 5 0 1,82E‐03 3,84E‐07 4,41E‐02 227,002 5 0 1,82E‐03 3,64E‐07 4,44E‐02 228,002 5 0 1,82E‐03 3,44E‐07 4,47E‐02 229,002 5 0 1,82E‐03 3,24E‐07 4,50E‐02 230,002 5 0 1,82E‐03 3,04E‐07 4,53E‐02 225,002 4 0 1,82E‐03 8,07E‐07 4,38E‐02 226,002 4 0 1,82E‐03 7,67E‐07 4,41E‐02 227,002 4 0 1,82E‐03 7,27E‐07 4,44E‐02 228,002 4 0 1,82E‐03 6,87E‐07 4,47E‐02 229,002 4 0 1,82E‐03 6,47E‐07 4,50E‐02 230,002 4 0 1,82E‐03 6,07E‐07 4,53E‐02 225,002 3 0 1,82E‐03 1,21E‐06 4,38E‐02 226,002 3 0 1,82E‐03 1,15E‐06 4,41E‐02 227,002 3 0 1,82E‐03 1,09E‐06 4,44E‐02 228,002 3 0 1,82E‐03 1,03E‐06 4,47E‐02 229,002 3 0 1,82E‐03 9,71E‐07 4,50E‐02 230,002 3 0 1,82E‐03 9,11E‐07 4,53E‐02

Table 9.1.1.1 Displacement of the Nodes at the superficial face

81

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 3,26E‐08 7,08E‐08 1,29E‐11 226,002 8 0 1,16E‐06 ‐3,84E‐07 ‐3,83E‐07 3,25E‐08 7,07E‐08 2,31E‐11 227,002 8 0 1,10E‐06 ‐3,63E‐07 ‐3,63E‐07 3,23E‐08 7,06E‐08 3,92E‐11 228,002 8 0 1,04E‐06 ‐3,43E‐07 ‐3,43E‐07 3,20E‐08 7,04E‐08 6,33E‐11 229,002 8 0 9,79E‐07 ‐3,23E‐07 ‐3,23E‐07 3,17E‐08 6,99E‐08 9,81E‐11 230,002 8 0 9,18E‐07 ‐3,04E‐07 ‐3,03E‐07 3,15E‐08 6,92E‐08 1,46E‐10 225,002 7 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 1,72E‐08 6,94E‐08 8,96E‐12 226,002 7 0 1,16E‐06 ‐3,84E‐07 ‐3,83E‐07 1,71E‐08 6,94E‐08 1,55E‐11 227,002 7 0 1,10E‐06 ‐3,64E‐07 ‐3,63E‐07 1,70E‐08 6,92E‐08 2,55E‐11 228,002 7 0 1,04E‐06 ‐3,44E‐07 ‐3,43E‐07 1,69E‐08 6,90E‐08 4,03E‐11 229,002 7 0 9,79E‐07 ‐3,24E‐07 ‐3,23E‐07 1,68E‐08 6,85E‐08 6,13E‐11 230,002 7 0 9,19E‐07 ‐3,04E‐07 ‐3,03E‐07 1,66E‐08 6,78E‐08 8,95E‐11 225,002 6 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 ‐2,22E‐16 6,90E‐08 ‐1,39E‐17 226,002 6 0 1,16E‐06 ‐3,84E‐07 ‐3,83E‐07 2,22E‐16 6,89E‐08 0 227,002 6 0 1,10E‐06 ‐3,64E‐07 ‐3,63E‐07 ‐2,22E‐16 6,88E‐08 0 228,002 6 0 1,04E‐06 ‐3,44E‐07 ‐3,43E‐07 6,66E‐16 6,85E‐08 ‐4,16E‐17 229,002 6 0 9,79E‐07 ‐3,24E‐07 ‐3,23E‐07 2,22E‐16 6,81E‐08 2,78E‐17 230,002 6 0 9,19E‐07 ‐3,04E‐07 ‐3,03E‐07 8,88E‐16 6,73E‐08 ‐4,16E‐17 225,002 5 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 ‐1,72E‐08 6,94E‐08 ‐8,96E‐12 226,002 5 0 1,16E‐06 ‐3,84E‐07 ‐3,83E‐07 ‐1,71E‐08 6,94E‐08 ‐1,55E‐11 227,002 5 0 1,10E‐06 ‐3,64E‐07 ‐3,63E‐07 ‐1,70E‐08 6,92E‐08 ‐2,55E‐11 228,002 5 0 1,04E‐06 ‐3,44E‐07 ‐3,43E‐07 ‐1,69E‐08 6,90E‐08 ‐4,03E‐11 229,002 5 0 9,79E‐07 ‐3,24E‐07 ‐3,23E‐07 ‐1,68E‐08 6,85E‐08 ‐6,13E‐11 230,002 5 0 9,19E‐07 ‐3,04E‐07 ‐3,03E‐07 ‐1,66E‐08 6,78E‐08 ‐8,95E‐11 225,002 4 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 ‐3,26E‐08 7,08E‐08 ‐1,29E‐11 226,002 4 0 1,16E‐06 ‐3,84E‐07 ‐3,83E‐07 ‐3,25E‐08 7,07E‐08 ‐2,31E‐11 227,002 4 0 1,10E‐06 ‐3,63E‐07 ‐3,63E‐07 ‐3,23E‐08 7,06E‐08 ‐3,92E‐11 228,002 4 0 1,04E‐06 ‐3,43E‐07 ‐3,43E‐07 ‐3,20E‐08 7,04E‐08 ‐6,33E‐11 229,002 4 0 9,79E‐07 ‐3,23E‐07 ‐3,23E‐07 ‐3,17E‐08 6,99E‐08 ‐9,81E‐11 230,002 4 0 9,18E‐07 ‐3,04E‐07 ‐3,03E‐07 ‐3,15E‐08 6,92E‐08 ‐1,46E‐10 225,002 3 0 1,22E‐06 ‐4,04E‐07 ‐4,04E‐07 ‐4,39E‐08 7,33E‐08 ‐8,96E‐12 226,002 3 0 1,16E‐06 ‐3,83E‐07 ‐3,83E‐07 ‐4,38E‐08 7,32E‐08 ‐1,80E‐11 227,002 3 0 1,10E‐06 ‐3,63E‐07 ‐3,63E‐07 ‐4,36E‐08 7,31E‐08 ‐3,28E‐11 228,002 3 0 1,04E‐06 ‐3,43E‐07 ‐3,43E‐07 ‐4,32E‐08 7,29E‐08 ‐5,61E‐11 229,002 3 0 9,78E‐07 ‐3,23E‐07 ‐3,23E‐07 ‐4,28E‐08 7,25E‐08 ‐9,08E‐11 230,002 3 0 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 ‐4,24E‐08 7,18E‐08 ‐1,40E‐10

Table 9.1.1.2 Strain of the Nodes at the superficial face

82

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 0 8,44E‐02 ‐6,27E‐06 5,72E‐07 8,45E‐04 1,84E‐03 3,35E‐07226,002 8 0 8,02E‐02 ‐1,29E‐05 1,59E‐06 8,42E‐04 1,83E‐03 6,00E‐07227,002 8 0 7,59E‐02 ‐2,12E‐05 3,26E‐06 8,37E‐04 1,83E‐03 1,02E‐06228,002 8 0 7,17E‐02 ‐2,99E‐05 5,87E‐06 8,31E‐04 1,83E‐03 1,64E‐06229,002 8 0 6,75E‐02 ‐3,58E‐05 9,79E‐06 8,23E‐04 1,81E‐03 2,54E‐06230,002 8 0 6,34E‐02 ‐3,26E‐05 1,55E‐05 8,16E‐04 1,79E‐03 3,78E‐06225,002 7 0 8,44E‐02 ‐7,36E‐06 6,28E‐07 4,46E‐04 1,80E‐03 2,32E‐07226,002 7 0 8,02E‐02 ‐1,49E‐05 1,65E‐06 4,45E‐04 1,80E‐03 4,01E‐07227,002 7 0 7,59E‐02 ‐2,40E‐05 3,30E‐06 4,42E‐04 1,80E‐03 6,61E‐07228,002 7 0 7,17E‐02 ‐3,30E‐05 5,86E‐06 4,39E‐04 1,79E‐03 1,05E‐06229,002 7 0 6,75E‐02 ‐3,79E‐05 9,65E‐06 4,35E‐04 1,78E‐03 1,59E‐06230,002 7 0 6,34E‐02 ‐3,10E‐05 1,51E‐05 4,32E‐04 1,76E‐03 2,32E‐06225,002 6 0 8,44E‐02 ‐7,73E‐06 6,46E‐07 1,46E‐11 1,79E‐03 0226,002 6 0 8,02E‐02 ‐1,55E‐05 1,67E‐06 1,09E‐11 1,79E‐03 0227,002 6 0 7,59E‐02 ‐2,49E‐05 3,31E‐06 3,64E‐12 1,78E‐03 0228,002 6 0 7,17E‐02 ‐3,39E‐05 5,85E‐06 0 1,78E‐03 0229,002 6 0 6,75E‐02 ‐3,84E‐05 9,58E‐06 0 1,77E‐03 9,09E‐13230,002 6 0 6,34E‐02 ‐3,00E‐05 1,49E‐05 ‐7,28E‐12 1,75E‐03 ‐9,09E‐13225,002 5 0 8,44E‐02 ‐7,36E‐06 6,28E‐07 ‐4,46E‐04 1,80E‐03 ‐2,32E‐07226,002 5 0 8,02E‐02 ‐1,49E‐05 1,65E‐06 ‐4,45E‐04 1,80E‐03 ‐4,01E‐07227,002 5 0 7,59E‐02 ‐2,40E‐05 3,30E‐06 ‐4,42E‐04 1,80E‐03 ‐6,61E‐07228,002 5 0 7,17E‐02 ‐3,30E‐05 5,86E‐06 ‐4,39E‐04 1,79E‐03 ‐1,05E‐06229,002 5 0 6,75E‐02 ‐3,79E‐05 9,65E‐06 ‐4,35E‐04 1,78E‐03 ‐1,59E‐06230,002 5 0 6,34E‐02 ‐3,10E‐05 1,51E‐05 ‐4,32E‐04 1,76E‐03 ‐2,32E‐06225,002 4 0 8,44E‐02 ‐6,27E‐06 5,72E‐07 ‐8,45E‐04 1,84E‐03 ‐3,35E‐07226,002 4 0 8,02E‐02 ‐1,29E‐05 1,59E‐06 ‐8,42E‐04 1,83E‐03 ‐6,00E‐07227,002 4 0 7,59E‐02 ‐2,12E‐05 3,26E‐06 ‐8,37E‐04 1,83E‐03 ‐1,02E‐06228,002 4 0 7,17E‐02 ‐2,99E‐05 5,87E‐06 ‐8,31E‐04 1,83E‐03 ‐1,64E‐06229,002 4 0 6,75E‐02 ‐3,58E‐05 9,79E‐06 ‐8,23E‐04 1,81E‐03 ‐2,54E‐06230,002 4 0 6,34E‐02 ‐3,26E‐05 1,55E‐05 ‐8,16E‐04 1,79E‐03 ‐3,78E‐06225,002 3 0 8,44E‐02 ‐4,58E‐06 4,70E‐07 ‐1,14E‐03 1,90E‐03 ‐2,32E‐07226,002 3 0 8,02E‐02 ‐9,80E‐06 1,46E‐06 ‐1,14E‐03 1,90E‐03 ‐4,67E‐07227,002 3 0 7,59E‐02 ‐1,65E‐05 3,14E‐06 ‐1,13E‐03 1,90E‐03 ‐8,52E‐07228,002 3 0 7,17E‐02 ‐2,40E‐05 5,80E‐06 ‐1,12E‐03 1,89E‐03 ‐1,46E‐06229,002 3 0 6,75E‐02 ‐3,03E‐05 9,87E‐06 ‐1,11E‐03 1,88E‐03 ‐2,36E‐06230,002 3 0 6,33E‐02 ‐3,11E‐05 1,59E‐05 ‐1,10E‐03 1,86E‐03 ‐3,63E‐06

Table 9.1.1.3 Stress of the Nodes at the superficial face

83

9.1.2- Section at 1 mm below the top surface

X Y Z U1 U2 U3 225,002 8 1 1,52E‐03 ‐6,73E‐07 4,38E‐02 226,002 8 1 1,52E‐03 ‐6,40E‐07 4,41E‐02 227,002 8 1 1,52E‐03 ‐6,06E‐07 4,44E‐02 228,002 8 1 1,52E‐03 ‐5,73E‐07 4,47E‐02 229,002 8 1 1,52E‐03 ‐5,40E‐07 4,50E‐02 230,002 8 1 1,53E‐03 5,65E‐07 4,99E‐02 225,002 7 1 1,52E‐03 ‐3,37E‐07 4,38E‐02 226,002 7 1 1,52E‐03 ‐3,20E‐07 4,41E‐02 227,002 7 1 1,52E‐03 ‐3,03E‐07 4,44E‐02 228,002 7 1 1,52E‐03 ‐2,87E‐07 4,47E‐02 229,002 7 1 1,52E‐03 ‐2,70E‐07 4,50E‐02 230,002 7 1 1,53E‐03 2,77E‐07 4,99E‐02 225,002 6 1 1,52E‐03 ‐5,12E‐12 4,38E‐02 226,002 6 1 1,52E‐03 ‐5,14E‐12 4,41E‐02 227,002 6 1 1,52E‐03 ‐5,17E‐12 4,44E‐02 228,002 6 1 1,52E‐03 ‐5,20E‐12 4,47E‐02 229,002 6 1 1,52E‐03 ‐5,23E‐12 4,50E‐02 230,002 6 1 1,53E‐03 ‐5,67E‐12 4,99E‐02 225,002 5 1 1,52E‐03 3,36E‐07 4,38E‐02 226,002 5 1 1,52E‐03 3,20E‐07 4,41E‐02 227,002 5 1 1,52E‐03 3,03E‐07 4,44E‐02 228,002 5 1 1,52E‐03 2,87E‐07 4,47E‐02 229,002 5 1 1,52E‐03 2,70E‐07 4,50E‐02 230,002 5 1 1,53E‐03 ‐2,77E‐07 4,99E‐02 225,002 4 1 1,52E‐03 6,73E‐07 4,38E‐02 226,002 4 1 1,52E‐03 6,40E‐07 4,41E‐02 227,002 4 1 1,52E‐03 6,06E‐07 4,44E‐02 228,002 4 1 1,52E‐03 5,73E‐07 4,47E‐02 229,002 4 1 1,52E‐03 5,40E‐07 4,50E‐02 230,002 4 1 1,53E‐03 ‐5,65E‐07 4,99E‐02 225,002 3 1 1,52E‐03 1,01E‐06 4,38E‐02 226,002 3 1 1,52E‐03 9,59E‐07 4,41E‐02 227,002 3 1 1,52E‐03 9,09E‐07 4,44E‐02 228,002 3 1 1,52E‐03 8,59E‐07 4,47E‐02 229,002 3 1 1,52E‐03 8,10E‐07 4,50E‐02 230,002 3 1 1,53E‐03 ‐7,91E‐07 4,99E‐02

Table 9.1.2.1 Displacement of the Nodes at 1 mm

84

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 1 1,02E‐06 ‐3,36E‐07 ‐3,36E‐07 2,09E‐08 1,34E‐07 2,82E‐11 226,002 8 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 2,08E‐08 1,34E‐07 4,77E‐11 227,002 8 1 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 2,06E‐08 1,34E‐07 7,79E‐11 228,002 8 1 8,67E‐07 ‐2,86E‐07 ‐2,86E‐07 2,03E‐08 1,33E‐07 1,23E‐10 229,002 8 1 8,16E‐07 ‐2,70E‐07 ‐2,69E‐07 1,99E‐08 1,32E‐07 1,89E‐10 230,002 8 1 ‐6,34E‐08 2,67E‐07 ‐1,14E‐07 8,79E‐08 ‐8,96E‐08 1,22E‐07 225,002 7 1 1,02E‐06 ‐3,36E‐07 ‐3,36E‐07 1,13E‐08 1,32E‐07 1,82E‐11 226,002 7 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 1,12E‐08 1,31E‐07 3,02E‐11 227,002 7 1 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 1,11E‐08 1,31E‐07 4,87E‐11 228,002 7 1 8,67E‐07 ‐2,87E‐07 ‐2,86E‐07 1,10E‐08 1,31E‐07 7,59E‐11 229,002 7 1 8,17E‐07 ‐2,70E‐07 ‐2,69E‐07 1,08E‐08 1,30E‐07 1,15E‐10 230,002 7 1 ‐8,29E‐08 2,81E‐07 ‐9,73E‐08 4,65E‐08 4,23E‐09 1,23E‐08 225,002 6 1 1,02E‐06 ‐3,37E‐07 ‐3,36E‐07 5,55E‐17 1,31E‐07 0 226,002 6 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 ‐5,55E‐16 1,31E‐07 6,94E‐18 227,002 6 1 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 1,11E‐16 1,30E‐07 ‐6,94E‐18 228,002 6 1 8,67E‐07 ‐2,87E‐07 ‐2,86E‐07 1,11E‐16 1,30E‐07 1,39E‐17 229,002 6 1 8,17E‐07 ‐2,70E‐07 ‐2,69E‐07 2,22E‐16 1,29E‐07 ‐1,39E‐17 230,002 6 1 ‐7,83E‐08 2,75E‐07 ‐9,93E‐08 4,44E‐16 ‐1,61E‐08 2,22E‐16 225,002 5 1 1,02E‐06 ‐3,36E‐07 ‐3,36E‐07 ‐1,13E‐08 1,32E‐07 ‐1,82E‐11 226,002 5 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 ‐1,12E‐08 1,31E‐07 ‐3,02E‐11 227,002 5 1 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 ‐1,11E‐08 1,31E‐07 ‐4,87E‐11 228,002 5 1 8,67E‐07 ‐2,87E‐07 ‐2,86E‐07 ‐1,10E‐08 1,31E‐07 ‐7,59E‐11 229,002 5 1 8,17E‐07 ‐2,70E‐07 ‐2,69E‐07 ‐1,08E‐08 1,30E‐07 ‐1,15E‐10 230,002 5 1 ‐8,29E‐08 2,81E‐07 ‐9,73E‐08 ‐4,65E‐08 4,23E‐09 ‐1,23E‐08 225,002 4 1 1,02E‐06 ‐3,36E‐07 ‐3,36E‐07 ‐2,09E‐08 1,34E‐07 ‐2,82E‐11 226,002 4 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 ‐2,08E‐08 1,34E‐07 ‐4,77E‐11 227,002 4 1 9,18E‐07 ‐3,03E‐07 ‐3,03E‐07 ‐2,06E‐08 1,34E‐07 ‐7,79E‐11 228,002 4 1 8,67E‐07 ‐2,86E‐07 ‐2,86E‐07 ‐2,03E‐08 1,33E‐07 ‐1,23E‐10 229,002 4 1 8,16E‐07 ‐2,70E‐07 ‐2,69E‐07 ‐1,99E‐08 1,32E‐07 ‐1,89E‐10 230,002 4 1 ‐6,34E‐08 2,67E‐07 ‐1,14E‐07 ‐8,79E‐08 ‐8,96E‐08 ‐1,22E‐07 225,002 3 1 1,02E‐06 ‐3,36E‐07 ‐3,36E‐07 ‐2,69E‐08 1,39E‐07 ‐2,54E‐11 226,002 3 1 9,68E‐07 ‐3,20E‐07 ‐3,19E‐07 ‐2,67E‐08 1,39E‐07 ‐4,44E‐11 227,002 3 1 9,17E‐07 ‐3,03E‐07 ‐3,03E‐07 ‐2,65E‐08 1,39E‐07 ‐7,49E‐11 228,002 3 1 8,67E‐07 ‐2,86E‐07 ‐2,86E‐07 ‐2,61E‐08 1,38E‐07 ‐1,22E‐10 229,002 3 1 8,16E‐07 ‐2,70E‐07 ‐2,69E‐07 ‐2,56E‐08 1,37E‐07 ‐1,92E‐10 230,002 3 1 ‐1,15E‐07 2,97E‐07 ‐1,05E‐07 ‐1,54E‐07 2,26E‐07 ‐8,57E‐08

