Khoja of Science in Engineering in the University of Nairobi

INSTABILITY PROBLEMS OF CIRCULAR SAWS USED IN TIMBER

INDUSTRYw

Fatehally Ebrahim\Khoja

A thesis submitted in part fulfilment for the Degree of Master

of Science in Engineering in the University of Nairobi

(May 30th, 1973)

UNIVERSITY O F NAIROBI LIBRARYi i i in i i m i n0146992 3

I hereby declare that this thesis has not been

submitted for a degree in any other University

F.E. Khoja

______ ___________

ACKNOWLEDGEMENTS

The writer wishes to express his gratitude to

Mr. M. Vasudevan, Senior Lecturer in Strength of Materials and

Experimental Stress Analysis, who suggested the approach dealt

in this thesis for the study of the instability of saw blades and

gave helpful suggestions and guidance during the progress of the

investigation. The writer also takes the opportunity to thank

Mrs. Y.F.E. Khoja who typed the manuscript.

Acknowledgement is made to Timsales Limited Kenya and in

particular to Mr. T. Marroncelli for allowing the writer to make

use of the facilities of the Company in getting the saws cut the

timber at various intervals and for trying out the improved

technique suggested by the findings of this thesis. The writer is

also indebted to the workshop technicians of the Department of

Mechanical Engineering, University of Nairobi, for helping to build

the test rig and allied experimental accessories.

TABLE OF CONTENTS

PageList of figures ........................................... ^ )

List of Tables .............................................

Abstract ................................................... (v)

CHAPTER 1 INTRODUCTION ............................. 1

CHAPTER 2 REVIEW OF PREVIOUS WORI ....................... 6

2.1 Tensioning of circular saws ................. 6

2.2 Delaying the onset of Plastic Buckling ...... 12

CHAPTER 3 SCOPE OF THE PRESENT INVESTIGATION ............... 16

CHAPTER 4 EXPERIMENTAL W O R K .......................... 18

4.1 Experimental Programme ...................... 18

4.2 General Description of Apparatus ............ 18

4.2.1 Description of saw mill equipment.......... 18

4.2.2 Description of laboratory equipment ........ 19

4.3 Experimental procedure ....................... 23

4.3.1 Topographical pattern of virgin blade ...... 23

4.3.2 Topographical pattern after 150 and 300hours of use ................................ 24

4.3.3 Topographical pattern for a three pindriven blade ................................

4.3.4 Calibration of shackle ......................

4.3.5 Elastic analysis ............................ 25

4.3.6 Photoelastic analysis ....................... 27

CHAPTER 5 EXPERIMENTAL RESULTS ........................ 40

5.1 Linear plots from Topographical pattern..... 40

5.1.1 Linear plots for one pin driven blade ....... 40

5.1.2 Linear plots for three pin driven blade ..... 40

5.2 Southwell's plot 41

Page5.2.1 One pin supported blade.................... 41

5.2.2 Three pin supported blade............. 41

5.3 Photoelastic stress patterns ............... 41

5.3.1 One pin and three pin supported disc (5") ... 41

5.3.2 One pin and three pin supported disc (4") ... 41

CHAPTER 6 ANALYSIS AND DISCUSSION...................... 97

6.1 Analysis of the linear plots for one pindriven blade ............................... 97

6.1.1 Region of deformation ...................... 97

6.1.2 Region of maximum deformation .............. 97

6.2 Analysis of the linear plots for three pindriven blade..... .......................... 98

6.2.1 Region of deformation ................. 98

6.3 Discussion of linear plots ................. 98

6.4 Analysis and discussion of Southwell plots .. .100

6.5 Analysis of photoelastic patterns anddiscussions ................................ 100

6.5.1 Large disc - single pin drive .............. 100

6.5.2 Small disc - single pin drive .............. 101

6.5.3 Three pin drive for large and small discs ... 102

6.6 Conclusions ................................ 102

CHAPTER 7 RECOMMENDATIONS FOR FUTURE WORK .............. 106

APPENDICES :

Appendix 1 TABLES OF OBSERVATIONS AND RESULTS ....... 107

Appendix 2(i) CONNECTION OF BRIDGE FOR CANCELLING EFFECT 125

Appendix 2(ii) FORCE AT TOOTH .......................... 128

Appendix 2(iii) BACKGROUND TO SOUTHWELL'S PLOT .......... 129

REFERENCES ................................................