Table 9.1.2.2 Strain of the Nodes at 1 mm

85

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 1 7,03E‐02 ‐5,00E‐06 1,37E‐06 5,42E‐04 3,48E‐03 7,33E‐07 226,002 8 1 6,68E‐02 ‐1,00E‐05 3,64E‐06 5,39E‐04 3,48E‐03 1,24E‐06 227,002 8 1 6,33E‐02 ‐1,61E‐05 7,36E‐06 5,34E‐04 3,47E‐03 2,02E‐06 228,002 8 1 5,98E‐02 ‐2,21E‐05 1,32E‐05 5,26E‐04 3,46E‐03 3,20E‐06 229,002 8 1 5,63E‐02 ‐2,55E‐05 2,20E‐05 5,16E‐04 3,44E‐03 4,91E‐06 230,002 8 1 1,20E‐03 1,83E‐02 ‐1,42E‐03 2,28E‐03 ‐2,33E‐03 3,16E‐03 225,002 7 1 7,03E‐02 ‐6,27E‐06 1,39E‐06 2,93E‐04 3,41E‐03 4,72E‐07 226,002 7 1 6,68E‐02 ‐1,22E‐05 3,62E‐06 2,91E‐04 3,41E‐03 7,84E‐07 227,002 7 1 6,33E‐02 ‐1,94E‐05 7,27E‐06 2,88E‐04 3,40E‐03 1,26E‐06 228,002 7 1 5,98E‐02 ‐2,63E‐05 1,30E‐05 2,84E‐04 3,39E‐03 1,97E‐06 229,002 7 1 5,64E‐02 ‐2,98E‐05 2,16E‐05 2,79E‐04 3,37E‐03 2,98E‐06 230,002 7 1 7,90E‐04 1,97E‐02 4,41E‐05 1,21E‐03 1,10E‐04 3,18E‐04 225,002 6 1 7,03E‐02 ‐6,74E‐06 1,39E‐06 ‐5,46E‐12 3,39E‐03 ‐6,82E‐13 226,002 6 1 6,68E‐02 ‐1,30E‐05 3,61E‐06 7,28E‐12 3,39E‐03 0 227,002 6 1 6,33E‐02 ‐2,05E‐05 7,24E‐06 ‐1,09E‐11 3,38E‐03 2,27E‐13 228,002 6 1 5,98E‐02 ‐2,77E‐05 1,29E‐05 3,64E‐12 3,37E‐03 0 229,002 6 1 5,64E‐02 ‐3,12E‐05 2,14E‐05 ‐4,55E‐12 3,34E‐03 ‐4,55E‐13 230,002 6 1 8,61E‐04 1,92E‐02 ‐2,30E‐04 0 ‐4,19E‐04 0 225,002 5 1 7,03E‐02 ‐6,27E‐06 1,39E‐06 ‐2,93E‐04 3,41E‐03 ‐4,72E‐07 226,002 5 1 6,68E‐02 ‐1,22E‐05 3,62E‐06 ‐2,91E‐04 3,41E‐03 ‐7,84E‐07 227,002 5 1 6,33E‐02 ‐1,94E‐05 7,27E‐06 ‐2,88E‐04 3,40E‐03 ‐1,26E‐06 228,002 5 1 5,98E‐02 ‐2,63E‐05 1,30E‐05 ‐2,84E‐04 3,39E‐03 ‐1,97E‐06 229,002 5 1 5,64E‐02 ‐2,98E‐05 2,16E‐05 ‐2,79E‐04 3,37E‐03 ‐2,98E‐06 230,002 5 1 7,90E‐04 1,97E‐02 4,41E‐05 ‐1,21E‐03 1,10E‐04 ‐3,18E‐04 225,002 4 1 7,03E‐02 ‐5,00E‐06 1,37E‐06 ‐5,42E‐04 3,48E‐03 ‐7,33E‐07 226,002 4 1 6,68E‐02 ‐1,00E‐05 3,64E‐06 ‐5,39E‐04 3,48E‐03 ‐1,24E‐06 227,002 4 1 6,33E‐02 ‐1,61E‐05 7,36E‐06 ‐5,34E‐04 3,47E‐03 ‐2,02E‐06 228,002 4 1 5,98E‐02 ‐2,21E‐05 1,32E‐05 ‐5,26E‐04 3,46E‐03 ‐3,20E‐06 229,002 4 1 5,63E‐02 ‐2,55E‐05 2,20E‐05 ‐5,16E‐04 3,44E‐03 ‐4,91E‐06 230,002 4 1 1,20E‐03 1,83E‐02 ‐1,42E‐03 ‐2,28E‐03 ‐2,33E‐03 ‐3,16E‐03 225,002 3 1 7,03E‐02 ‐3,29E‐06 1,30E‐06 ‐6,97E‐04 3,60E‐03 ‐6,59E‐07 226,002 3 1 6,68E‐02 ‐6,83E‐06 3,59E‐06 ‐6,93E‐04 3,60E‐03 ‐1,15E‐06 227,002 3 1 6,33E‐02 ‐1,12E‐05 7,40E‐06 ‐6,86E‐04 3,59E‐03 ‐1,94E‐06 228,002 3 1 5,98E‐02 ‐1,56E‐05 1,34E‐05 ‐6,77E‐04 3,58E‐03 ‐3,16E‐06 229,002 3 1 5,63E‐02 ‐1,83E‐05 2,25E‐05 ‐6,64E‐04 3,56E‐03 ‐4,99E‐06 230,002 3 1 ‐2,11E‐03 1,93E‐02 ‐1,55E‐03 ‐3,99E‐03 5,87E‐03 ‐2,22E‐03

Table 9.1.2.3 Stress of the Nodes at 1 mm

86


X Y Z U1 U2 U3 225,002 8 2 1,21E‐03 ‐5,39E‐07 4,38E‐02 226,002 8 2 1,21E‐03 ‐5,12E‐07 4,41E‐02 227,002 8 2 1,21E‐03 ‐4,85E‐07 4,44E‐02 228,002 8 2 1,21E‐03 ‐4,59E‐07 4,47E‐02 229,002 8 2 1,22E‐03 ‐4,33E‐07 4,50E‐02 230,002 8 2 1,22E‐03 ‐3,07E‐08 4,99E‐02 225,002 7 2 1,21E‐03 ‐2,69E‐07 4,38E‐02 226,002 7 2 1,21E‐03 ‐2,56E‐07 4,41E‐02 227,002 7 2 1,21E‐03 ‐2,43E‐07 4,44E‐02 228,002 7 2 1,21E‐03 ‐2,30E‐07 4,47E‐02 229,002 7 2 1,22E‐03 ‐2,16E‐07 4,50E‐02 230,002 7 2 1,22E‐03 2,06E‐08 4,99E‐02 225,002 6 2 1,21E‐03 ‐5,12E‐12 4,38E‐02 226,002 6 2 1,21E‐03 ‐5,14E‐12 4,41E‐02 227,002 6 2 1,21E‐03 ‐5,17E‐12 4,44E‐02 228,002 6 2 1,21E‐03 ‐5,20E‐12 4,47E‐02 229,002 6 2 1,22E‐03 ‐5,23E‐12 4,50E‐02 230,002 6 2 1,22E‐03 ‐5,67E‐12 4,99E‐02 225,002 5 2 1,21E‐03 2,69E‐07 4,38E‐02 226,002 5 2 1,21E‐03 2,56E‐07 4,41E‐02 227,002 5 2 1,21E‐03 2,43E‐07 4,44E‐02 228,002 5 2 1,21E‐03 2,30E‐07 4,47E‐02 229,002 5 2 1,22E‐03 2,16E‐07 4,50E‐02 230,002 5 2 1,22E‐03 ‐2,06E‐08 4,99E‐02 225,002 4 2 1,21E‐03 5,39E‐07 4,38E‐02 226,002 4 2 1,21E‐03 5,12E‐07 4,41E‐02 227,002 4 2 1,21E‐03 4,85E‐07 4,44E‐02 228,002 4 2 1,21E‐03 4,59E‐07 4,47E‐02 229,002 4 2 1,22E‐03 4,33E‐07 4,50E‐02 230,002 4 2 1,22E‐03 3,07E‐08 4,99E‐02 225,002 3 2 1,21E‐03 8,08E‐07 4,38E‐02 226,002 3 2 1,21E‐03 7,68E‐07 4,41E‐02 227,002 3 2 1,21E‐03 7,28E‐07 4,44E‐02 228,002 3 2 1,21E‐03 6,88E‐07 4,47E‐02 229,002 3 2 1,22E‐03 6,49E‐07 4,50E‐02 230,002 3 2 1,22E‐03 1,40E‐07 4,99E‐02


87

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 1,28E‐08 2,49E‐07 6,53E‐11 226,002 8 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 1,27E‐08 2,49E‐07 1,05E‐10 227,002 8 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 1,24E‐08 2,48E‐07 1,66E‐10 228,002 8 2 6,94E‐07 ‐2,29E‐07 ‐2,28E‐07 1,21E‐08 2,48E‐07 2,57E‐10 229,002 8 2 6,53E‐07 ‐2,16E‐07 ‐2,15E‐07 1,17E‐08 2,46E‐07 3,88E‐10 230,002 8 2 4,90E‐08 ‐9,23E‐08 ‐2,61E‐08 ‐4,81E‐08 6,31E‐08 2,60E‐07 225,002 7 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 7,07E‐09 2,44E‐07 3,94E‐11 226,002 7 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 6,99E‐09 2,44E‐07 6,31E‐11 227,002 7 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 6,87E‐09 2,43E‐07 9,88E‐11 228,002 7 2 6,94E‐07 ‐2,30E‐07 ‐2,28E‐07 6,70E‐09 2,43E‐07 1,52E‐10 229,002 7 2 6,54E‐07 ‐2,16E‐07 ‐2,15E‐07 6,47E‐09 2,41E‐07 2,27E‐10 230,002 7 2 1,61E‐08 ‐1,04E‐08 ‐2,47E‐08 ‐2,89E‐09 ‐6,74E‐09 1,06E‐07 225,002 6 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 2,78E‐17 2,43E‐07 ‐2,78E‐17 226,002 6 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 5,55E‐17 2,42E‐07 ‐1,39E‐17 227,002 6 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 ‐8,33E‐17 2,42E‐07 6,94E‐18 228,002 6 2 6,94E‐07 ‐2,30E‐07 ‐2,28E‐07 1,94E‐16 2,41E‐07 ‐6,94E‐18 229,002 6 2 6,54E‐07 ‐2,16E‐07 ‐2,15E‐07 ‐8,33E‐17 2,39E‐07 ‐2,78E‐17 230,002 6 2 3,00E‐09 2,12E‐08 ‐2,58E‐08 ‐1,39E‐16 1,42E‐08 8,88E‐16 225,002 5 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 ‐7,07E‐09 2,44E‐07 ‐3,94E‐11 226,002 5 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 ‐6,99E‐09 2,44E‐07 ‐6,31E‐11 227,002 5 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 ‐6,87E‐09 2,43E‐07 ‐9,88E‐11 228,002 5 2 6,94E‐07 ‐2,30E‐07 ‐2,28E‐07 ‐6,70E‐09 2,43E‐07 ‐1,52E‐10 229,002 5 2 6,54E‐07 ‐2,16E‐07 ‐2,15E‐07 ‐6,47E‐09 2,41E‐07 ‐2,27E‐10 230,002 5 2 1,61E‐08 ‐1,04E‐08 ‐2,47E‐08 2,89E‐09 ‐6,74E‐09 ‐1,06E‐07 225,002 4 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 ‐1,28E‐08 2,49E‐07 ‐6,53E‐11 226,002 4 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 ‐1,27E‐08 2,49E‐07 ‐1,05E‐10 227,002 4 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 ‐1,24E‐08 2,48E‐07 ‐1,66E‐10 228,002 4 2 6,94E‐07 ‐2,29E‐07 ‐2,28E‐07 ‐1,21E‐08 2,48E‐07 ‐2,57E‐10 229,002 4 2 6,53E‐07 ‐2,16E‐07 ‐2,15E‐07 ‐1,17E‐08 2,46E‐07 ‐3,88E‐10 230,002 4 2 4,90E‐08 ‐9,23E‐08 ‐2,61E‐08 4,81E‐08 6,31E‐08 ‐2,60E‐07 225,002 3 2 8,16E‐07 ‐2,69E‐07 ‐2,69E‐07 ‐1,57E‐08 2,58E‐07 ‐6,90E‐11 226,002 3 2 7,75E‐07 ‐2,56E‐07 ‐2,56E‐07 ‐1,56E‐08 2,58E‐07 ‐1,12E‐10 227,002 3 2 7,34E‐07 ‐2,43E‐07 ‐2,42E‐07 ‐1,53E‐08 2,57E‐07 ‐1,79E‐10 228,002 3 2 6,94E‐07 ‐2,29E‐07 ‐2,28E‐07 ‐1,49E‐08 2,56E‐07 ‐2,82E‐10 229,002 3 2 6,53E‐07 ‐2,16E‐07 ‐2,15E‐07 ‐1,44E‐08 2,55E‐07 ‐4,33E‐10 230,002 3 2 2,38E‐07 ‐5,03E‐07 ‐2,93E‐08 1,02E‐07 ‐1,22E‐07 ‐9,23E‐07


88

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 2 5,63E‐02 ‐4,04E‐06 3,86E‐06 3,32E‐04 6,46E‐03 1,69E‐06 226,002 8 2 5,35E‐02 ‐8,24E‐06 9,53E‐06 3,28E‐04 6,46E‐03 2,73E‐06 227,002 8 2 5,07E‐02 ‐1,35E‐05 1,88E‐05 3,22E‐04 6,44E‐03 4,30E‐06 228,002 8 2 4,79E‐02 ‐1,92E‐05 3,33E‐05 3,14E‐04 6,42E‐03 6,66E‐06 229,002 8 2 4,51E‐02 ‐2,37E‐05 5,50E‐05 3,03E‐04 6,38E‐03 1,01E‐05 230,002 8 2 ‐9,58E‐04 ‐8,29E‐03 ‐4,85E‐03 ‐1,25E‐03 1,64E‐03 6,75E‐03 225,002 7 2 5,63E‐02 ‐5,19E‐06 3,81E‐06 1,83E‐04 6,34E‐03 1,02E‐06 226,002 7 2 5,35E‐02 ‐1,03E‐05 9,38E‐06 1,81E‐04 6,33E‐03 1,64E‐06 227,002 7 2 5,07E‐02 ‐1,67E‐05 1,85E‐05 1,78E‐04 6,32E‐03 2,56E‐06 228,002 7 2 4,79E‐02 ‐2,35E‐05 3,27E‐05 1,74E‐04 6,29E‐03 3,93E‐06 229,002 7 2 4,51E‐02 ‐2,87E‐05 5,41E‐05 1,68E‐04 6,25E‐03 5,89E‐06 230,002 7 2 ‐1,24E‐04 ‐1,50E‐03 ‐2,24E‐03 ‐7,49E‐05 ‐1,75E‐04 2,75E‐03 225,002 6 2 5,63E‐02 ‐5,62E‐06 3,79E‐06 ‐2,73E‐12 6,30E‐03 ‐2,27E‐13 226,002 6 2 5,35E‐02 ‐1,11E‐05 9,31E‐06 ‐1,82E‐12 6,29E‐03 2,27E‐13 227,002 6 2 5,07E‐02 ‐1,79E‐05 1,84E‐05 5,46E‐12 6,27E‐03 ‐4,55E‐13 228,002 6 2 4,79E‐02 ‐2,50E‐05 3,25E‐05 ‐9,09E‐13 6,25E‐03 ‐2,27E‐13 229,002 6 2 4,51E‐02 ‐3,04E‐05 5,38E‐05 9,09E‐12 6,21E‐03 ‐6,82E‐13 230,002 6 2 7,68E‐05 1,02E‐03 ‐1,42E‐03 8,19E‐12 3,68E‐04 0 225,002 5 2 5,63E‐02 ‐5,19E‐06 3,81E‐06 ‐1,83E‐04 6,34E‐03 ‐1,02E‐06 226,002 5 2 5,35E‐02 ‐1,03E‐05 9,38E‐06 ‐1,81E‐04 6,33E‐03 ‐1,64E‐06 227,002 5 2 5,07E‐02 ‐1,67E‐05 1,85E‐05 ‐1,78E‐04 6,32E‐03 ‐2,56E‐06 228,002 5 2 4,79E‐02 ‐2,35E‐05 3,27E‐05 ‐1,74E‐04 6,29E‐03 ‐3,93E‐06 229,002 5 2 4,51E‐02 ‐2,87E‐05 5,41E‐05 ‐1,68E‐04 6,25E‐03 ‐5,89E‐06 230,002 5 2 ‐1,24E‐04 ‐1,50E‐03 ‐2,24E‐03 7,49E‐05 ‐1,75E‐04 ‐2,75E‐03 225,002 4 2 5,63E‐02 ‐4,04E‐06 3,86E‐06 ‐3,32E‐04 6,46E‐03 ‐1,69E‐06 226,002 4 2 5,35E‐02 ‐8,24E‐06 9,53E‐06 ‐3,28E‐04 6,46E‐03 ‐2,73E‐06 227,002 4 2 5,07E‐02 ‐1,35E‐05 1,88E‐05 ‐3,22E‐04 6,44E‐03 ‐4,30E‐06 228,002 4 2 4,79E‐02 ‐1,92E‐05 3,33E‐05 ‐3,14E‐04 6,42E‐03 ‐6,66E‐06 229,002 4 2 4,51E‐02 ‐2,37E‐05 5,50E‐05 ‐3,03E‐04 6,38E‐03 ‐1,01E‐05 230,002 4 2 ‐9,58E‐04 ‐8,29E‐03 ‐4,85E‐03 1,25E‐03 1,64E‐03 ‐6,75E‐03 225,002 3 2 5,63E‐02 ‐2,54E‐06 3,81E‐06 ‐4,08E‐04 6,69E‐03 ‐1,79E‐06 226,002 3 2 5,35E‐02 ‐5,43E‐06 9,62E‐06 ‐4,04E‐04 6,69E‐03 ‐2,91E‐06 227,002 3 2 5,07E‐02 ‐9,10E‐06 1,91E‐05 ‐3,97E‐04 6,67E‐03 ‐4,65E‐06 228,002 3 2 4,79E‐02 ‐1,31E‐05 3,39E‐05 ‐3,87E‐04 6,65E‐03 ‐7,31E‐06 229,002 3 2 4,51E‐02 ‐1,63E‐05 5,62E‐05 ‐3,73E‐04 6,62E‐03 ‐1,12E‐05 230,002 3 2 ‐2,48E‐03 ‐4,09E‐02 ‐1,63E‐02 2,65E‐03 ‐3,15E‐03 ‐2,39E‐02


89




90

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 7,47E‐09 3,40E‐07 1,11E‐10 226,002 8 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 7,34E‐09 3,40E‐07 1,73E‐10 227,002 8 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 7,12E‐09 3,39E‐07 2,66E‐10 228,002 8 3 5,20E‐07 ‐1,72E‐07 ‐1,71E‐07 6,80E‐09 3,38E‐07 4,03E‐10 229,002 8 3 4,90E‐07 ‐1,63E‐07 ‐1,60E‐07 6,36E‐09 3,36E‐07 6,02E‐10 230,002 8 3 1,43E‐07 ‐2,61E‐07 ‐4,26E‐08 ‐7,13E‐08 4,11E‐08 5,20E‐07 225,002 7 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 4,20E‐09 3,33E‐07 6,43E‐11 226,002 7 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 4,13E‐09 3,33E‐07 9,98E‐11 227,002 7 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 4,00E‐09 3,32E‐07 1,53E‐10 228,002 7 3 5,20E‐07 ‐1,72E‐07 ‐1,71E‐07 3,83E‐09 3,31E‐07 2,31E‐10 229,002 7 3 4,90E‐07 ‐1,63E‐07 ‐1,60E‐07 3,58E‐09 3,29E‐07 3,42E‐10 230,002 7 3 7,65E‐08 ‐1,63E‐07 ‐4,15E‐10 ‐3,40E‐08 5,33E‐08 1,90E‐07 225,002 6 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 2,78E‐17 3,31E‐07 ‐4,16E‐17 226,002 6 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 1,39E‐16 3,31E‐07 6,94E‐18 227,002 6 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 ‐9,71E‐17 3,30E‐07 ‐6,94E‐18 228,002 6 3 5,21E‐07 ‐1,72E‐07 ‐1,71E‐07 6,94E‐17 3,29E‐07 2,78E‐17 229,002 6 3 4,90E‐07 ‐1,63E‐07 ‐1,60E‐07 1,39E‐16 3,27E‐07 ‐6,25E‐17 230,002 6 3 6,32E‐08 ‐1,42E‐07 8,39E‐09 ‐6,66E‐16 3,16E‐08 3,55E‐15 225,002 5 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 ‐4,20E‐09 3,33E‐07 ‐6,43E‐11 226,002 5 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 ‐4,13E‐09 3,33E‐07 ‐9,98E‐11 227,002 5 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 ‐4,00E‐09 3,32E‐07 ‐1,53E‐10 228,002 5 3 5,20E‐07 ‐1,72E‐07 ‐1,71E‐07 ‐3,83E‐09 3,31E‐07 ‐2,31E‐10 229,002 5 3 4,90E‐07 ‐1,63E‐07 ‐1,60E‐07 ‐3,58E‐09 3,29E‐07 ‐3,42E‐10 230,002 5 3 7,65E‐08 ‐1,63E‐07 ‐4,15E‐10 3,40E‐08 5,33E‐08 ‐1,90E‐07 225,002 4 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 ‐7,47E‐09 3,40E‐07 ‐1,11E‐10 226,002 4 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 ‐7,34E‐09 3,40E‐07 ‐1,73E‐10 227,002 4 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 ‐7,12E‐09 3,39E‐07 ‐2,66E‐10 228,002 4 3 5,20E‐07 ‐1,72E‐07 ‐1,71E‐07 ‐6,80E‐09 3,38E‐07 ‐4,03E‐10 229,002 4 3 4,90E‐07 ‐1,63E‐07 ‐1,60E‐07 ‐6,36E‐09 3,36E‐07 ‐6,02E‐10 230,002 4 3 1,43E‐07 ‐2,61E‐07 ‐4,26E‐08 7,13E‐08 4,11E‐08 ‐5,20E‐07 225,002 3 3 6,12E‐07 ‐2,02E‐07 ‐2,02E‐07 ‐8,95E‐09 3,52E‐07 ‐1,27E‐10 226,002 3 3 5,81E‐07 ‐1,92E‐07 ‐1,92E‐07 ‐8,79E‐09 3,51E‐07 ‐1,98E‐10 227,002 3 3 5,51E‐07 ‐1,82E‐07 ‐1,81E‐07 ‐8,53E‐09 3,51E‐07 ‐3,08E‐10 228,002 3 3 5,20E‐07 ‐1,72E‐07 ‐1,71E‐07 ‐8,13E‐09 3,50E‐07 ‐4,71E‐10 229,002 3 3 4,89E‐07 ‐1,62E‐07 ‐1,60E‐07 ‐7,58E‐09 3,48E‐07 ‐7,12E‐10 230,002 3 3 2,51E‐07 ‐3,09E‐07 ‐2,06E‐07 2,85E‐07 2,11E‐07 ‐1,15E‐06


91

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 3 4,22E‐02 ‐3,39E‐06 7,69E‐06 1,94E‐04 8,82E‐03 2,87E‐06 226,002 8 3 4,01E‐02 ‐7,36E‐06 1,80E‐05 1,90E‐04 8,81E‐03 4,48E‐06 227,002 8 3 3,80E‐02 ‐1,27E‐05 3,47E‐05 1,85E‐04 8,80E‐03 6,89E‐06 228,002 8 3 3,59E‐02 ‐1,94E‐05 6,05E‐05 1,76E‐04 8,77E‐03 1,05E‐05 229,002 8 3 3,38E‐02 ‐2,67E‐05 9,91E‐05 1,65E‐04 8,72E‐03 1,56E‐05 230,002 8 3 ‐6,64E‐04 ‐2,16E‐02 ‐1,03E‐02 ‐1,85E‐03 1,07E‐03 1,35E‐02 225,002 7 3 4,22E‐02 ‐4,36E‐06 7,60E‐06 1,09E‐04 8,65E‐03 1,67E‐06 226,002 7 3 4,01E‐02 ‐9,20E‐06 1,77E‐05 1,07E‐04 8,64E‐03 2,59E‐06 227,002 7 3 3,80E‐02 ‐1,57E‐05 3,41E‐05 1,04E‐04 8,62E‐03 3,96E‐06 228,002 7 3 3,59E‐02 ‐2,36E‐05 5,96E‐05 9,93E‐05 8,59E‐03 5,98E‐06 229,002 7 3 3,38E‐02 ‐3,21E‐05 9,78E‐05 9,30E‐05 8,54E‐03 8,88E‐06 230,002 7 3 ‐4,02E‐04 ‐1,28E‐02 ‐4,39E‐03 ‐8,83E‐04 1,38E‐03 4,93E‐03 225,002 6 3 4,22E‐02 ‐4,73E‐06 7,55E‐06 ‐2,27E‐12 8,59E‐03 ‐4,55E‐13 226,002 6 3 4,01E‐02 ‐9,88E‐06 1,76E‐05 4,55E‐13 8,58E‐03 ‐4,55E‐13 227,002 6 3 3,80E‐02 ‐1,68E‐05 3,39E‐05 5,46E‐12 8,56E‐03 0 228,002 6 3 3,59E‐02 ‐2,52E‐05 5,93E‐05 3,87E‐12 8,53E‐03 0 229,002 6 3 3,38E‐02 ‐3,41E‐05 9,73E‐05 2,50E‐12 8,49E‐03 ‐1,36E‐12 230,002 6 3 ‐2,72E‐04 ‐1,09E‐02 ‐3,11E‐03 ‐3,64E‐12 8,19E‐04 ‐8,73E‐11 225,002 5 3 4,22E‐02 ‐4,36E‐06 7,60E‐06 ‐1,09E‐04 8,65E‐03 ‐1,67E‐06 226,002 5 3 4,01E‐02 ‐9,20E‐06 1,77E‐05 ‐1,07E‐04 8,64E‐03 ‐2,59E‐06 227,002 5 3 3,80E‐02 ‐1,57E‐05 3,41E‐05 ‐1,04E‐04 8,62E‐03 ‐3,96E‐06 228,002 5 3 3,59E‐02 ‐2,36E‐05 5,96E‐05 ‐9,93E‐05 8,59E‐03 ‐5,98E‐06 229,002 5 3 3,38E‐02 ‐3,21E‐05 9,78E‐05 ‐9,30E‐05 8,54E‐03 ‐8,88E‐06 230,002 5 3 ‐4,02E‐04 ‐1,28E‐02 ‐4,39E‐03 8,83E‐04 1,38E‐03 ‐4,93E‐03 225,002 4 3 4,22E‐02 ‐3,39E‐06 7,69E‐06 ‐1,94E‐04 8,82E‐03 ‐2,87E‐06 226,002 4 3 4,01E‐02 ‐7,36E‐06 1,80E‐05 ‐1,90E‐04 8,81E‐03 ‐4,48E‐06 227,002 4 3 3,80E‐02 ‐1,27E‐05 3,47E‐05 ‐1,85E‐04 8,80E‐03 ‐6,89E‐06 228,002 4 3 3,59E‐02 ‐1,94E‐05 6,05E‐05 ‐1,76E‐04 8,77E‐03 ‐1,05E‐05 229,002 4 3 3,38E‐02 ‐2,67E‐05 9,91E‐05 ‐1,65E‐04 8,72E‐03 ‐1,56E‐05 230,002 4 3 ‐6,64E‐04 ‐2,16E‐02 ‐1,03E‐02 1,85E‐03 1,07E‐03 ‐1,35E‐02 225,002 3 3 4,22E‐02 ‐2,14E‐06 7,64E‐06 ‐2,32E‐04 9,12E‐03 ‐3,28E‐06 226,002 3 3 4,01E‐02 ‐4,92E‐06 1,82E‐05 ‐2,28E‐04 9,12E‐03 ‐5,14E‐06 227,002 3 3 3,80E‐02 ‐8,73E‐06 3,52E‐05 ‐2,21E‐04 9,10E‐03 ‐7,98E‐06 228,002 3 3 3,59E‐02 ‐1,35E‐05 6,15E‐05 ‐2,11E‐04 9,08E‐03 ‐1,22E‐05 229,002 3 3 3,38E‐02 ‐1,89E‐05 1,01E‐04 ‐1,97E‐04 9,04E‐03 ‐1,85E‐05 230,002 3 3 ‐3,68E‐04 ‐2,94E‐02 ‐2,41E‐02 7,39E‐03 5,48E‐03 ‐2,97E‐02


92




93

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 4 4,08E‐07 ‐1,35E‐07 ‐1,34E‐07 4,07E‐09 4,06E‐07 1,58E‐10 226,002 8 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 3,95E‐09 4,05E‐07 2,41E‐10 227,002 8 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 3,75E‐09 4,05E‐07 3,65E‐10 228,002 8 4 3,47E‐07 ‐1,15E‐07 ‐1,13E‐07 3,45E‐09 4,04E‐07 5,45E‐10 229,002 8 4 3,26E‐07 ‐1,09E‐07 ‐1,06E‐07 3,01E‐09 4,02E‐07 8,05E‐10 230,002 8 4 1,53E‐07 ‐2,06E‐07 ‐1,05E‐07 ‐9,00E‐08 1,19E‐07 5,64E‐07 225,002 7 4 4,08E‐07 ‐1,35E‐07 ‐1,34E‐07 2,31E‐09 3,98E‐07 8,99E‐11 226,002 7 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 2,24E‐09 3,97E‐07 1,37E‐10 227,002 7 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 2,13E‐09 3,97E‐07 2,06E‐10 228,002 7 4 3,47E‐07 ‐1,15E‐07 ‐1,13E‐07 1,96E‐09 3,96E‐07 3,07E‐10 229,002 7 4 3,26E‐07 ‐1,09E‐07 ‐1,06E‐07 1,72E‐09 3,94E‐07 4,51E‐10 230,002 7 4 1,08E‐07 ‐2,08E‐07 ‐1,63E‐08 ‐3,08E‐08 7,06E‐08 2,46E‐07 225,002 6 4 4,08E‐07 ‐1,35E‐07 ‐1,34E‐07 3,82E‐17 3,95E‐07 ‐2,78E‐17 226,002 6 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 1,08E‐16 3,95E‐07 ‐6,94E‐18 227,002 6 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 5,55E‐17 3,94E‐07 ‐2,08E‐17 228,002 6 4 3,47E‐07 ‐1,15E‐07 ‐1,13E‐07 1,08E‐16 3,93E‐07 6,94E‐18 229,002 6 4 3,26E‐07 ‐1,09E‐07 ‐1,06E‐07 9,94E‐17 3,91E‐07 ‐6,94E‐18 230,002 6 4 9,15E‐08 ‐1,97E‐07 6,47E‐09 0 7,00E‐08 ‐7,11E‐15 225,002 5 4 4,08E‐07 ‐1,35E‐07 ‐1,34E‐07 ‐2,31E‐09 3,98E‐07 ‐8,99E‐11 226,002 5 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 ‐2,24E‐09 3,97E‐07 ‐1,37E‐10 227,002 5 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 ‐2,13E‐09 3,97E‐07 ‐2,06E‐10 228,002 5 4 3,47E‐07 ‐1,15E‐07 ‐1,13E‐07 ‐1,96E‐09 3,96E‐07 ‐3,07E‐10 229,002 5 4 3,26E‐07 ‐1,09E‐07 ‐1,06E‐07 ‐1,72E‐09 3,94E‐07 ‐4,51E‐10 230,002 5 4 1,08E‐07 ‐2,08E‐07 ‐1,63E‐08 3,08E‐08 7,06E‐08 ‐2,46E‐07 225,002 4 4 4,08E‐07 ‐1,35E‐07 ‐1,34E‐07 ‐4,07E‐09 4,06E‐07 ‐1,58E‐10 226,002 4 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 ‐3,95E‐09 4,05E‐07 ‐2,41E‐10 227,002 4 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 ‐3,75E‐09 4,05E‐07 ‐3,65E‐10 228,002 4 4 3,47E‐07 ‐1,15E‐07 ‐1,13E‐07 ‐3,45E‐09 4,04E‐07 ‐5,45E‐10 229,002 4 4 3,26E‐07 ‐1,09E‐07 ‐1,06E‐07 ‐3,01E‐09 4,02E‐07 ‐8,05E‐10 230,002 4 4 1,53E‐07 ‐2,06E‐07 ‐1,05E‐07 9,00E‐08 1,19E‐07 ‐5,64E‐07 225,002 3 4 4,08E‐07 ‐1,35E‐07 ‐1,35E‐07 ‐4,81E‐09 4,19E‐07 ‐1,87E‐10 226,002 3 4 3,88E‐07 ‐1,28E‐07 ‐1,28E‐07 ‐4,67E‐09 4,19E‐07 ‐2,87E‐10 227,002 3 4 3,67E‐07 ‐1,22E‐07 ‐1,20E‐07 ‐4,43E‐09 4,18E‐07 ‐4,36E‐10 228,002 3 4 3,46E‐07 ‐1,15E‐07 ‐1,13E‐07 ‐4,06E‐09 4,18E‐07 ‐6,57E‐10 229,002 3 4 3,26E‐07 ‐1,09E‐07 ‐1,05E‐07 ‐3,52E‐09 4,16E‐07 ‐9,79E‐10 230,002 3 4 2,48E‐07 ‐1,12E‐07 ‐3,85E‐07 9,29E‐08 2,07E‐07 ‐9,52E‐07