(i)

LIST OF FIGURES

Caption

The Wadkin Saw ..................................

The Experimental Rig ............................

The Photoelastic Bench ..........................

The Radial Arm ..................................

The Perspex Collars and Supports ................

The Radial Arm on Blade .........................

The Blade on Lathe ..............................

The Loading Shackle .............................

The Shaft, Collars and Nuts .....................

The Dillon Machine for Calibration of Shackle ....

Calibration Curve ...............................

Topography of one pin driven Blade :

(i) Virgin Blade ............................

(ii) After 150 hours of use ..................

(iii) After further 150 hours of use ..........

Linear plot of one pin driven blade .............

Linear Plot of one pin driven blade .............

Linear plot of one pin driven blade .............

Topography of three pin driven Blade :

(i) Virgin Blade ............................

(ii) After 150 hours of use ..................

Linear plot of three pin driven blade ...........



Southwell Plot (one pin drive, pin at 0° position)

Figure Paae

Ka) 29

1(b) 30

1(c) 31

2(a) 32

2(b) 33

3 34

4 35

5 36

6 37

7 38

8 39

9(a) 42

9(b) 43

9(c) 44

10(a) 45

10(b) 46

10(c) 47

11(a) 48

11(b) 4912(a) 50

12(b) 51

12(c) 52

13(a) 53

(ii)






















Southwell Plot (three pin drive)....................

Southwell Plot (three pin drive) ..................





Figure Page

13(b) 54

13(c) 55

13(d) 57

13(e) 57

13(f) 58

13(g) 59

14(a) 60

14(b) 61

14(c) 62

14(d) 63

14(e) 64

14(f) 65

14(g) 66

15(a) 67

15(b) 68

15(c) 69

15(d) 70

15(e) 71

15(f) 72

15(g) 73

15(h) 74

16(«). 7516(b) 76

16(c) 7716(d) 78

16(e) 7916(f) 80

(H i )

Fringe pattern for 5" disc (one pin drive, pin at 0 position) ....................................

Fringe pattern for 5" disc (one pin drive, pin at 60 position) ..................................

Frigge pattern for 5" disc (one pin drive, pin at 120 position) .................................

Fringe pattern for 5" disc (one pin drive, pin at 180 position) ...................................

Fringe pattern for 5" disc (one pin drive, pin at240 position) ...................................


Fringe pattern for 5" disc (three pins at 0°,120° and 240°) ..................................

Fringe pattern for 5" disc (three pins at 60°,180 and 300 ) ..................................

Fringe pattern for 4" disc (one pin drive, pin at0 position) ....................................


Fringe pattern for 4" disc (one pin drive, pin at 120 position) ..................................


Frigge pattern for 4" disc (one pin drive, pin at240 position) ...................................


Fringe pattern for 4" disc (three pins at 0 ,120 and 240 ) ...................................

Fringe pattern for 4" disc (three pins at 60 ,180° and 300°) ...................................

Figure Page

17(«) 81

17(b) 82

17(c) 83

17(d) 84

17(e) 85

17(f) 86

18(a) 87

18(b) 88

19(a) 89

19(b) 90

19(c) 91

19(d) 92

19(e) 93

19(f) 94

20(a) 95

20(b) 96

LIST OF TABLES

-ap.tion No. Page

Shackle calibration data........................... 1 108

Southwell plot d a t a ................................ 2(a) 109

Southwell plot d a t a ................................ 2(b) 110

Southwell plot d a t a ................................ 2(c) 111

Buckling loads from Southwell plots ............... 3 112










Southwell plot d a t a ............................... 8(b) 122

Southwell plot d a t a ............................... 8(c) 123


(iv)

(V)

ABSTRACT

In this thesis, a study has been made regarding the serious

problem of buckling of circular saw blades used in timber industry.

Measurements of deformations of a buckled saw blade (buckled under

actual working conditions) has been made in detail and topographical

patterns obtained. These have been plotted linearly. Analysis of

these plots showed that the pin or pins were operative at all times

and not only when encountering a hard knot. It also showed that

a three pin driven blade could be much more stable than a one pin

driven blade (the usual commercial practice). This was tried out in

an actual sawmill and found successful. A beginning at least has

also been made to understand the problem from a more scientific back

ground by trying to apply Southwell’s approach of determination of

buckling limit load of structures to the present problem.