94

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 4 2,82E‐02 ‐2,84E‐06 1,18E‐05 1,06E‐04 1,05E‐02 4,10E‐06 226,002 8 4 2,68E‐02 ‐6,84E‐06 2,66E‐05 1,02E‐04 1,05E‐02 6,26E‐06 227,002 8 4 2,53E‐02 ‐1,27E‐05 5,05E‐05 9,73E‐05 1,05E‐02 9,46E‐06 228,002 8 4 2,39E‐02 ‐2,07E‐05 8,72E‐05 8,94E‐05 1,05E‐02 1,41E‐05 229,002 8 4 2,25E‐02 ‐3,10E‐05 1,42E‐04 7,81E‐05 1,04E‐02 2,09E‐05 230,002 8 4 ‐2,64E‐05 ‐1,87E‐02 ‐1,34E‐02 ‐2,33E‐03 3,09E‐03 1,46E‐02 225,002 7 4 2,82E‐02 ‐3,64E‐06 1,17E‐05 6,00E‐05 1,03E‐02 2,33E‐06 226,002 7 4 2,68E‐02 ‐8,47E‐06 2,63E‐05 5,82E‐05 1,03E‐02 3,55E‐06 227,002 7 4 2,54E‐02 ‐1,54E‐05 4,99E‐05 5,52E‐05 1,03E‐02 5,35E‐06 228,002 7 4 2,39E‐02 ‐2,49E‐05 8,62E‐05 5,08E‐05 1,03E‐02 7,96E‐06 229,002 7 4 2,25E‐02 ‐3,71E‐05 1,40E‐04 4,45E‐05 1,02E‐02 1,17E‐05 230,002 7 4 ‐2,29E‐04 ‐1,67E‐02 ‐6,70E‐03 ‐7,98E‐04 1,83E‐03 6,37E‐03 225,002 6 4 2,82E‐02 ‐3,95E‐06 1,17E‐05 ‐1,14E‐13 1,03E‐02 0 226,002 6 4 2,68E‐02 ‐9,08E‐06 2,62E‐05 2,39E‐12 1,02E‐02 2,27E‐13 227,002 6 4 2,54E‐02 ‐1,65E‐05 4,96E‐05 1,76E‐12 1,02E‐02 ‐2,27E‐13 228,002 6 4 2,39E‐02 ‐2,65E‐05 8,58E‐05 1,42E‐12 1,02E‐02 ‐6,82E‐13 229,002 6 4 2,25E‐02 ‐3,93E‐05 1,40E‐04 ‐4,74E‐13 1,02E‐02 ‐6,82E‐13 230,002 6 4 ‐2,47E‐04 ‐1,52E‐02 ‐4,66E‐03 3,64E‐11 1,82E‐03 5,82E‐11 225,002 5 4 2,82E‐02 ‐3,64E‐06 1,17E‐05 ‐6,00E‐05 1,03E‐02 ‐2,33E‐06 226,002 5 4 2,68E‐02 ‐8,47E‐06 2,63E‐05 ‐5,82E‐05 1,03E‐02 ‐3,55E‐06 227,002 5 4 2,54E‐02 ‐1,54E‐05 4,99E‐05 ‐5,52E‐05 1,03E‐02 ‐5,35E‐06 228,002 5 4 2,39E‐02 ‐2,49E‐05 8,62E‐05 ‐5,08E‐05 1,03E‐02 ‐7,96E‐06 229,002 5 4 2,25E‐02 ‐3,71E‐05 1,40E‐04 ‐4,45E‐05 1,02E‐02 ‐1,17E‐05 230,002 5 4 ‐2,29E‐04 ‐1,67E‐02 ‐6,70E‐03 7,98E‐04 1,83E‐03 ‐6,37E‐03 225,002 4 4 2,82E‐02 ‐2,84E‐06 1,18E‐05 ‐1,06E‐04 1,05E‐02 ‐4,10E‐06 226,002 4 4 2,68E‐02 ‐6,84E‐06 2,66E‐05 ‐1,02E‐04 1,05E‐02 ‐6,26E‐06 227,002 4 4 2,53E‐02 ‐1,27E‐05 5,05E‐05 ‐9,73E‐05 1,05E‐02 ‐9,46E‐06 228,002 4 4 2,39E‐02 ‐2,07E‐05 8,72E‐05 ‐8,94E‐05 1,05E‐02 ‐1,41E‐05 229,002 4 4 2,25E‐02 ‐3,10E‐05 1,42E‐04 ‐7,81E‐05 1,04E‐02 ‐2,09E‐05 230,002 4 4 ‐2,64E‐05 ‐1,87E‐02 ‐1,34E‐02 2,33E‐03 3,09E‐03 ‐1,46E‐02 225,002 3 4 2,82E‐02 ‐1,83E‐06 1,17E‐05 ‐1,25E‐04 1,09E‐02 ‐4,85E‐06 226,002 3 4 2,68E‐02 ‐4,69E‐06 2,69E‐05 ‐1,21E‐04 1,09E‐02 ‐7,44E‐06 227,002 3 4 2,53E‐02 ‐8,91E‐06 5,11E‐05 ‐1,15E‐04 1,09E‐02 ‐1,13E‐05 228,002 3 4 2,39E‐02 ‐1,48E‐05 8,84E‐05 ‐1,05E‐04 1,08E‐02 ‐1,70E‐05 229,002 3 4 2,25E‐02 ‐2,25E‐05 1,43E‐04 ‐9,14E‐05 1,08E‐02 ‐2,54E‐05 230,002 3 4 2,97E‐04 ‐1,83E‐02 ‐3,25E‐02 2,41E‐03 5,38E‐03 ‐2,47E‐02


95




96

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 5 2,04E‐07 ‐6,75E‐08 ‐6,72E‐08 1,81E‐09 4,45E‐07 2,00E‐10 226,002 8 5 1,94E‐07 ‐6,43E‐08 ‐6,35E‐08 1,70E‐09 4,45E‐07 3,00E‐10 227,002 8 5 1,83E‐07 ‐6,11E‐08 ‐5,97E‐08 1,53E‐09 4,44E‐07 4,48E‐10 228,002 8 5 1,73E‐07 ‐5,80E‐08 ‐5,56E‐08 1,26E‐09 4,44E‐07 6,61E‐10 229,002 8 5 1,62E‐07 ‐5,50E‐08 ‐5,11E‐08 8,49E‐10 4,43E‐07 9,66E‐10 230,002 8 5 1,41E‐07 ‐1,36E‐07 ‐1,49E‐07 ‐4,53E‐08 1,27E‐07 4,68E‐07 225,002 7 5 2,04E‐07 ‐6,75E‐08 ‐6,71E‐08 1,03E‐09 4,37E‐07 1,12E‐10 226,002 7 5 1,94E‐07 ‐6,43E‐08 ‐6,35E‐08 9,69E‐10 4,36E‐07 1,69E‐10 227,002 7 5 1,84E‐07 ‐6,11E‐08 ‐5,97E‐08 8,70E‐10 4,36E‐07 2,51E‐10 228,002 7 5 1,73E‐07 ‐5,81E‐08 ‐5,56E‐08 7,15E‐10 4,35E‐07 3,69E‐10 229,002 7 5 1,62E‐07 ‐5,52E‐08 ‐5,11E‐08 4,86E‐10 4,34E‐07 5,37E‐10 230,002 7 5 1,07E‐07 ‐1,77E‐07 ‐4,07E‐08 ‐2,76E‐08 9,77E‐08 2,30E‐07 225,002 6 5 2,04E‐07 ‐6,75E‐08 ‐6,71E‐08 7,63E‐17 4,34E‐07 ‐6,94E‐18 226,002 6 5 1,94E‐07 ‐6,43E‐08 ‐6,35E‐08 2,08E‐17 4,34E‐07 1,39E‐17 227,002 6 5 1,84E‐07 ‐6,12E‐08 ‐5,97E‐08 1,04E‐16 4,33E‐07 ‐6,94E‐18 228,002 6 5 1,73E‐07 ‐5,81E‐08 ‐5,56E‐08 7,63E‐17 4,32E‐07 ‐2,78E‐17 229,002 6 5 1,62E‐07 ‐5,52E‐08 ‐5,11E‐08 9,71E‐17 4,31E‐07 2,78E‐17 230,002 6 5 9,67E‐08 ‐1,87E‐07 ‐1,18E‐08 ‐2,22E‐16 8,57E‐08 ‐1,78E‐15 225,002 5 5 2,04E‐07 ‐6,75E‐08 ‐6,71E‐08 ‐1,03E‐09 4,37E‐07 ‐1,12E‐10 226,002 5 5 1,94E‐07 ‐6,43E‐08 ‐6,35E‐08 ‐9,69E‐10 4,36E‐07 ‐1,69E‐10 227,002 5 5 1,84E‐07 ‐6,11E‐08 ‐5,97E‐08 ‐8,70E‐10 4,36E‐07 ‐2,51E‐10 228,002 5 5 1,73E‐07 ‐5,81E‐08 ‐5,56E‐08 ‐7,15E‐10 4,35E‐07 ‐3,69E‐10 229,002 5 5 1,62E‐07 ‐5,52E‐08 ‐5,11E‐08 ‐4,86E‐10 4,34E‐07 ‐5,37E‐10 230,002 5 5 1,07E‐07 ‐1,77E‐07 ‐4,07E‐08 2,76E‐08 9,77E‐08 ‐2,30E‐07 225,002 4 5 2,04E‐07 ‐6,75E‐08 ‐6,72E‐08 ‐1,81E‐09 4,45E‐07 ‐2,00E‐10 226,002 4 5 1,94E‐07 ‐6,43E‐08 ‐6,35E‐08 ‐1,70E‐09 4,45E‐07 ‐3,00E‐10 227,002 4 5 1,83E‐07 ‐6,11E‐08 ‐5,97E‐08 ‐1,53E‐09 4,44E‐07 ‐4,48E‐10 228,002 4 5 1,73E‐07 ‐5,80E‐08 ‐5,56E‐08 ‐1,26E‐09 4,44E‐07 ‐6,61E‐10 229,002 4 5 1,62E‐07 ‐5,50E‐08 ‐5,11E‐08 ‐8,49E‐10 4,43E‐07 ‐9,66E‐10 230,002 4 5 1,41E‐07 ‐1,36E‐07 ‐1,49E‐07 4,53E‐08 1,27E‐07 ‐4,68E‐07 225,002 3 5 2,04E‐07 ‐6,75E‐08 ‐6,72E‐08 ‐2,13E‐09 4,60E‐07 ‐2,40E‐10 226,002 3 5 1,94E‐07 ‐6,42E‐08 ‐6,35E‐08 ‐2,01E‐09 4,60E‐07 ‐3,63E‐10 227,002 3 5 1,83E‐07 ‐6,11E‐08 ‐5,97E‐08 ‐1,81E‐09 4,59E‐07 ‐5,44E‐10 228,002 3 5 1,73E‐07 ‐5,79E‐08 ‐5,56E‐08 ‐1,48E‐09 4,59E‐07 ‐8,07E‐10 229,002 3 5 1,62E‐07 ‐5,49E‐08 ‐5,10E‐08 ‐9,92E‐10 4,58E‐07 ‐1,19E‐09 230,002 3 5 1,95E‐07 ‐4,29E‐08 ‐3,55E‐07 6,59E‐08 1,80E‐07 ‐6,49E‐07


97

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 5 1,41E‐02 ‐2,25E‐06 1,48E‐05 4,69E‐05 1,15E‐02 5,18E‐06 226,002 8 5 1,34E‐02 ‐6,31E‐06 3,29E‐05 4,42E‐05 1,15E‐02 7,79E‐06 227,002 8 5 1,27E‐02 ‐1,26E‐05 6,17E‐05 3,97E‐05 1,15E‐02 1,16E‐05 228,002 8 5 1,20E‐02 ‐2,17E‐05 1,06E‐04 3,26E‐05 1,15E‐02 1,71E‐05 229,002 8 5 1,12E‐02 ‐3,47E‐05 1,71E‐04 2,20E‐05 1,15E‐02 2,50E‐05 230,002 8 5 5,61E‐05 ‐1,43E‐02 ‐1,50E‐02 ‐1,18E‐03 3,28E‐03 1,21E‐02 225,002 7 5 1,41E‐02 ‐2,91E‐06 1,48E‐05 2,67E‐05 1,13E‐02 2,92E‐06 226,002 7 5 1,34E‐02 ‐7,81E‐06 3,26E‐05 2,51E‐05 1,13E‐02 4,38E‐06 227,002 7 5 1,27E‐02 ‐1,53E‐05 6,12E‐05 2,26E‐05 1,13E‐02 6,50E‐06 228,002 7 5 1,20E‐02 ‐2,62E‐05 1,05E‐04 1,86E‐05 1,13E‐02 9,57E‐06 229,002 7 5 1,12E‐02 ‐4,14E‐05 1,70E‐04 1,26E‐05 1,13E‐02 1,39E‐05 230,002 7 5 ‐5,62E‐05 ‐1,48E‐02 ‐7,70E‐03 ‐7,15E‐04 2,53E‐03 5,98E‐03 225,002 6 5 1,41E‐02 ‐3,17E‐06 1,47E‐05 1,71E‐13 1,13E‐02 0 226,002 6 5 1,34E‐02 ‐8,36E‐06 3,25E‐05 2,16E‐12 1,12E‐02 2,27E‐13 227,002 6 5 1,27E‐02 ‐1,63E‐05 6,09E‐05 3,41E‐13 1,12E‐02 ‐2,27E‐13 228,002 6 5 1,20E‐02 ‐2,78E‐05 1,05E‐04 6,82E‐13 1,12E‐02 ‐1,14E‐12 229,002 6 5 1,12E‐02 ‐4,39E‐05 1,69E‐04 1,48E‐12 1,12E‐02 ‐4,55E‐13 230,002 6 5 ‐1,16E‐04 ‐1,48E‐02 ‐5,75E‐03 0 2,22E‐03 2,33E‐10 225,002 5 5 1,41E‐02 ‐2,91E‐06 1,48E‐05 ‐2,67E‐05 1,13E‐02 ‐2,92E‐06 226,002 5 5 1,34E‐02 ‐7,81E‐06 3,26E‐05 ‐2,51E‐05 1,13E‐02 ‐4,38E‐06 227,002 5 5 1,27E‐02 ‐1,53E‐05 6,12E‐05 ‐2,26E‐05 1,13E‐02 ‐6,50E‐06 228,002 5 5 1,20E‐02 ‐2,62E‐05 1,05E‐04 ‐1,86E‐05 1,13E‐02 ‐9,57E‐06 229,002 5 5 1,12E‐02 ‐4,14E‐05 1,70E‐04 ‐1,26E‐05 1,13E‐02 ‐1,39E‐05 230,002 5 5 ‐5,62E‐05 ‐1,48E‐02 ‐7,70E‐03 7,15E‐04 2,53E‐03 ‐5,98E‐03 225,002 4 5 1,41E‐02 ‐2,25E‐06 1,48E‐05 ‐4,69E‐05 1,15E‐02 ‐5,18E‐06 226,002 4 5 1,34E‐02 ‐6,31E‐06 3,29E‐05 ‐4,42E‐05 1,15E‐02 ‐7,79E‐06 227,002 4 5 1,27E‐02 ‐1,26E‐05 6,17E‐05 ‐3,97E‐05 1,15E‐02 ‐1,16E‐05 228,002 4 5 1,20E‐02 ‐2,17E‐05 1,06E‐04 ‐3,26E‐05 1,15E‐02 ‐1,71E‐05 229,002 4 5 1,12E‐02 ‐3,47E‐05 1,71E‐04 ‐2,20E‐05 1,15E‐02 ‐2,50E‐05 230,002 4 5 5,61E‐05 ‐1,43E‐02 ‐1,50E‐02 1,18E‐03 3,28E‐03 ‐1,21E‐02 225,002 3 5 1,41E‐02 ‐1,46E‐06 1,47E‐05 ‐5,53E‐05 1,19E‐02 ‐6,24E‐06 226,002 3 5 1,34E‐02 ‐4,39E‐06 3,31E‐05 ‐5,22E‐05 1,19E‐02 ‐9,42E‐06 227,002 3 5 1,27E‐02 ‐8,95E‐06 6,24E‐05 ‐4,69E‐05 1,19E‐02 ‐1,41E‐05 228,002 3 5 1,20E‐02 ‐1,57E‐05 1,07E‐04 ‐3,85E‐05 1,19E‐02 ‐2,09E‐05 229,002 3 5 1,12E‐02 ‐2,53E‐05 1,73E‐04 ‐2,57E‐05 1,19E‐02 ‐3,08E‐05 230,002 3 5 ‐1,25E‐04 ‐1,25E‐02 ‐2,86E‐02 1,71E‐03 4,67E‐03 ‐1,68E‐02


98


X Y Z U1 U2 U3 225,002 8 6 1,01E‐09 ‐3,39E‐10 4,38E‐02 226,002 8 6 1,12E‐09 ‐6,47E‐10 4,41E‐02 227,002 8 6 1,08E‐09 ‐1,10E‐09 4,44E‐02 228,002 8 6 7,25E‐10 ‐1,75E‐09 4,47E‐02 229,002 8 6 ‐1,69E‐10 ‐2,64E‐09 4,50E‐02 230,002 8 6 ‐8,94E‐08 ‐2,52E‐07 4,99E‐02 225,002 7 6 7,76E‐10 ‐1,71E‐10 4,38E‐02 226,002 7 6 8,52E‐10 ‐3,30E‐10 4,41E‐02 227,002 7 6 7,93E‐10 ‐5,67E‐10 4,44E‐02 228,002 7 6 4,66E‐10 ‐9,05E‐10 4,47E‐02 229,002 7 6 ‐3,39E‐10 ‐1,37E‐09 4,50E‐02 230,002 7 6 ‐8,21E‐08 ‐1,39E‐07 4,99E‐02 225,002 6 6 6,96E‐10 ‐5,12E‐12 4,38E‐02 226,002 6 6 7,62E‐10 ‐5,14E‐12 4,41E‐02 227,002 6 6 7,02E‐10 ‐5,17E‐12 4,44E‐02 228,002 6 6 3,88E‐10 ‐5,20E‐12 4,47E‐02 229,002 6 6 ‐3,82E‐10 ‐5,23E‐12 4,50E‐02 230,002 6 6 ‐7,94E‐08 ‐5,67E‐12 4,99E‐02 225,002 5 6 7,76E‐10 1,60E‐10 4,38E‐02 226,002 5 6 8,52E‐10 3,19E‐10 4,41E‐02 227,002 5 6 7,93E‐10 5,56E‐10 4,44E‐02 228,002 5 6 4,66E‐10 8,95E‐10 4,47E‐02 229,002 5 6 ‐3,39E‐10 1,36E‐09 4,50E‐02 230,002 5 6 ‐8,21E‐08 1,39E‐07 4,99E‐02 225,002 4 6 1,01E‐09 3,29E‐10 4,38E‐02 226,002 4 6 1,12E‐09 6,37E‐10 4,41E‐02 227,002 4 6 1,08E‐09 1,09E‐09 4,44E‐02 228,002 4 6 7,25E‐10 1,74E‐09 4,47E‐02 229,002 4 6 ‐1,69E‐10 2,63E‐09 4,50E‐02 230,002 4 6 ‐8,94E‐08 2,52E‐07 4,99E‐02 225,002 3 6 1,40E‐09 5,06E‐10 4,38E‐02 226,002 3 6 1,58E‐09 9,46E‐10 4,41E‐02 227,002 3 6 1,58E‐09 1,59E‐09 4,44E‐02 228,002 3 6 1,22E‐09 2,50E‐09 4,47E‐02 229,002 3 6 2,34E‐10 3,75E‐09 4,50E‐02 230,002 3 6 ‐9,99E‐08 3,04E‐07 4,99E‐02