1

CHAPTER 1

INTRODUCTION :

1.1 Many problems are associated with the economical use of the

various types of saws used in the timber industry. There are

basically three types of saws in use : the Frame Saw, the Band Saw

and the Circular Saw.

The frame saw, as the name implies, is basically a framework

type of structure into which the blades, for cutting the timber,

are fixed. It is capable of making multiple cuts and hence has a

high production rate and is easy to service. However, its capital

cost is very high. The band saw has many advantages from the point

of view of accuracy of sawing, the power required and the relative

ease of servicing the saw. Unfortunately this type of saw is also

very expensive (about £5,000 as compared to about £1,500 for a

circular saw) and would be impractical in a country where labour costs

are low and where a large number of small saw mills is desirable.

The ideal type of saw for such circumstances is the circular saw.

The circular saw, however, has very many serious problems regarding

the servicing of the blade.

In order to understand the problems associated with the

circular saw blade, the author consulted a local saw-milling firm

(l)* and received the following information in connection with a

Wadkin sawing machine used by them. The blade most frequently used

♦Numbers in curved brackets refer to references

2

on this machine was a straight cut, spring set 20 inches diameter

12 G thickness blade. It was mainly used for cutting timber to size.

The blade was driven through one pin by a 5.5 H.P., 3 P.H., 50 c/s

motor running at 1420 R.P.M.

The blade required servicing after every 5 to 6 cutting hours.

During servicing the teeth were sharpened and set, care being taken

that the correct bevel angle, clearance angle and set were obtained

(2). If necessary, gulleting (2) also had to be done. At each of

these intervals, a check was also made for the straightness or flat

ness of the blade to see whether the blade had permanently buckled.

If the blade had buckled, it was straightened out by means of

manually directed hammer blows by a skilled craftsman. It is this

problem viz the straightening of the circular blade which is still

wrapped up in the mysteries of a craft, which is the most baffling

one in the timber industry. It is technically known as "saw tensioning".

This problem also occurs in the blades used in the frame saw and the

band saw. However, in these cases, a scientific and systematic

method has been previously formulated.

On an average, a blade of the above size requires tensioning

about once every week. A badly distorted blade would require about

an hour for tensioning and a less damaged one about half an hour.

This, in terms of loss of timber cut, tensioning cost and overheads

of a workshop, amounts to about Shs. 6,000/- a year for one circular

blade of this size. The saw mill in question had three of these saws

in operation at the time of this investigation. The cost of tension

ing the above size of blade was about Shs. lo/-. A blade of say

3 ft. or 4 ft. in diameter would require tensioning about twice or

3

thrice per week and the cost in this case would be about Shs. 20/-

for each tensioning. It is needless to say that an investigation of

the problems of tensioning circular saw blades is very much needed.

The above problem has to some extent been overcome by the use

of ’inserted saw tooth' blades. However, these increase the kerf

width (2) and are very expensive. The increase of the kerf width gives

a low conversion ratio (the ratio of volume of timber cut to the

volume prior to cutting) and is undesirable.

1.2 In the case of the circular saw blade the thickness of the

blade is indeed very small compared to the diameter. This means

that the blade cannot be, laterally, very rigid. The effects of

temperature and intermittent cutting forces acting simultaneously

during cutting will only add to the problem of buckling, even if this

be gradual. In view of this, it is erroneous to believe that the

problem of tensioning can be got rid of altogether. However, it seems

that there are two areas or fields of approaching the problem. One,

is to study the problem of tensioning with a view to placing it on

a scientific and systematic basis as exists for band saw blades. The

other is to study the problem with the intention of trying to delay

the onset of buckling.

Dugdale (3, 4, 5) in his investigations on the effects of

internal stress on elastic and flexural stiffnesses covers to some

extent both of these areas. He also looks into the mechanics of saw

tensioning. Mote (6) investigates the problem from the standpoint

of total potential energy. Johnston (8) seems to be the only person

as far as the author is aware, who has attempted to delay if

4

not eradicate the need for tensioning. However, his two-piece

experimental saw is still in the development stage.

The aim of the present work is to show that the onset of

buckling can be delayed by the variation of the method of support of

the blade. Three ways have been adopted to show that this is

possible, (a) The most practical one viz actually using the blade

supported at multiple points to cut timber under industrial working

conditions and measuring the plastic deformations after successive

use and comparing the patterns. The conclusions arrived at cannot

be very comprehensive or convincing due to the obvious necessity of

conducting such tests for limited hours only and the limited number

of different sizes of blades that can be used in the industrial

facility available, (b) An indirect method of conducting static

tests in the laboratory on the blade supported in different ways,

measuring deformation patterns and predicting loads to start buckling.

The method of prediction is on the lines of a method suggested by

Southwell in his note "On the Analysis of Experimental Observations

in Problems of Elastic Stability". Details of Southwell's method

and review of work done by various authors using this method is given

in Chapter II. Conclusions arrived at using Southwell’s method is

again limited since it is done under laboratory conditions not

simulating all the practical working conditions of the blade and

deformation patterns are only elastic. It may be asked what value

the determination of such a stability limit has to the present

problem. The answer is (i) plastic instability is preceeded by

elastic instability, (ii) by comparing elastic instability loads

under different support conditions a relative picture of the

5

advantages of such support conditions can be obtained without

deforming the structure plastically. (c) An indirect method

of obtaining elastic stress patterns of the blade supported under

different support conditions and different points of load applica

tion using the Photo Elasticity technique. This method has again

the same limitation as (b) above. The only justification of

conclusions from (b) and (c) is that they seem to be practical ways

to study a problem, a complete solution of which requires much more

advanced theoretical and experimental work.

6

CHAPTER 2

REVIEW OF PREVIOUS WORK :

2.1 Tensioning of circular saws :

It was mentioned in the previous chapter that the art of

tensioning was wrapped up in the mysteries of a craft. This is in

fact very true. The practice of the craft varies from one saw

doctor to another. Being a craft without any scientific or systematic

explanation, it can sometimes be annoyingly slow and expensive,

particularly where larger blades are involved. In view of this,

various authors have investigated the problem in its several aspects.

Basically, the art of tensioning is the careful manner of imposition-

ing internal stresses into the blade, so as to get a stiffer and

flatter blade. It is this reasoning which has tempted many authors

to investigate the effect of internal stress on a blade. Others have

persued the matter from the point of view of temperature effects and

temperature stresses.

Dugdale (3) investigates the effects of internal stress on the

flexural stiffness of discs. A stiffness coefficient is defined for

experimental use. He also derives a stiffness coefficient to be used

for theoretical calculation of stiffness. The latter coefficient is

derived using the energy method and considering internal stresses,

membrane stresses and the interaction between the two.

Experimental values of stiffness coefficients are obtained by

two methods, both of which basically involve the loading of the disc

laterally and measuring lateral displacements. The values of load

7

and deflections are substituted into the expression for the

experimental stiffness coefficient and values of stiffness

coefficients obtained for various modes of deflection of the disc.

For the theoretical calculation of the stiffness coefficients a prior

knowledge of the internal stresses was essential. He experimented

with four discs each of 12 inches diameter but having different

thicknesses and different mechanical (i.e. hammerings) and heat

treatments. Foil-type resistance strain gauges were fixed circum

ferentially at various radii in pairs, one opposite the other on

either face of the disc to obtain an average strain. Radial and

circumferential saw cuts were then made around each gauge to release

internal strains. Radial and hoop stresses were deduced from

circumferential strains by making use of the equation of radial stress

equilibrium. He then uses these internal stress values to calculate

the stiffness coefficients.

The same author in another paper (4) again investigates the

effect of internal stress on elastic stiffness. However this time

he uses not only internal stresses but also radial gradients for

calculating the stiffness coefficients. He does this for one disc

of the same diameter as in the previous case and investigates the

disc in the stress-free condition and after the impositioning of

internal stress. The investigation was carried out for various modes

of deflection.

The results for the stiffness coefficients from both the above

investigations indicate that internal stress affects the stiffness

of discs. The extent to which the stiffness coefficients for the

various modes of flexure are affected due to the varying heat and

8

mechanical treatments to the different discs can be used as a useful

guide for tensioning a blade.

In his paper (5), on the theory of circular saw tensioning,

Dugdale looks into the mechanics of saw tensioning. He derives ways

of computing interned circumferential and internal radial stresses

for two types of blow (hammering) distributions, viz radial and

circumferential over an annulus of finite width. To ensure that the

hammerings were of equal strength, he used a spring actuated

mechanism. The blows were applied with a hammer head having a flat

face in the shape of a narrow rectangle so as to induce only plane-

strain. He uses the computed values of these stresses for both types

of hammerings and using the stiffness coefficient derived in earlier

investigations, calculates the stiffness for the different cases.