99

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 6 1,50E‐10 ‐1,70E‐10 1,63E‐10 6,10E‐11 4,58E‐07 2,29E‐10 226,002 8 6 4,81E‐11 ‐3,11E‐10 4,69E‐10 ‐1,87E‐11 4,58E‐07 3,41E‐10 227,002 8 6 ‐1,73E‐10 ‐5,17E‐10 9,78E‐10 ‐1,62E‐10 4,58E‐07 5,02E‐10 228,002 8 6 ‐5,86E‐10 ‐8,05E‐10 1,78E‐09 ‐3,97E‐10 4,58E‐07 7,32E‐10 229,002 8 6 ‐1,29E‐09 ‐1,20E‐09 2,99E‐09 ‐7,61E‐10 4,58E‐07 1,06E‐09 230,002 8 6 1,10E‐07 ‐8,83E‐08 ‐1,37E‐07 ‐2,98E‐08 1,22E‐07 3,35E‐07 225,002 7 6 1,13E‐10 ‐1,65E‐10 1,79E‐10 2,87E‐11 4,50E‐07 1,28E‐10 226,002 7 6 2,35E‐11 ‐3,20E‐10 4,84E‐10 ‐1,82E‐11 4,49E‐07 1,90E‐10 227,002 7 6 ‐1,70E‐10 ‐5,50E‐10 9,88E‐10 ‐1,01E‐10 4,49E‐07 2,80E‐10 228,002 7 6 ‐5,32E‐10 ‐8,78E‐10 1,78E‐09 ‐2,36E‐10 4,49E‐07 4,07E‐10 229,002 7 6 ‐1,16E‐09 ‐1,33E‐09 2,97E‐09 ‐4,43E‐10 4,48E‐07 5,87E‐10 230,002 7 6 8,98E‐08 ‐1,31E‐07 ‐5,24E‐08 ‐1,56E‐08 1,02E‐07 1,80E‐07 225,002 6 6 1,01E‐10 ‐1,64E‐10 1,84E‐10 3,47E‐17 4,47E‐07 ‐1,39E‐17 226,002 6 6 1,72E‐11 ‐3,24E‐10 4,88E‐10 5,55E‐17 4,47E‐07 1,39E‐17 227,002 6 6 ‐1,65E‐10 ‐5,63E‐10 9,90E‐10 6,94E‐17 4,46E‐07 ‐6,94E‐18 228,002 6 6 ‐5,10E‐10 ‐9,06E‐10 1,78E‐09 8,33E‐17 4,46E‐07 ‐2,78E‐17 229,002 6 6 ‐1,10E‐09 ‐1,38E‐09 2,97E‐09 1,39E‐17 4,45E‐07 6,94E‐18 230,002 6 6 8,25E‐08 ‐1,45E‐07 ‐2,42E‐08 0 9,60E‐08 0 225,002 5 6 1,13E‐10 ‐1,65E‐10 1,79E‐10 ‐2,87E‐11 4,50E‐07 ‐1,28E‐10 226,002 5 6 2,35E‐11 ‐3,20E‐10 4,84E‐10 1,82E‐11 4,49E‐07 ‐1,90E‐10 227,002 5 6 ‐1,70E‐10 ‐5,50E‐10 9,88E‐10 1,01E‐10 4,49E‐07 ‐2,80E‐10 228,002 5 6 ‐5,32E‐10 ‐8,78E‐10 1,78E‐09 2,36E‐10 4,49E‐07 ‐4,07E‐10 229,002 5 6 ‐1,16E‐09 ‐1,33E‐09 2,97E‐09 4,43E‐10 4,48E‐07 ‐5,87E‐10 230,002 5 6 8,98E‐08 ‐1,31E‐07 ‐5,24E‐08 1,56E‐08 1,02E‐07 ‐1,80E‐07 225,002 4 6 1,50E‐10 ‐1,70E‐10 1,63E‐10 ‐6,10E‐11 4,58E‐07 ‐2,29E‐10 226,002 4 6 4,81E‐11 ‐3,11E‐10 4,69E‐10 1,87E‐11 4,58E‐07 ‐3,41E‐10 227,002 4 6 ‐1,73E‐10 ‐5,17E‐10 9,78E‐10 1,62E‐10 4,58E‐07 ‐5,02E‐10 228,002 4 6 ‐5,86E‐10 ‐8,05E‐10 1,78E‐09 3,97E‐10 4,58E‐07 ‐7,32E‐10 229,002 4 6 ‐1,29E‐09 ‐1,20E‐09 2,99E‐09 7,61E‐10 4,58E‐07 ‐1,06E‐09 230,002 4 6 1,10E‐07 ‐8,83E‐08 ‐1,37E‐07 2,98E‐08 1,22E‐07 ‐3,35E‐07 225,002 3 6 2,23E‐10 ‐1,85E‐10 1,33E‐10 ‐9,35E‐11 4,73E‐07 ‐2,79E‐10 226,002 3 6 1,07E‐10 ‐3,08E‐10 4,41E‐10 ‐5,07E‐12 4,73E‐07 ‐4,16E‐10 227,002 3 6 ‐1,50E‐10 ‐4,82E‐10 9,56E‐10 1,59E‐10 4,73E‐07 ‐6,15E‐10 228,002 3 6 ‐6,30E‐10 ‐7,17E‐10 1,77E‐09 4,32E‐10 4,73E‐07 ‐9,00E‐10 229,002 3 6 ‐1,45E‐09 ‐1,02E‐09 3,00E‐09 8,60E‐10 4,73E‐07 ‐1,31E‐09 230,002 3 6 1,40E‐07 ‐1,40E‐08 ‐2,77E‐07 3,53E‐08 1,52E‐07 ‐4,30E‐07


100

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 6 1,50E‐05 ‐1,63E‐06 1,57E‐05 1,58E‐06 1,19E‐02 5,94E‐06 226,002 8 6 1,29E‐05 ‐5,74E‐06 3,48E‐05 ‐4,85E‐07 1,19E‐02 8,84E‐06 227,002 8 6 5,51E‐06 ‐1,23E‐05 6,52E‐05 ‐4,20E‐06 1,19E‐02 1,30E‐05 228,002 8 6 ‐1,09E‐05 ‐2,23E‐05 1,12E‐04 ‐1,03E‐05 1,19E‐02 1,90E‐05 229,002 8 6 ‐4,19E‐05 ‐3,69E‐05 1,80E‐04 ‐1,97E‐05 1,19E‐02 2,75E‐05 230,002 8 6 ‐9,14E‐05 ‐1,04E‐02 ‐1,29E‐02 ‐7,73E‐04 3,16E‐03 8,70E‐03 225,002 7 6 1,22E‐05 ‐2,20E‐06 1,56E‐05 7,45E‐07 1,17E‐02 3,33E‐06 226,002 7 6 1,06E‐05 ‐7,17E‐06 3,45E‐05 ‐4,73E‐07 1,17E‐02 4,94E‐06 227,002 7 6 4,69E‐06 ‐1,50E‐05 6,47E‐05 ‐2,63E‐06 1,17E‐02 7,26E‐06 228,002 7 6 ‐9,01E‐06 ‐2,69E‐05 1,11E‐04 ‐6,12E‐06 1,16E‐02 1,06E‐05 229,002 7 6 ‐3,54E‐05 ‐4,43E‐05 1,79E‐04 ‐1,15E‐05 1,16E‐02 1,52E‐05 230,002 7 6 ‐5,74E‐05 ‐1,15E‐02 ‐7,44E‐03 ‐4,06E‐04 2,65E‐03 4,66E‐03 225,002 6 6 1,13E‐05 ‐2,43E‐06 1,56E‐05 0 1,16E‐02 ‐6,82E‐13 226,002 6 6 1,00E‐05 ‐7,71E‐06 3,44E‐05 6,82E‐13 1,16E‐02 0 227,002 6 6 4,59E‐06 ‐1,61E‐05 6,45E‐05 9,09E‐13 1,16E‐02 0 228,002 6 6 ‐8,10E‐06 ‐2,86E‐05 1,11E‐04 1,59E‐12 1,16E‐02 ‐4,55E‐13 229,002 6 6 ‐3,28E‐05 ‐4,70E‐05 1,79E‐04 1,82E‐12 1,16E‐02 ‐9,09E‐13 230,002 6 6 ‐6,79E‐05 ‐1,19E‐02 ‐5,60E‐03 7,28E‐12 2,49E‐03 ‐1,75E‐10 225,002 5 6 1,22E‐05 ‐2,20E‐06 1,56E‐05 ‐7,45E‐07 1,17E‐02 ‐3,33E‐06 226,002 5 6 1,06E‐05 ‐7,17E‐06 3,45E‐05 4,73E‐07 1,17E‐02 ‐4,94E‐06 227,002 5 6 4,69E‐06 ‐1,50E‐05 6,47E‐05 2,63E‐06 1,17E‐02 ‐7,26E‐06 228,002 5 6 ‐9,01E‐06 ‐2,69E‐05 1,11E‐04 6,12E‐06 1,16E‐02 ‐1,06E‐05 229,002 5 6 ‐3,54E‐05 ‐4,43E‐05 1,79E‐04 1,15E‐05 1,16E‐02 ‐1,52E‐05 230,002 5 6 ‐5,74E‐05 ‐1,15E‐02 ‐7,44E‐03 4,06E‐04 2,65E‐03 ‐4,66E‐03 225,002 4 6 1,50E‐05 ‐1,63E‐06 1,57E‐05 ‐1,58E‐06 1,19E‐02 ‐5,94E‐06 226,002 4 6 1,29E‐05 ‐5,74E‐06 3,48E‐05 4,85E‐07 1,19E‐02 ‐8,84E‐06 227,002 4 6 5,51E‐06 ‐1,23E‐05 6,52E‐05 4,20E‐06 1,19E‐02 ‐1,30E‐05 228,002 4 6 ‐1,09E‐05 ‐2,23E‐05 1,12E‐04 1,03E‐05 1,19E‐02 ‐1,90E‐05 229,002 4 6 ‐4,19E‐05 ‐3,69E‐05 1,80E‐04 1,97E‐05 1,19E‐02 ‐2,75E‐05 230,002 4 6 ‐9,14E‐05 ‐1,04E‐02 ‐1,29E‐02 7,73E‐04 3,16E‐03 ‐8,70E‐03 225,002 3 6 2,02E‐05 ‐9,92E‐07 1,55E‐05 ‐2,43E‐06 1,23E‐02 ‐7,23E‐06 226,002 3 6 1,76E‐05 ‐3,94E‐06 3,49E‐05 ‐1,32E‐07 1,23E‐02 ‐1,08E‐05 227,002 3 6 8,51E‐06 ‐8,69E‐06 6,59E‐05 4,11E‐06 1,23E‐02 ‐1,59E‐05 228,002 3 6 ‐1,15E‐05 ‐1,60E‐05 1,13E‐04 1,12E‐05 1,23E‐02 ‐2,33E‐05 229,002 3 6 ‐4,89E‐05 ‐2,67E‐05 1,82E‐04 2,23E‐05 1,23E‐02 ‐3,39E‐05 230,002 3 6 ‐2,83E‐04 ‐8,29E‐03 ‐2,19E‐02 9,15E‐04 3,95E‐03 ‐1,12E‐02


101


X Y Z U1 U2 U3 225,002 8 7 ‐3,03E‐04 1,34E‐07 4,38E‐02 226,002 8 7 ‐3,03E‐04 1,27E‐07 4,41E‐02 227,002 8 7 ‐3,03E‐04 1,20E‐07 4,44E‐02 228,002 8 7 ‐3,04E‐04 1,13E‐07 4,47E‐02 229,002 8 7 ‐3,04E‐04 1,05E‐07 4,50E‐02 230,002 8 7 ‐3,05E‐04 ‐1,72E‐07 4,99E‐02 225,002 7 7 ‐3,03E‐04 6,72E‐08 4,38E‐02 226,002 7 7 ‐3,03E‐04 6,37E‐08 4,41E‐02 227,002 7 7 ‐3,03E‐04 6,01E‐08 4,44E‐02 228,002 7 7 ‐3,04E‐04 5,64E‐08 4,47E‐02 229,002 7 7 ‐3,04E‐04 5,25E‐08 4,50E‐02 230,002 7 7 ‐3,05E‐04 ‐9,74E‐08 4,99E‐02 225,002 6 7 ‐3,03E‐04 ‐5,12E‐12 4,38E‐02 226,002 6 7 ‐3,03E‐04 ‐5,14E‐12 4,41E‐02 227,002 6 7 ‐3,03E‐04 ‐5,17E‐12 4,44E‐02 228,002 6 7 ‐3,04E‐04 ‐5,20E‐12 4,47E‐02 229,002 6 7 ‐3,04E‐04 ‐5,23E‐12 4,50E‐02 230,002 6 7 ‐3,05E‐04 ‐5,67E‐12 4,99E‐02 225,002 5 7 ‐3,03E‐04 ‐6,72E‐08 4,38E‐02 226,002 5 7 ‐3,03E‐04 ‐6,37E‐08 4,41E‐02 227,002 5 7 ‐3,03E‐04 ‐6,01E‐08 4,44E‐02 228,002 5 7 ‐3,04E‐04 ‐5,64E‐08 4,47E‐02 229,002 5 7 ‐3,04E‐04 ‐5,25E‐08 4,50E‐02 230,002 5 7 ‐3,05E‐04 9,74E‐08 4,99E‐02 225,002 4 7 ‐3,03E‐04 ‐1,34E‐07 4,38E‐02 226,002 4 7 ‐3,03E‐04 ‐1,27E‐07 4,41E‐02 227,002 4 7 ‐3,03E‐04 ‐1,20E‐07 4,44E‐02 228,002 4 7 ‐3,04E‐04 ‐1,13E‐07 4,47E‐02 229,002 4 7 ‐3,04E‐04 ‐1,05E‐07 4,50E‐02 230,002 4 7 ‐3,05E‐04 1,72E‐07 4,99E‐02 225,002 3 7 ‐3,03E‐04 ‐2,02E‐07 4,38E‐02 226,002 3 7 ‐3,03E‐04 ‐1,91E‐07 4,41E‐02 227,002 3 7 ‐3,03E‐04 ‐1,80E‐07 4,44E‐02 228,002 3 7 ‐3,04E‐04 ‐1,69E‐07 4,47E‐02 229,002 3 7 ‐3,04E‐04 ‐1,58E‐07 4,50E‐02 230,002 3 7 ‐3,05E‐04 2,03E‐07 4,99E‐02


102

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 7 ‐2,04E‐07 6,72E‐08 6,75E‐08 ‐1,68E‐09 4,45E‐07 2,42E‐10 226,002 8 7 ‐1,94E‐07 6,37E‐08 6,44E‐08 ‐1,73E‐09 4,45E‐07 3,56E‐10 227,002 8 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 ‐1,85E‐09 4,45E‐07 5,19E‐10 228,002 8 7 ‐1,74E‐07 5,64E‐08 5,89E‐08 ‐2,05E‐09 4,45E‐07 7,49E‐10 229,002 8 7 ‐1,64E‐07 5,27E‐08 5,66E‐08 ‐2,37E‐09 4,46E‐07 1,07E‐09 230,002 8 7 7,57E‐08 ‐5,63E‐08 ‐1,02E‐07 ‐1,70E‐08 1,08E‐07 2,27E‐07 225,002 7 7 ‐2,04E‐07 6,72E‐08 6,75E‐08 ‐9,67E‐10 4,36E‐07 1,35E‐10 226,002 7 7 ‐1,94E‐07 6,37E‐08 6,44E‐08 ‐1,00E‐09 4,36E‐07 1,98E‐10 227,002 7 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 ‐1,07E‐09 4,37E‐07 2,89E‐10 228,002 7 7 ‐1,74E‐07 5,64E‐08 5,89E‐08 ‐1,18E‐09 4,37E‐07 4,16E‐10 229,002 7 7 ‐1,64E‐07 5,26E‐08 5,67E‐08 ‐1,37E‐09 4,37E‐07 5,94E‐10 230,002 7 7 6,39E‐08 ‐9,00E‐08 ‐4,28E‐08 ‐9,76E‐09 9,67E‐08 1,26E‐07 225,002 6 7 ‐2,04E‐07 6,72E‐08 6,75E‐08 1,39E‐17 4,34E‐07 ‐4,16E‐17 226,002 6 7 ‐1,94E‐07 6,37E‐08 6,44E‐08 ‐2,78E‐17 4,34E‐07 ‐6,94E‐18 227,002 6 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 6,94E‐18 4,34E‐07 0 228,002 6 7 ‐1,74E‐07 5,64E‐08 5,89E‐08 5,90E‐17 4,34E‐07 ‐2,08E‐17 229,002 6 7 ‐1,64E‐07 5,25E‐08 5,67E‐08 1,02E‐16 4,34E‐07 ‐1,39E‐17 230,002 6 7 5,98E‐08 ‐1,01E‐07 ‐2,27E‐08 ‐1,11E‐16 9,25E‐08 ‐8,88E‐16 225,002 5 7 ‐2,04E‐07 6,72E‐08 6,75E‐08 9,67E‐10 4,36E‐07 ‐1,35E‐10 226,002 5 7 ‐1,94E‐07 6,37E‐08 6,44E‐08 1,00E‐09 4,36E‐07 ‐1,98E‐10 227,002 5 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 1,07E‐09 4,37E‐07 ‐2,89E‐10 228,002 5 7 ‐1,74E‐07 5,64E‐08 5,89E‐08 1,18E‐09 4,37E‐07 ‐4,16E‐10 229,002 5 7 ‐1,64E‐07 5,26E‐08 5,67E‐08 1,37E‐09 4,37E‐07 ‐5,94E‐10 230,002 5 7 6,39E‐08 ‐9,00E‐08 ‐4,28E‐08 9,76E‐09 9,67E‐08 ‐1,26E‐07 225,002 4 7 ‐2,04E‐07 6,72E‐08 6,75E‐08 1,68E‐09 4,45E‐07 ‐2,42E‐10 226,002 4 7 ‐1,94E‐07 6,37E‐08 6,44E‐08 1,73E‐09 4,45E‐07 ‐3,56E‐10 227,002 4 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 1,85E‐09 4,45E‐07 ‐5,19E‐10 228,002 4 7 ‐1,74E‐07 5,64E‐08 5,89E‐08 2,05E‐09 4,45E‐07 ‐7,49E‐10 229,002 4 7 ‐1,64E‐07 5,27E‐08 5,66E‐08 2,37E‐09 4,46E‐07 ‐1,07E‐09 230,002 4 7 7,57E‐08 ‐5,63E‐08 ‐1,02E‐07 1,70E‐08 1,08E‐07 ‐2,27E‐07 225,002 3 7 ‐2,04E‐07 6,71E‐08 6,74E‐08 1,94E‐09 4,60E‐07 ‐2,96E‐10 226,002 3 7 ‐1,94E‐07 6,37E‐08 6,43E‐08 1,99E‐09 4,60E‐07 ‐4,37E‐10 227,002 3 7 ‐1,84E‐07 6,01E‐08 6,15E‐08 2,12E‐09 4,60E‐07 ‐6,39E‐10 228,002 3 7 ‐1,74E‐07 5,65E‐08 5,88E‐08 2,34E‐09 4,60E‐07 ‐9,24E‐10 229,002 3 7 ‐1,64E‐07 5,28E‐08 5,66E‐08 2,71E‐09 4,61E‐07 ‐1,33E‐09 230,002 3 7 9,26E‐08 ‐3,92E‐09 ‐1,91E‐07 2,01E‐08 1,26E‐07 ‐2,75E‐07