Comparison of stiffness values with those found experimentally show

that theory was consistently accurate in respect of circumferential

blows and less accurate in respect of radial blows.

The purpose of this last investigation was to follow in exact

terms the effect of various axi-symmetric distributions of hammer

blows. The main purpose, however, was to gain a better understanding

of the terminology and ways of thinking of skilled operatives, so

that existing practice could be codified and possibly improved.

Mote (6) examines the problem from the viewpoint of energy.

He advocates that the blade be treated as a structural stability problem.

The total energy is divided into four parts viz (i) bending stiffness,

(ii) thermal gradients, (iii) rotational and (iv) tensioning. He

claims that in many instances the blade reaches an undesirable

condition prior to actual buckling. The blade undergoes large

9

amplitude transverse vibrations causing loss of dimensional

tolerance of the work-piece, increased power consumption and heating

of the blade. Thus resonance usually precedes actual buckling and

may indirectly account for the actual buckling itself. He proposes

to overcome this situation by increasing the fundamental frequency

of the blade, which otherwise would be quite close to the forcing

frequency. Toted energy increases or decreases with frequency and

the problem becomes one of increasing the total potential energy.

In the case of rotation, the in-plane stresses are always

positive and can only increase the angular kinetic energy of the

toted energy. The other terms in the expression for total energy

increase or decrease depending upon the nature of the internal

stresses. He investigates internal hoop and radial stresses for

these other potential energy terms and applies it to a blade for

optimum tensioning and stability. He concludes that the blade can

be investigated as a structural stability problem and that it is

optimally tensioned if the fundamental frequency of oscillation is

as large as possible for specified operating conditions.

Lindholm (7) considers the buckling of the circular saw under

symmetrical heat distribution. He is of the opinion that when cutting

timber, the teeth of the saw blade get hot due to friction whereas

the disc otherwise is kept cool by air-cooling. Because of this un

even temperature distribution, temperature stresses occur and cause

the disc to lose its rigidity, and it starts cutting unevenly. He

suggests two solutions for the problem. One, he says, is to heat the

disc at the central part by some sort of felt pads, and cause an even

temperature distribution. The other is to 'hammer' or tension the

10

blade so that initial tensions are introduced and these will

compensate the temperature stresses and cause the disc to remain

stable. In order to find the optimum method of hammering, he

proposes a theoretical study and works through to find a theoretical

buckling temperature.

Johnston (8), as far as the author is aware, is the only person

who has attempted to do away with tensioning altogether. He quotes

Berolzheimer and Best (9) in stating that heat generated at the

teeth of a saw blade while cutting causes high tangential compressive

stress at the rim because of the restraint offered to free expansion

of the heated rim by the cooler central portion. One of the ways to

overcome this would be to induce compressive stresses in the central

portion by hand hammering. This would augment the tensile stresses

at the rim due to centrifugal forces of rotation and allow a greater

thermal expansion of the rim before compressive stress becomes high

enough to cause buckling. However during the process of manufacture

a blade is already subjected to heat treatment and hammering which

leaves the blade in highly complex state of residual stress. Some

of these are so high that any slight hammering for compensation

purposes would cause permanent deformation by exceeding the yield

strength of the material. He therefore claims that a method of

compensation, more accurate than hand hammering, would improve saw

blade action.

On the grounds of having examined the stresses on a blade whilst

cutting, he suggests that if a rim of a saw blade were mounted on a

central disc so as to be able to expand freely with temperature

increase, the cause of instability could be removed. He conducted

11

experiments with two 30 inch diameter, 36 teeth saw blades obtained

from a manufacturer. One was a standard saw blade to which no

alterations were made. The other was processed from a blank without

any hammering or heat treatment. This latter saw was cut and

rejoined so as to form a ring and central disc. The two pieces were

rejoined so as to allow a diametral movement of 0.010 inches without

restraint.

The two piece saw when used to cut an 8 inch cant of white

pine, became erratic and heated seriously in the centre section from

rubbing in the cant. The centre section had to be flattened by

hammering and thereafter smooth operation was obtained in cutting

white pine. Since the hammering of the blade was at cross-purposes

to the purpose of the investigation, a new centre section was

constructed. It was made up of fine gauge mild steel sheet pieces

stuck together with epoxy resin in a glue press and no tensioning

whatsoever was applied. The standard and experimental blade were

each fixed with a strain gauge and a thermocouple at 12^ inch radius.