103

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 7 ‐1,41E‐02 ‐1,18E‐06 1,40E‐05 ‐4,35E‐05 1,15E‐02 6,28E‐06 226,002 8 7 ‐1,34E‐02 ‐5,37E‐06 3,17E‐05 ‐4,50E‐05 1,15E‐02 9,25E‐06 227,002 8 7 ‐1,27E‐02 ‐1,22E‐05 6,00E‐05 ‐4,79E‐05 1,15E‐02 1,35E‐05 228,002 8 7 ‐1,20E‐02 ‐2,27E‐05 1,03E‐04 ‐5,31E‐05 1,16E‐02 1,94E‐05 229,002 8 7 ‐1,13E‐02 ‐3,84E‐05 1,67E‐04 ‐6,14E‐05 1,16E‐02 2,78E‐05 230,002 8 7 ‐2,10E‐04 ‐7,06E‐03 ‐9,40E‐03 ‐4,42E‐04 2,81E‐03 5,88E‐03 225,002 7 7 ‐1,41E‐02 ‐1,73E‐06 1,41E‐05 ‐2,51E‐05 1,13E‐02 3,50E‐06 226,002 7 7 ‐1,34E‐02 ‐6,85E‐06 3,16E‐05 ‐2,60E‐05 1,13E‐02 5,15E‐06 227,002 7 7 ‐1,27E‐02 ‐1,51E‐05 5,97E‐05 ‐2,77E‐05 1,13E‐02 7,49E‐06 228,002 7 7 ‐1,20E‐02 ‐2,78E‐05 1,03E‐04 ‐3,07E‐05 1,13E‐02 1,08E‐05 229,002 7 7 ‐1,13E‐02 ‐4,66E‐05 1,66E‐04 ‐3,55E‐05 1,13E‐02 1,54E‐05 230,002 7 7 ‐1,47E‐04 ‐8,13E‐03 ‐5,68E‐03 ‐2,53E‐04 2,51E‐03 3,26E‐03 225,002 6 7 ‐1,41E‐02 ‐1,96E‐06 1,41E‐05 9,09E‐13 1,13E‐02 ‐6,82E‐13 226,002 6 7 ‐1,34E‐02 ‐7,41E‐06 3,15E‐05 9,66E‐13 1,12E‐02 ‐4,55E‐13 227,002 6 7 ‐1,27E‐02 ‐1,62E‐05 5,96E‐05 1,02E‐12 1,13E‐02 ‐2,27E‐13 228,002 6 7 ‐1,20E‐02 ‐2,96E‐05 1,02E‐04 1,48E‐12 1,13E‐02 ‐9,09E‐13 229,002 6 7 ‐1,13E‐02 ‐4,96E‐05 1,66E‐04 1,56E‐12 1,13E‐02 1,14E‐13 230,002 6 7 ‐1,32E‐04 ‐8,49E‐03 ‐4,41E‐03 9,09E‐12 2,40E‐03 ‐2,91E‐11 225,002 5 7 ‐1,41E‐02 ‐1,73E‐06 1,41E‐05 2,51E‐05 1,13E‐02 ‐3,50E‐06 226,002 5 7 ‐1,34E‐02 ‐6,85E‐06 3,16E‐05 2,60E‐05 1,13E‐02 ‐5,15E‐06 227,002 5 7 ‐1,27E‐02 ‐1,51E‐05 5,97E‐05 2,77E‐05 1,13E‐02 ‐7,49E‐06 228,002 5 7 ‐1,20E‐02 ‐2,78E‐05 1,03E‐04 3,07E‐05 1,13E‐02 ‐1,08E‐05 229,002 5 7 ‐1,13E‐02 ‐4,66E‐05 1,66E‐04 3,55E‐05 1,13E‐02 ‐1,54E‐05 230,002 5 7 ‐1,47E‐04 ‐8,13E‐03 ‐5,68E‐03 2,53E‐04 2,51E‐03 ‐3,26E‐03 225,002 4 7 ‐1,41E‐02 ‐1,18E‐06 1,40E‐05 4,35E‐05 1,15E‐02 ‐6,28E‐06 226,002 4 7 ‐1,34E‐02 ‐5,37E‐06 3,17E‐05 4,50E‐05 1,15E‐02 ‐9,25E‐06 227,002 4 7 ‐1,27E‐02 ‐1,22E‐05 6,00E‐05 4,79E‐05 1,15E‐02 ‐1,35E‐05 228,002 4 7 ‐1,20E‐02 ‐2,27E‐05 1,03E‐04 5,31E‐05 1,16E‐02 ‐1,94E‐05 229,002 4 7 ‐1,13E‐02 ‐3,84E‐05 1,67E‐04 6,14E‐05 1,16E‐02 ‐2,78E‐05 230,002 4 7 ‐2,10E‐04 ‐7,06E‐03 ‐9,40E‐03 4,42E‐04 2,81E‐03 ‐5,88E‐03 225,002 3 7 ‐1,41E‐02 ‐5,80E‐07 1,39E‐05 5,02E‐05 1,19E‐02 ‐7,68E‐06 226,002 3 7 ‐1,34E‐02 ‐3,54E‐06 3,18E‐05 5,17E‐05 1,19E‐02 ‐1,13E‐05 227,002 3 7 ‐1,27E‐02 ‐8,39E‐06 6,06E‐05 5,49E‐05 1,19E‐02 ‐1,66E‐05 228,002 3 7 ‐1,20E‐02 ‐1,59E‐05 1,04E‐04 6,07E‐05 1,19E‐02 ‐2,40E‐05 229,002 3 7 ‐1,13E‐02 ‐2,73E‐05 1,68E‐04 7,02E‐05 1,20E‐02 ‐3,44E‐05 230,002 3 7 ‐3,50E‐04 ‐5,36E‐03 ‐1,51E‐02 5,22E‐04 3,27E‐03 ‐7,14E‐03


104


X Y Z U1 U2 U3 225,002 8 8 ‐6,06E‐04 2,69E‐07 4,38E‐02 226,002 8 8 ‐6,07E‐04 2,55E‐07 4,41E‐02 227,002 8 8 ‐6,07E‐04 2,41E‐07 4,44E‐02 228,002 8 8 ‐6,07E‐04 2,27E‐07 4,47E‐02 229,002 8 8 ‐6,08E‐04 2,13E‐07 4,50E‐02 230,002 8 8 ‐6,10E‐04 ‐1,13E‐07 4,99E‐02 225,002 7 8 ‐6,06E‐04 1,35E‐07 4,38E‐02 226,002 7 8 ‐6,07E‐04 1,28E‐07 4,41E‐02 227,002 7 8 ‐6,07E‐04 1,21E‐07 4,44E‐02 228,002 7 8 ‐6,07E‐04 1,14E‐07 4,47E‐02 229,002 7 8 ‐6,08E‐04 1,06E‐07 4,50E‐02 230,002 7 8 ‐6,10E‐04 ‐6,41E‐08 4,99E‐02 225,002 6 8 ‐6,06E‐04 ‐5,12E‐12 4,38E‐02 226,002 6 8 ‐6,07E‐04 ‐5,14E‐12 4,41E‐02 227,002 6 8 ‐6,07E‐04 ‐5,17E‐12 4,44E‐02 228,002 6 8 ‐6,07E‐04 ‐5,20E‐12 4,47E‐02 229,002 6 8 ‐6,08E‐04 ‐5,23E‐12 4,50E‐02 230,002 6 8 ‐6,10E‐04 ‐5,67E‐12 4,99E‐02 225,002 5 8 ‐9,07E‐04 ‐2,42E‐07 4,38E‐02 226,002 5 8 ‐6,07E‐04 ‐1,28E‐07 4,41E‐02 227,002 5 8 ‐6,07E‐04 ‐1,21E‐07 4,44E‐02 228,002 5 8 ‐6,07E‐04 ‐1,14E‐07 4,47E‐02 229,002 5 8 ‐6,08E‐04 ‐1,06E‐07 4,50E‐02 230,002 5 8 ‐6,10E‐04 6,41E‐08 4,99E‐02 225,002 4 8 ‐6,06E‐04 ‐2,69E‐07 4,26E‐02 226,002 4 8 ‐6,07E‐04 ‐2,55E‐07 4,41E‐02 227,002 4 8 ‐6,07E‐04 ‐2,41E‐07 4,44E‐02 228,002 4 8 ‐6,07E‐04 ‐2,27E‐07 4,47E‐02 229,002 4 8 ‐6,08E‐04 ‐2,13E‐07 4,50E‐02 230,002 4 8 ‐6,10E‐04 1,13E‐07 4,99E‐02 225,002 3 8 ‐6,06E‐04 ‐4,04E‐07 4,38E‐02 226,002 3 8 ‐6,07E‐04 ‐3,83E‐07 4,41E‐02 227,002 3 8 ‐6,07E‐04 ‐3,62E‐07 4,44E‐02 228,002 3 8 ‐6,07E‐04 ‐3,41E‐07 4,47E‐02 229,002 3 8 ‐6,08E‐04 ‐3,20E‐07 4,50E‐02 230,002 3 8 ‐6,10E‐04 1,32E‐07 4,99E‐02


105

X Y Z E11 E22 E33 E12 E13 E23 225,002 8 8 ‐4,08E‐07 1,35E‐07 1,35E‐07 ‐3,92E‐09 4,05E‐07 2,35E‐10 226,002 8 8 ‐3,88E‐07 1,28E‐07 1,28E‐07 ‐3,96E‐09 4,05E‐07 3,44E‐10 227,002 8 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 ‐4,05E‐09 4,06E‐07 4,96E‐10 228,002 8 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 ‐4,23E‐09 4,06E‐07 7,09E‐10 229,002 8 8 ‐3,27E‐07 1,07E‐07 1,10E‐07 ‐4,52E‐09 4,07E‐07 1,00E‐09 230,002 8 8 4,36E‐08 ‐3,59E‐08 ‐5,89E‐08 ‐1,02E‐08 8,93E‐08 1,48E‐07 225,002 7 8 ‐4,08E‐07 1,35E‐07 1,35E‐07 ‐2,24E‐09 3,97E‐07 1,31E‐10 226,002 7 8 ‐3,88E‐07 1,28E‐07 1,28E‐07 ‐2,27E‐09 3,98E‐07 1,91E‐10 227,002 7 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 ‐2,32E‐09 3,98E‐07 2,75E‐10 228,002 7 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 ‐2,42E‐09 3,98E‐07 3,93E‐10 229,002 7 8 ‐3,27E‐07 1,06E‐07 1,10E‐07 ‐2,59E‐09 3,99E‐07 5,56E‐10 230,002 7 8 3,72E‐08 ‐5,88E‐08 ‐2,19E‐08 ‐5,83E‐09 8,23E‐08 8,39E‐08 225,002 6 8 ‐4,08E‐07 1,35E‐07 1,35E‐07 0 3,95E‐07 ‐4,16E‐17 226,002 6 8 ‐3,88E‐07 1,28E‐07 1,28E‐07 ‐1,73E‐17 3,95E‐07 ‐6,94E‐18 227,002 6 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 3,47E‐18 3,95E‐07 ‐6,94E‐18 228,002 6 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 4,16E‐17 3,96E‐07 ‐2,08E‐17 229,002 6 8 ‐3,27E‐07 1,06E‐07 1,10E‐07 9,71E‐17 3,96E‐07 3,47E‐17 230,002 6 8 3,48E‐08 ‐6,68E‐08 ‐8,86E‐09 ‐5,55E‐17 7,98E‐08 ‐1,78E‐15 225,002 5 8 ‐7,34E‐07 2,42E‐07 2,42E‐07 4,15E‐09 3,33E‐07 ‐2,33E‐11 226,002 5 8 ‐3,88E‐07 1,28E‐07 1,28E‐07 2,27E‐09 3,98E‐07 ‐1,91E‐10 227,002 5 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 2,32E‐09 3,98E‐07 ‐2,75E‐10 228,002 5 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 2,42E‐09 3,98E‐07 ‐3,93E‐10 229,002 5 8 ‐3,27E‐07 1,06E‐07 1,10E‐07 2,59E‐09 3,99E‐07 ‐5,56E‐10 230,002 5 8 3,72E‐08 ‐5,88E‐08 ‐2,19E‐08 5,83E‐09 8,23E‐08 ‐8,39E‐08 225,002 4 8 ‐4,08E‐07 1,35E‐07 1,35E‐07 3,92E‐09 4,05E‐07 ‐2,35E‐10 226,002 4 8 ‐3,88E‐07 1,28E‐07 1,28E‐07 3,96E‐09 4,05E‐07 ‐3,44E‐10 227,002 4 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 4,05E‐09 4,06E‐07 ‐4,96E‐10 228,002 4 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 4,23E‐09 4,06E‐07 ‐7,09E‐10 229,002 4 8 ‐3,27E‐07 1,07E‐07 1,10E‐07 4,52E‐09 4,07E‐07 ‐1,00E‐09 230,002 4 8 4,36E‐08 ‐3,59E‐08 ‐5,89E‐08 1,02E‐08 8,93E‐08 ‐1,48E‐07 225,002 3 8 ‐4,08E‐07 1,34E‐07 1,35E‐07 4,59E‐09 4,19E‐07 ‐2,89E‐10 226,002 3 8 ‐3,87E‐07 1,28E‐07 1,28E‐07 4,62E‐09 4,19E‐07 ‐4,23E‐10 227,002 3 8 ‐3,67E‐07 1,21E‐07 1,22E‐07 4,71E‐09 4,20E‐07 ‐6,13E‐10 228,002 3 8 ‐3,47E‐07 1,14E‐07 1,16E‐07 4,90E‐09 4,20E‐07 ‐8,77E‐10 229,002 3 8 ‐3,27E‐07 1,07E‐07 1,10E‐07 5,23E‐09 4,22E‐07 ‐1,24E‐09 230,002 3 8 5,25E‐08 ‐2,19E‐09 ‐1,13E‐07 1,18E‐08 9,97E‐08 ‐1,73E‐07


106

X Y Z S11 S22 S33 S12 S13 S23 225,002 8 8 ‐2,81E‐02 ‐1,23E‐06 1,05E‐05 ‐1,02E‐04 1,05E‐02 6,11E‐06 226,002 8 8 ‐2,67E‐02 ‐5,73E‐06 2,47E‐05 ‐1,03E‐04 1,05E‐02 8,92E‐06 227,002 8 8 ‐2,53E‐02 ‐1,30E‐05 4,76E‐05 ‐1,05E‐04 1,05E‐02 1,29E‐05 228,002 8 8 ‐2,39E‐02 ‐2,43E‐05 8,28E‐05 ‐1,10E‐04 1,05E‐02 1,84E‐05 229,002 8 8 ‐2,25E‐02 ‐4,10E‐05 1,34E‐04 ‐1,17E‐04 1,06E‐02 2,60E‐05 230,002 8 8 ‐3,19E‐04 ‐4,44E‐03 ‐5,63E‐03 ‐2,64E‐04 2,32E‐03 3,83E‐03 225,002 7 8 ‐2,81E‐02 ‐1,89E‐06 1,06E‐05 ‐5,81E‐05 1,03E‐02 3,40E‐06 226,002 7 8 ‐2,67E‐02 ‐7,43E‐06 2,47E‐05 ‐5,88E‐05 1,03E‐02 4,96E‐06 227,002 7 8 ‐2,53E‐02 ‐1,64E‐05 4,75E‐05 ‐6,02E‐05 1,03E‐02 7,14E‐06 228,002 7 8 ‐2,39E‐02 ‐3,00E‐05 8,25E‐05 ‐6,29E‐05 1,03E‐02 1,02E‐05 229,002 7 8 ‐2,25E‐02 ‐5,02E‐05 1,34E‐04 ‐6,73E‐05 1,04E‐02 1,44E‐05 230,002 7 8 ‐2,64E‐04 ‐5,24E‐03 ‐3,33E‐03 ‐1,51E‐04 2,13E‐03 2,18E‐03 225,002 6 8 ‐2,81E‐02 ‐2,15E‐06 1,06E‐05 1,14E‐13 1,02E‐02 ‐9,09E‐13 226,002 6 8 ‐2,67E‐02 ‐8,08E‐06 2,47E‐05 6,82E‐13 1,02E‐02 ‐2,27E‐13 227,002 6 8 ‐2,53E‐02 ‐1,76E‐05 4,75E‐05 1,82E‐12 1,03E‐02 ‐6,82E‐13 228,002 6 8 ‐2,39E‐02 ‐3,21E‐05 8,24E‐05 1,93E‐12 1,03E‐02 ‐4,55E‐13 229,002 6 8 ‐2,25E‐02 ‐5,36E‐05 1,34E‐04 1,14E‐12 1,03E‐02 ‐5,68E‐13 230,002 6 8 ‐2,47E‐04 ‐5,52E‐03 ‐2,51E‐03 0 2,07E‐03 ‐7,28E‐11 225,002 5 8 ‐5,07E‐02 3,72E‐06 ‐3,39E‐06 1,08E‐04 8,64E‐03 ‐6,04E‐07 226,002 5 8 ‐2,67E‐02 ‐7,43E‐06 2,47E‐05 5,88E‐05 1,03E‐02 ‐4,96E‐06 227,002 5 8 ‐2,53E‐02 ‐1,64E‐05 4,75E‐05 6,02E‐05 1,03E‐02 ‐7,14E‐06 228,002 5 8 ‐2,39E‐02 ‐3,00E‐05 8,25E‐05 6,29E‐05 1,03E‐02 ‐1,02E‐05 229,002 5 8 ‐2,25E‐02 ‐5,02E‐05 1,34E‐04 6,73E‐05 1,04E‐02 ‐1,44E‐05 230,002 5 8 ‐2,64E‐04 ‐5,24E‐03 ‐3,33E‐03 1,51E‐04 2,13E‐03 ‐2,18E‐03 225,002 4 8 ‐2,81E‐02 ‐1,23E‐06 1,05E‐05 1,02E‐04 1,05E‐02 ‐6,11E‐06 226,002 4 8 ‐2,67E‐02 ‐5,73E‐06 2,47E‐05 1,03E‐04 1,05E‐02 ‐8,92E‐06 227,002 4 8 ‐2,53E‐02 ‐1,30E‐05 4,76E‐05 1,05E‐04 1,05E‐02 ‐1,29E‐05 228,002 4 8 ‐2,39E‐02 ‐2,43E‐05 8,28E‐05 1,10E‐04 1,05E‐02 ‐1,84E‐05 229,002 4 8 ‐2,25E‐02 ‐4,10E‐05 1,34E‐04 1,17E‐04 1,06E‐02 ‐2,60E‐05 230,002 4 8 ‐3,19E‐04 ‐4,44E‐03 ‐5,63E‐03 2,64E‐04 2,32E‐03 ‐3,83E‐03 225,002 3 8 ‐2,81E‐02 ‐4,93E‐07 1,03E‐05 1,19E‐04 1,09E‐02 ‐7,49E‐06 226,002 3 8 ‐2,67E‐02 ‐3,59E‐06 2,47E‐05 1,20E‐04 1,09E‐02 ‐1,10E‐05 227,002 3 8 ‐2,53E‐02 ‐8,67E‐06 4,80E‐05 1,22E‐04 1,09E‐02 ‐1,59E‐05 228,002 3 8 ‐2,39E‐02 ‐1,66E‐05 8,35E‐05 1,27E‐04 1,09E‐02 ‐2,27E‐05 229,002 3 8 ‐2,25E‐02 ‐2,84E‐05 1,36E‐04 1,36E‐04 1,09E‐02 ‐3,22E‐05 230,002 3 8 ‐4,19E‐04 ‐3,25E‐03 ‐8,99E‐03 3,05E‐04 2,59E‐03 ‐4,48E‐03


107

9.2- Appendix II. Matlab’s Program and Results.

In this appendix it is shown the program implemented in Matlab to run the Dänliker

Algorithms.

So, the program for the extrapolation is the next:

%coordinates of the points% PA=[226.002 6 0]; PxA=PA(1,1); PyA=PA(1,2); PzA=PA(1,3); PB=[227.002 6 0]; PxB=PB(1,1); PyB=PB(1,2); PzB=PB(1,3); PC=[228.002 6 0]; PxC=PC(1,1); PyC=PC(1,2); PzC=PC(1,3); PD=[227.002 5 0]; PxD=PD(1,1); PyD=PD(1,2); PzD=PD(1,3); PE=[227.002 7 0]; PxE=PE(1,1); PyE=PE(1,2); PzE=PE(1,3); PF=[225.002 6 0]; PxF=PF(1,1); PyF=PF(1,2); PzF=PF(1,3); PG=[229.002 6 0]; PxG=PG(1,1); PyG=PG(1,2); PzG=PG(1,3); PH=[226.002 5 0]; PxH=PH(1,1); PyH=PH(1,2); PzH=PH(1,3); PI=[228.002 5 0]; PxI=PI(1,1); PyI=PI(1,2); PzI=PI(1,3);

108

PJ=[226.002 7 0]; PxJ=PJ(1,1); PyJ=PJ(1,2); PzJ=PJ(1,3); PK=[228.002 7 0]; PxK=PK(1,1); PyK=PK(1,2); PzK=PK(1,3); PL=[227.002 4 0]; PxL=PL(1,1); PyL=PL(1,2); PzL=PL(1,3); PM=[227.002 8 0]; PxM=PM(1,1); PyM=PM(1,2); PzM=PM(1,3); %displacement of the points% DpA=[0.00182042 -0.0000000000051445 0.0440751]; UA=DpA(1,1); VA=DpA(1,2); WA=DpA(1,3); DpB=[0.00182155 -0.00000000000517246 0.0443788]; UB=DpB(1,1); VB=DpB(1,2); WB=DpB(1,3); DpC=[0.00182262 -0.00000000000520042 0.0446828]; UC=DpC(1,1); VC=DpC(1,2); WC=DpC(1,3); DpD=[0.00182155 0.00000036352 0.0443788]; UD=DpD(1,1); VD=DpD(1,2); WD=DpD(1,3); DpE=[0.00182155 -0.00000036353 0.0443788]; UE=DpE(1,1); VE=DpE(1,2); WE=DpE(1,3); DpF=[0.00181923 -0.00000000000511648 0.0437715]; UF=DpF(1,1); VF=DpF(1,2); WF=DpF(1,3); DpG=[0.00182363 -0.00000000000522835 0.0449869]; UG=DpG(1,1); VG=DpG(1,2); WG=DpG(1,3);

109

DpH=[0.00182042 0.000000383571 0.0440751]; UH=DpH(1,1); VH=DpH(1,2); WH=DpH(1,3); DpI=[0.00182262 0.000000343535 0.0446828]; UI=DpI(1,1); VI=DpI(1,2); WI=DpI(1,3); DpJ=[0.00182042 -0.000000383581 0.0440751]; UJ=DpJ(1,1); VJ=DpJ(1,2); WJ=DpJ(1,3); DpK=[0.00182262 -0.000000343545 0.0446828]; UK=DpK(1,1); VK=DpK(1,2); WK=DpK(1,3); DpL=[0.00182155 0.000000727004 0.0443787]; UL=DpL(1,1); VL=DpL(1,2); WL=DpL(1,3); DpM=[0.00182155 -0.000000727014 0.0443787]; UM=DpM(1,1); VM=DpM(1,2); WM=DpM(1,3); %matrixes of first derivates% format long NA(1,1)=(UB-UF)/2; NA(1,2)=(VB-VF)/2; NA(1,3)=(WB-WF)/2; NA(2,1)=(UJ-UH)/2; NA(2,2)=(VJ-VH)/2; NA(2,3)=(WJ-WH)/2; NA(3,1)=0; NA(3,2)=0; NA(3,3)=0; NB(1,1)=(UC-UA)/2; NB(1,2)=(VC-VA)/2; NB(1,3)=(WC-WA)/2; NB(2,1)=(UE-UD)/2; NB(2,2)=(VE-VD)/2; NB(2,3)=(WE-WD)/2; NB(3,1)=0; NB(3,2)=0; NB(3,3)=0;

110

NC(1,1)=(UG-UB)/2; NC(1,2)=(VG-VB)/2; NC(1,3)=(WG-WB)/2; NC(2,1)=(UK-UI)/2; NC(2,2)=(VK-VI)/2; NC(2,3)=(WK-WI)/2; NC(3,1)=0; NC(3,2)=0; NC(3,3)=0; ND(1,1)=(UI-UH)/2; ND(1,2)=(VI-VH)/2; ND(1,3)=(WI-WH)/2; ND(2,1)=(UB-UL)/2; ND(2,2)=(VB-VL)/2; ND(2,3)=(WB-WL)/2; ND(3,1)=0; ND(3,2)=0; ND(3,3)=0; NE(1,1)=(UK-UJ)/2; NE(1,2)=(VK-VJ)/2; NE(1,3)=(WK-WJ)/2; NE(2,1)=(UM-UB)/2; NE(2,2)=(VM-VB)/2; NE(2,3)=(WM-WB)/2; NE(3,1)=0; NE(3,2)=0; NE(3,3)=0; %matrix of second derivates% MB(1,1)=(NC(1,3)-NA(1,3))/2; MB(1,2)=(NE(1,3)-ND(1,3))/2; MB(1,3)=-(((NC(1,1)-NA(1,1))/2+(NE(1,2)-ND(1,2))/2)+1/(1-2*0.33)*((NC(1,1)-NA(1,1))/2+(NC(2,2)-NA(2,2))/2)); MB(2,1)=(NC(2,3)-NA(2,3))/2; MB(2,2)=(NE(2,3)-ND(2,3))/2; MB(2,3)=-(((NC(2,1)-NA(2,1))/2+(NE(2,2)-ND(2,2))/2)+1/(1-2*0.33)*((NE(1,1)-ND(1,1))/2+(NE(2,2)-ND(2,2))/2)); MB(3,1)=0; MB(3,2)=0; MB(3,3)=1/(2*(1-0.33))*((NC(1,3)-NA(1,3))/2+(NE(2,3)-ND(2,3))/2); %extrapolation% NN=NB; MM=MB; NN1=NN+MM; MM1=2*MM; NN2=NN1+MM1; MM2=MM+2*MM1; NN3=NN2+MM2; MM3=MM1+2*MM2; NN4=NN3+MM3; MM4=MM2+2*MM3;

111

NN5=NN4+MM4; %strain and stress% E0(1,1)=NB(1,1); E0(1,2)=0.5*NB(1,2)+0.5*NB(2,1); E0(1,3)=0.5*NB(1,3)+0.5*NB(3,1); E0(2,1)=0.5*NB(1,2)+0.5*NB(2,1); E0(2,2)=NB(2,2); E0(2,3)=0.5*NB(3,2)+0.5*NB(2,3); E0(3,1)=0.5*NB(1,3)+0.5*NB(3,1); E0(3,2)=0.5*NB(3,2)+0.5*NB(2,3); E0(3,3)=NB(3,3); E0 E1(1,1)=NN1(1,1); E1(1,2)=0.5*NN1(1,2)+0.5*NN1(2,1); E1(1,3)=0.5*NN1(1,3)+0.5*NN1(3,1); E1(2,1)=0.5*NN1(1,2)+0.5*NN1(2,1); E1(2,2)=NN1(2,2); E1(2,3)=0.5*NN1(3,2)+0.5*NN1(2,3); E1(3,1)=0.5*NN1(1,3)+0.5*NN1(3,1); E1(3,2)=0.5*NN1(3,2)+0.5*NN1(2,3); E1(3,3)=NN1(3,3); E1 E2(1,1)=NN2(1,1); E2(1,2)=0.5*NN2(1,2)+0.5*NN2(2,1); E2(1,3)=0.5*NN2(1,3)+0.5*NN2(3,1); E2(2,1)=0.5*NN2(1,2)+0.5*NN2(2,1); E2(2,2)=NN2(2,2); E2(2,3)=0.5*NN2(3,2)+0.5*NN2(2,3); E2(3,1)=0.5*NN2(1,3)+0.5*NN2(3,1); E2(3,2)=0.5*NN2(3,2)+0.5*NN2(2,3); E2(3,3)=NN2(3,3); E2 E3(1,1)=NN3(1,1); E3(1,2)=0.5*NN3(1,2)+0.5*NN3(2,1); E3(1,3)=0.5*NN3(1,3)+0.5*NN3(3,1); E3(2,1)=0.5*NN3(1,2)+0.5*NN3(2,1); E3(2,2)=NN3(2,2); E3(2,3)=0.5*NN3(3,2)+0.5*NN3(2,3); E3(3,1)=0.5*NN3(1,3)+0.5*NN3(3,1); E3(3,2)=0.5*NN3(3,2)+0.5*NN3(2,3); E3(3,3)=NN3(3,3); E3 E4(1,1)=NN4(1,1); E4(1,2)=0.5*NN4(1,2)+0.5*NN4(2,1); E4(1,3)=0.5*NN4(1,3)+0.5*NN4(3,1); E4(2,1)=0.5*NN4(1,2)+0.5*NN4(2,1); E4(2,2)=NN4(2,2); E4(2,3)=0.5*NN4(3,2)+0.5*NN4(2,3); E4(3,1)=0.5*NN4(1,3)+0.5*NN4(3,1); E4(3,2)=0.5*NN4(3,2)+0.5*NN4(2,3); E4(3,3)=NN4(3,3); E4

112

E5(1,1)=NN5(1,1); E5(1,2)=0.5*NN5(1,2)+0.5*NN5(2,1); E5(1,3)=0.5*NN5(1,3)+0.5*NN5(3,1); E5(2,1)=0.5*NN5(1,2)+0.5*NN5(2,1); E5(2,2)=NN5(2,2); E5(2,3)=0.5*NN5(3,2)+0.5*NN5(2,3); E5(3,1)=0.5*NN5(1,3)+0.5*NN5(3,1); E5(3,2)=0.5*NN5(3,2)+0.5*NN5(2,3); E5(3,3)=NN5(3,3); E5 A=0.33/(1-2*0.33); S0(1,1)=69000/(1+0.33)*((1+A)*E0(1,1)+A*E0(2,2)+A*E0(3,3)); S0(1,2)=69000/(1+0.33)*0.5*E0(1,2); S0(1,3)=69000/(1+0.33)*0.5*E0(1,3); S0(2,1)=69000/(1+0.33)*0.5*E0(2,1); S0(2,2)=69000/(1+0.33)*(A*E0(1,1)+(1+A)*E0(2,2)+A*E0(3,3)); S0(2,3)=69000/(1+0.33)*0.5*E0(2,3); S0(3,1)=69000/(1+0.33)*0.5*E0(3,1); S0(3,2)=69000/(1+0.33)*0.5*E0(3,2); S0(3,3)=69000/(1+0.33)*(A*E0(1,1)+A*E0(2,2)+(1+A)*E0(3,3)); S0 S1(1,1)=69000/(1+0.33)*((1+A)*E1(1,1)+A*E1(2,2)+A*E1(3,3)); S1(1,2)=69000/(1+0.33)*0.5*E1(1,2); S1(1,3)=69000/(1+0.33)*0.5*E1(1,3); S1(2,1)=69000/(1+0.33)*0.5*E1(2,1); S1(2,2)=69000/(1+0.33)*(A*E1(1,1)+(1+A)*E1(2,2)+A*E1(3,3)); S1(2,3)=69000/(1+0.33)*0.5*E1(2,3); S1(3,1)=69000/(1+0.33)*0.5*E1(3,1); S1(3,2)=69000/(1+0.33)*0.5*E1(3,2); S1(3,3)=69000/(1+0.33)*(A*E1(1,1)+A*E1(2,2)+(1+A)*E1(3,3)); S1 S2(1,1)=69000/(1+0.33)*((1+A)*E2(1,1)+A*E2(2,2)+A*E2(3,3)); S2(1,2)=69000/(1+0.33)*0.5*E2(1,2); S2(1,3)=69000/(1+0.33)*0.5*E2(1,3); S2(2,1)=69000/(1+0.33)*0.5*E2(2,1); S2(2,2)=69000/(1+0.33)*(A*E2(1,1)+(1+A)*E2(2,2)+A*E2(3,3)); S2(2,3)=69000/(1+0.33)*0.5*E2(2,3); S2(3,1)=69000/(1+0.33)*0.5*E2(3,1); S2(3,2)=69000/(1+0.33)*0.5*E2(3,2); S2(3,3)=69000/(1+0.33)*(A*E2(1,1)+A*E2(2,2)+(1+A)*E2(3,3)); S2 S3(1,1)=69000/(1+0.33)*((1+A)*E3(1,1)+A*E3(2,2)+A*E3(3,3)); S3(1,2)=69000/(1+0.33)*0.5*E3(1,2); S3(1,3)=69000/(1+0.33)*0.5*E3(1,3); S3(2,1)=69000/(1+0.33)*0.5*E3(2,1); S3(2,2)=69000/(1+0.33)*(A*E3(1,1)+(1+A)*E3(2,2)+A*E3(3,3)); S3(2,3)=69000/(1+0.33)*0.5*E3(2,3); S3(3,1)=69000/(1+0.33)*0.5*E3(3,1); S3(3,2)=69000/(1+0.33)*0.5*E3(3,2); S3(3,3)=69000/(1+0.33)*(A*E3(1,1)+A*E3(2,2)+(1+A)*E3(3,3)); S3

113

S4(1,1)=69000/(1+0.33)*((1+A)*E4(1,1)+A*E4(2,2)+A*E4(3,3)); S4(1,2)=69000/(1+0.33)*0.5*E4(1,2); S4(1,3)=69000/(1+0.33)*0.5*E4(1,3); S4(2,1)=69000/(1+0.33)*0.5*E4(2,1); S4(2,2)=69000/(1+0.33)*(A*E4(1,1)+(1+A)*E4(2,2)+A*E4(3,3)); S4(2,3)=69000/(1+0.33)*0.5*E4(2,3); S4(3,1)=69000/(1+0.33)*0.5*E4(3,1); S4(3,2)=69000/(1+0.33)*0.5*E4(3,2); S4(3,3)=69000/(1+0.33)*(A*E4(1,1)+A*E4(2,2)+(1+A)*E4(3,3)); S4 S5(1,1)=69000/(1+0.33)*((1+A)*E5(1,1)+A*E5(2,2)+A*E5(3,3)); S5(1,2)=69000/(1+0.33)*0.5*E5(1,2); S5(1,3)=69000/(1+0.33)*0.5*E5(1,3); S5(2,1)=69000/(1+0.33)*0.5*E5(2,1); S5(2,2)=69000/(1+0.33)*(A*E5(1,1)+(1+A)*E5(2,2)+A*E5(3,3)); S5(2,3)=69000/(1+0.33)*0.5*E5(2,3); S5(3,1)=69000/(1+0.33)*0.5*E5(3,1); S5(3,2)=69000/(1+0.33)*0.5*E5(3,2); S5(3,3)=69000/(1+0.33)*(A*E5(1,1)+A*E5(2,2)+(1+A)*E5(3,3)); S5

Where:

• PA, PB, PC … PM are the points with their coordinates.

• DpA, DpB, DpC … DpM are the displacements of these points.

• NA, NB, NC, ND and NE are the matrix of the first derivates [μik] and NNi

(i = 1 … 5) the successive extrapolations for the point B.

• M is the initial matrix [μik,z] and MM (i =1 … 4) the successive extrapolations

for the point B.

• Ei (i=0, 1… 5) and Si (i=0, 1… 5) the components of the strain and stress

tensor respectively for point B.

These points are the points drawn in figure 3.2.1 Grid in the Finite Difference.

Note: the increment in the inside extrapolation is 1 mm (h=1mm).

So, the results for a point situated among these points (point B) are:

114

E0 =

1.0e-003 *

0.001100000000000 -0.000000000013980 0.151925000000001

-0.000000000013980 -0.000363525000000 0

0.151925000000001 0 0

E1 =

1.0e-003 *

0.001299999999999 -0.000000000013980 0.152003788058824

-0.000000000013980 -0.000413525000001 -0.000000000169924

0.152003788058824 -0.000000000169924 0.000111940298506

E2 =

1.0e-003 *

0.001699999999997 -0.000000000013980 0.152161364176471

-0.000000000013980 -0.000513525000004 -0.000000000509771

0.152161364176471 -0.000000000509771 0.000335820895517

E3 =

1.0e-003 *

0.002699999999991 -0.000000000013980 0.152555304470588

-0.000000000013980 -0.000763525000012 -0.000000001359391

0.152555304470588 -0.000000001359391 0.000895522388044

115

E4 =

1.0e-003 *

0.005099999999976 -0.000000000013980 0.153500761176470

-0.000000000013980 -0.001363525000029 -0.000000003398476

0.153500761176470 -0.000000003398476 0.002238805970110

E5 =

1.0e-003 *

0.010899999999942 -0.000000000013980 0.155785614882351

-0.000000000013980 -0.002813525000070 -0.000000008326267

0.155785614882351 -0.000000008326267 0.005485074626770

S0 =

0.094152002985413 -0.000000000362639 3.940911654135356

-0.000000000362639 0.018224766143304 0

3.940911654135356 0 0.037084333812476

S1 =

0.117717638980288 -0.000000000362639 3.942955404533408

-0.000000000362639 0.028820477326136 -0.000000004407798

3.942955404533408 -0.000000004407798 0.056081458978000

116

S2 =

0.164848910970039 -0.000000000362639 3.947042905329513

-0.000000000362639 0.050011899691801 -0.000000013223395

3.947042905329513 -0.000000013223395 0.094075709309046

S3 =

0.282677090944416 -0.000000000362639 3.957261657319775

-0.000000000362639 0.102990455605962 -0.000000035262387

3.957261657319775 -0.000000035262387 0.189061335136663

S4 =

0.565464722882921 -0.000000000362639 3.981786662096403

-0.000000000362639 0.230138989799951 -0.000000088155969

3.981786662096403 -0.000000088155969 0.417026837122942

S5 =

1.248868166734308 -0.000000000362639 4.041055423639922

-0.000000000362639 0.537414614102090 -0.000000215982123

4.041055423639922 -0.000000215982123 0.967943466923118

117

The Matlab’s program to know these news values in the section at 1 mm are:

%coordinates of the points% PA=[226.002 6 1]; PxA=PA(1,1); PyA=PA(1,2); PzA=PA(1,3); PB=[227.002 6 1]; PxB=PB(1,1); PyB=PB(1,2); PzB=PB(1,3); PC=[228.002 6 1]; PxC=PC(1,1); PyC=PC(1,2); PzC=PC(1,3); PD=[227.002 5 1]; PxD=PD(1,1); PyD=PD(1,2); PzD=PD(1,3); PE=[227.002 7 1]; PxE=PE(1,1); PyE=PE(1,2); PzE=PE(1,3); %displacement of the points% DpA=[0.00151682 -0.00000000000514437 0.0440747]; UA=DpA(1,1); VA=DpA(1,2); WA=DpA(1,3); DpB=[0.00151776 -0.00000000000517236 0.0443785]; UB=DpB(1,1); VB=DpB(1,2); WB=DpB(1,3); DpC=[0.00151866 -0.00000000000520031 0.0446825]; UC=DpC(1,1); VC=DpC(1,2); WC=DpC(1,3); DpD=[0.00151776 0.000000303136 0.0443785]; UD=DpD(1,1); VD=DpD(1,2); WD=DpD(1,3);

118

DpE=[0.00151776 -0.000000303146 0.0443785]; UE=DpE(1,1); VE=DpE(1,2); WE=DpE(1,3); %matrix of first derivates% format long NB(1,1)=(UC-UA)/2; NB(1,2)=(VC-VA)/2; NB(1,3)=(WC-WA)/2; NB(2,1)=(UE-UD)/2; NB(2,2)=(VE-VD)/2; NB(2,3)=(WE-WD)/2; NB(3,1)=0; NB(3,2)=0; NB(3,3)=0; %strain and stress% E0(1,1)=NB(1,1); E0(1,2)=0.5*NB(1,2)+0.5*NB(2,1); E0(1,3)=0.5*NB(1,3)+0.5*NB(3,1); E0(2,1)=0.5*NB(1,2)+0.5*NB(2,1); E0(2,2)=NB(2,2); E0(2,3)=0.5*NB(3,2)+0.5*NB(2,3); E0(3,1)=0.5*NB(1,3)+0.5*NB(3,1); E0(3,2)=0.5*NB(3,2)+0.5*NB(2,3); E0(3,3)=NB(3,3); E0

And the result for ε is:

E0 =

1.0e-003 *

0.000920000000000 -0.000000000013985 0.151950000000000

-0.000000000013985 -0.000303141000000 0

0.151950000000000 0 0

119

9.3- Appendix III. Laser Safety.

Before using a laser is necessary to know a little about laser safety and how to use in a right way the lasers and all the things that are involving this field.

The word laser is an acronym for Light Amplification by the Stimulated emission of Radiation. The ‘light’ produced by a laser, a form of non-ionising optical radiation, has a unique combination of spatial coherence (all the waves are in phase); monochromaticity (i.e. have just one colour or a narrow bandwidth) and usually high collimation (i.e. low angular divergence such that the beam does not ‘spread out’ significantly with distance). This combination of characteristics distinguishes laser radiation from all other light sources. Lasers come in various shapes and forms. They have many uses in teaching, research, manufacturing, medicine, dentistry, shop checkouts and most commonly at work in the office. Lasers emit radiation as narrow concentrated beams of light, not necessarily visible to the human eye. The optical and skin hazards presented by lasers vary markedly according to the wavelength an power of the output. The hazards of lasers are often associated with the ability of the laser to damage eyesight or burn skin, but quite often the radiation or optical hazards are not the ones that present the greatest risk, with associated risks from electrical supplies, cryogenic liquids or chemical dyes being more hazardous. There is a legal requirement to identify risks and take appropriate action to eliminate or control those risks. We all have a responsibility under the Health & Safety at Work etc. Act 1974 to ensure that work with lasers is carried out safely. Users have a duty to protect both themselves and others from the potential hazards involved, particularly when working with the more powerful lasers. The safety of laser products is covered by the British Standards Institute (BSi), BS EN 60825 series of documents. The BS standard is a ‘euronorm’ based upon the corresponding International Electrotechnical Commission’s IEC 60825 standard encompasses a range of standards for manufacturers on lasers, fibre optic systems, laser guards, components and user’s guide etc. Of particular importance for users is the Technical Report PD IEC TR 60825-14:2004 which is a new detailed user’s guide that incorporates a risk assessment approach to laser safety. Many Institutions have internet access to British Standards Online and have licence subscriptions allowing standards to be downloaded. Each institution should at least have access to Technical Report PD IEC TR 60825-14:2004 or its current version.

120

The previous classification system, which was based on five classes (1, 2, 3A, 3B and 4), has been replaced with a new system of seven classes (1, 1M, 2, 2M, 3R, 3B and 4). It should be noted however that most of the changes relate to the lower classes of laser and that there has been minimal changes to the requirements for Class 3B and Class 4 lasers.

First of all, the laser users have some responsibilities which are the next:

• To observe the Local Rules and Schemes of Work applicable to the lasers that will be used and to follow the guidance of supervisors and the Departmental Laser Safety Officer.

• Not to leave a laser experiment running unattended unless a risk assessment has established that it is safe to do so.

• For their own safety and that of others who may be affected by their acts or omissions.

• When working with Class 3B or Class 4 lasers and there is the possibility of stray laser beams that could damage the eyesight, the appropriate laser eyewear must be worn.

It is also necessary to know the different classes of lasers ant their hazards, and how works a laser before using it. Lasers produce electromagnetic radiation at wavelengths extending from 100 nm in the ultraviolet, through the visible (400-700 nm), and the near infrared (700-1400 nm), to the far infrared (1400 nm – 1 mm). Thus, the light emitted can be either visible or invisible. Lasers can be operated in a number of different modes. Some lasers produce a continuous output and are known as continuous wave or CW lasers. The power outputs of CW lasers are usually expressed in terms of watts (W). Others operate in a pulsed mode producing short bursts of radiation. The power of the laser output can vary from less than 1 mW to many watts in some CW devices. The energy output of pulsed lasers is generally expressed in joules (J) per pulse. Because of the wide ranges possible for the wavelength, energy content and pulse characteristics of laser beams, the hazards arising from their use varies widely. It is impossible to regard lasers as single group to which common safety limits can apply. A system of laser classification is used to indicate the level of laser beam hazard and maximum Accessible Emission Levels (AELs) have been determined for each class of laser. The previous classification system, which was based on five classes (1, 2, 3A, 3B and 4), has been replaced with a new system of seven classes (1, 1M, 2, 2M, 3R, 3B and 4) and these are described below [7].

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• Class 1: Lasers that are safe under reasonably foreseeable conditions of operation, either because of the inherently low emission of the laser itself, or because of its engineering design such that it is totally enclosed and human access to higher levels is not possible under normal operation. If access panels of a totally enclosed system are removed for servicing etc then the laser product is no longer Class 1 and the precautions applicable to the embedded laser must be applied until the panels are replaced.

• Class 1M: Lasers products emitting in the wavelength range 302.5 nm to 4000 nm, whose total output is in excess of that normally permitted for Class 1 laser products but because of their diverging beams or very low power density do not pose a hazard in normal use and comply with the measurement conditions for a Class 1M product. However they may be hazardous to the eyes under certain conditions if gathering optics are used with them, i.e.

a) With a diverging beam if optics are placed within 100 nm of the source to concentrate/collimate the beam.

b) With a large diameter collimated beam viewed with binoculars or a telescope.

• Class 2: Lasers that only emit visible radiation in the wavelength range from 400 nm to 700 nm and whose output is less than the appropriate AEL. They are safe for accidental viewing as eye protection is afforded by aversion responses, including the blink reflex. This reaction may be expected to provide adequate protection under reasonably foreseeable conditions of operation including the use of optical instruments for intrabeam viewing.

• Class 2M: Laser products that only emit visible radiation in the wavelength range 400 nm to 700 nm, whose total output is in excess of that normally permitted for Class 2 laser products but because of their diverging beams or very low power density are safe for accidental viewing during normal use and comply with the measurement conditions for a Class 2M product. However they may be hazardous to the eyes under certain conditions if gathering optics are used with them, i.e.

a) With a diverging beam if optics are placed within 100 nm of the source to concentrate/collimate the beam.

b) With a large diameter collimated beam viewed with binoculars or a telescope.

• Class 3R: Lasers that emit in the wavelength range from 302.5 nm to 1 mm where direct intrabeam viewing is potentially hazardous but the risk is lower than for Class 3B lasers, and fewer manufacturing requirements and control measures for the user apply. The AEL is restricted to no more than five times the AEL of Class 2 for visible wavelengths and no more than five times the AEL of Class 1 for other wavelengths.

• Class 3B: Lasers that are normally hazardous when direct intrabeam exposure occurs (i.e. within the Nominal Ocular Hazard Distance, which is the distance

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within which the beam irradiance or radiant exposure will exceed the appropriate MPE). Viewing diffuse reflections is normally safe. Output levels must be less than the appropriate AELs for Class 3B devices.

• Class 4: High power lasers that exceed the AELs for Class 3B products that are also capable of producing hazardous diffuse reflections. They may cause skin injuries, could also constitute a fire hazard and could cause hazardous fumes to be produced as well as being a hazard to the eyes. Their use requires extreme caution.

In this way, it is visible in the next table the different hazards, precaution, controls and Limits in each kind of laser class.

Laser Class

Hazard Precaution Controls Limits & Comments

1 Intrinsically safe under reasonably foreseeable conditions or by good engineering design.

Do not stare into beam. Protection by blink reflex.

No access allowed. Very low exposure

250 μJ in 0.25s 1 mW cw

1M Wavelength 302.5 – 4000 nm. Intrinsically safe under reasonably foreseeable conditions or by good engineering design. Potentially hazardous when viewed by an optical instrument.

Do not stare into beam. Protection by blink reflex BUT do not view with optical instruments.

No access allowed. Very low exposure

250 μJ in 0.25s 1 mW cw

2 Low power cw or pulsed lasers in visible region (400 nm - 700 nm)

Do not stare into beam. Protection by blink reflex.

Momentary exposure not hazardous

1 mW cw 1 mW peak power.

2M Low power cw or pulsed lasers in visible region (400 nm - 700 nm). Potentially hazardous when viewed by an optical instrument.

Do not stare into beam. Protection by blink reflex BUT do not view with optical instruments.

Momentary exposure not hazardous

1 mW cw 1 mW peak power.

3R Lasers in range 302.5 – 106

nm where direct intrabeam viewing is hazardous. Viewing with optical aids may be hazardous.

Do not stare into beam or view directly with optical instruments.

Momentary exposure not hazardous if optical aids not used

5 mW cw 6 J/m

2 in 0.25 s

25 W/m2 in

0.25 s

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3B Lasers in range 302.5 – 106

nm where direct intrabeam viewing is hazardous. Direct viewing hazardous. Diffuse reflections normally safe.

Avoid exposure to beam. Direct intra-beam viewing is hazardous.

Controlled area. Eye protection if direct/specular viewing possible.

30 mJ for ruby 0.5 W cw for HeNe, argon etc

4 High power devices above 0.5 W. Diffuse reflections hazardous. Skin injuries possible. Fire hazard

Avoid eye or skin exposure to direct or scattered radiation.

Controlled area. Extreme caution required. Enclosed beams. Remote control if possible.

CO2, ruby, YAG, argon, excimer...

Table 9.3.1 Classification of the lasers

Once classified the different classes of laser, it is necessary to speak a little about labeling of lasers, designation of laser areas and laboratory design.

Labelling of Lasers

Inherently safe lasers in Class 1 do not need warning labels but lasers which are Class 1 by engineering design and contain an embedded laser of higher power should be labeled as ‘Class 1’. Supplementary information describing the laser product as a ‘Totally Enclosed System’ with details of the embedded laser clearly displayed may be of value in situation where access to the embedded product is routinely required. All other laser products should carry the appropriate warning labels in accordance with BS EN 60825-1. Recently manufactured lasers should all conform to this Standard. Where lasers and laser systems are not adequately labeled (some American systems have very small labels that are hard to read and do not comply with our BS), they will need to be relabeled. Institutions should note that for mains powered equipment the labeling of lasers in future will have to comply with European Standards and any institution obtaining a laser directly from the US will assume the responsibilities of the importer and supplier.

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Designation of Laser Areas The points of access to areas in which Class 3B and Class 4 lasers are used must be marked with appropriate warning signs. There may also be experiments where open beam work with modified Class 1M/Class 2M or Class 3R lasers are used that will also warrant the display of appropriate warning signs. Laboratory Design The following considerations relate mainly to the use of Class 4 lasers but some may be appropriate for Class 3B devices as well, or, as general specifications for a laser laboratory. If practicable the laser laboratory should have a high level of illumination that will reduce the risk of stray laser light reaching the retina. Windows should be kept to a minimum and may need to be covered or protected by blinds. These should be non-reflective and may need to be fireproof where higher-powered lasers are used. Walls, ceilings and fittings should be painted with a light coloured matt paint to enhance illumination and minimize specular reflections. Reflecting surfaces such as the use of glass-fronted cupboards should be avoided. Ventilation is important especially with higher-powered lasers if cryogens are used, or if toxic fumes are produced that need to be extracted and in this case it is important that the extraction is very close to the source. Facilities may also be needed for the handling of toxic chemicals that are associated with some dye lasers. The laboratory should be equipped with appropriated firefighting equipment. Electrical supplies, switch and control gear should be sited in order to:

• Enable the laser to be shut down by a person standing next to the laser. • Enable the laser to be made safe in an emergency from outside the laser area if

reasonably practicable. • Prevent accidental firing of a laser. • Provide an indication of the state of readiness of the laser. • Enable personnel to stand in a safe place. • Provide sufficient and adequate power supplies for all ancillary equipment and

apparatus so that the use of trailing leads is minimized. Once, it is explained in a little way about the lasers and their hazards, it will be good write the general rules that have to be follow in the lab.

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• Never stare into a laser beam, unexpanded or not. • Never look at specular reflections of a laser beam.

If present try to minimize or eliminate If this is not possible understand the reflective nature of material

• Wear laser safety goggles at all times. Choose the appropriate eyewear for wavelengths you are working at. Choose the appropriate density for total or partial block.

• Never point a laser at anyone • Never move a laser while it is switched on.

Never move any associated equipment such as power supplies or gas bottles when laser is on.

• Never allow visitors into the lab without your permission. Rectrict number of visitors Ensure that visitors are properly briefed into the hazards. Ensure the visitors are sufficiently protected (eyes and skin).

• When entering a laser lab always knock loudly and wait for a signal to enter If no signal is given enter cautiously.

• Do not touch electrical supplies. • Keep walkways clear at all times.

In case of an emergency involving the laser, electrical equipment, personnel, fire, etc:

Lasers

• Switch off the unit immediately by hitting the RED stop button located on the

laser power supply and the laser head • Where possible isolate the laser from the supply by disengaging the main

isolator • If appropriate, sound the FIRE alarm • In case of injury due to exposure to laser radiation (particularly to the eye) or

suspected exposure, the injured person should be taken to A&E at ARI, Foresterhill and treated appropriately. Details of the laser should be taken along to help the diagnosis and treatment.

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Electrical equipment

• Isolate the equipment from the electrical supply immediately; beware of wet areas in the vicinity

• Do not touch injured party until the isolation is complete. YOU run the risk of injury if you fail to do so

• If appropriate sound the FIRE alarm • In case of injury, the injured person should be taken to A&E at ARI,

Foresterhill and treated appropriately.

And finally, a list of several important rules for lasers and optical systems.

• Ensure that anyone present in the room is wearing protective eyewear. • Operate the laser on its lowest possible setting (as indicated on the laser power

supply). • Remove watches and reflective jewellery before any alignment activities begin. • Check on the stability and rigidity of all optical mounts, beam blocks and stray

beam shields. • Ensure that the unexpanded beam cannot be directly accessed by the eye or with

optical aids. • Make sure the beam can never rise to eye level. • Do not bend down to eye level. • When placing mirrors into the beam, always start with mirrors facing

downwards. • Never move a mirror unless you know where the beam will go. • Do not direct laser beams outside the confines of the table. • Do not direct stray laser beams outside the confines of the table. • Enclose all beam paths as far as possible. • Ensure that stray laser light cannot leave the room. • The arrangement should be inspected by lab supervisor before the laser is turned

up to full power. • Post a warning sign outside alerting visitors and colleagues to your activities. • When using high power gas lasers (e.g. argon or CO2).

Never place highly reflective material such as copper or aluminium in the beam path unless you are sure of the requirements needed to process/study/use the material.

Never place material that will cause damage to the lens system e.g. wood, perspex, and brittle materials into the beam path.

• Never open lasers, controllers or transformers without supervision since most lasers use high voltage supplies.

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• Be aware of presence of high pressure tubes, toxic chemicals, gases, etc.

• Never attempt to operate a water or air-cooled laser with cooling turned off. When powering down a water-cooled or air-cooled laser the coolant is

turned off last.

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