Studies of strain due to centrifugal forces and temperature

gradients showed that tensile strains acted at the 12̂ - inch radius

for the experimental blade and compressive strains acted at the same

radius in the standard blade. Tensile strains have a beneficial

effect on stability and hence show the experimental blade to be a

definite improvement in theory at least. However on using the blade

to cut timber under identical conditions, the performance of the two

saws was similar. Johnston at this stage decided to change the design

of the experimental saw and no further work was done with it.

12

2.2 Delaying the onset of Plastic buckling :

In the foregoing investigations the effects of internal stresses

and temperature on a blade have been studied. However no work seems

to have been done by anyone to investigate what actually happens to

a virgin blade under successive hours of use. It is the intention

of the author to study a blade after certain periods of use and

from such investigations to find a way of delaying the onset of

plastic buckling of a blade.

Berolzheimer and Best (9) quote Skoglund (17) and Malcolm (18)

who both discuss the importance of adequate collars. The conclusion

is that if the largest possible collar diameter is adopted, the blade

will be better supported and therefore stiffer and less likely to

vibrate. Using very large collars would restrict the size of timber

that can be cut. The author therefore decided to change the method

of support in a different manner after studying what happens to a

blade driven by one pin in the conventional manner. To evaluate the

advantages of methods of support, it would be necessary to have some

method of at least relatively predicting the onset of buckling. To

this end, the author proposes to use Southwell's method for predicting

buckling limit loads. In 1931 R.V. Southwell (lO) devised a method

by which test data from an elastic strut, which was initially imperfect,

could be used to find the stability limit i.e. the critical buckling

load without actually reaching it. He used the usual second order

differential equation for the buckling of struts together with a

term for the initial imperfection. He solved the equation by assuming

a fourier series for the form of the centre line of the initial

imperfection and for the deflection and arrived at the same result as shown in Appendix (2 iii). Plots of ̂ /p against A ( A being the

13

maximum deflection and P the load causing it) could be a straight

line so long as P was not very much smaller than the theoretical

Euler critical load nor was it of such magnitude as to start yielding

at the maximum deflection point. Such plots have come to be known

as Southwell plots. The application of Southwell’s method of finding

the critical buckling load of a strut is well understood and known;

the applicability of Southwell's method to more complicated cases

has been investigated by many authors.

Southwell and Skan (19) theoretically analysed a flat strip

clamped along its edges and subjected to shearing forces so as to

calculate a critical shearing load. Gough and Cox (20) made an

attempt to (i) check the conclusions reached by Southwell and Skan

and (ii) to investigate the possibility of standardising some test

as an "acceptance test" for finding critical loads due to shear. In

their first series of tests the buckling load was determined purely

by visual examination as the load at which waves were first detected

by the distortion of images reflected in the surface of the strip.

This method of predicting the buckling load was not found to be very

successful. The second series of tests were based on the Southwell

plots. In these tests, the growth of the amplitude of waving was

taken as the deflection for increasing shearing loads. The authors

made no attempt to analyse the problem theoretically and to get the

result in the form of an equation which would be amenable to Southwell's

plot. They merely observed values of deflection, as stated above,

and shear loads and constructed Southwell plots. Specimens of the

same thickness but various widths were chosen for the tests. The

lengths of the various strips varied so as to have a constant length

to width ratio. In general, it was found that the buckling loads

found experimentally were in good agreement with theoretical values

14

found by using the expression derived by Southwell and Skan.

Donnell (ll) shows that Southwell's method can be applied to

cases where buckling does not introduce appreciable second-order

stresses. Even in cases where second-order stresses exist, the method

is roughly applicable.

As a first illustration, Donnell considers a hinged strut with

continuous elastic support. He uses the energy method and equates

the strain energy stored in the strut and elastic support to the work

done by the external load. He assumes a summation series for the

form of the centre line of the imperfect strut and a similar series

for deflections due to loading. Using these in the energy equation

mentioned above, he derives an equation exactly similar to that derived

by Southwell. As in the case of Southwell, the equation is a relation

between load, deflection due to load and deflection due to imperfec

tions, and can be exploited experimentally see Appendix (2 iii) .

He next considers a panel hinged on three sides and free on

the fourth. He assumes the deformed shape to be a developable surface

and as before applies the energy principle. However, this time the

assumptions for deflections due to imperfections and loading are

summation series as before but with a factor to account for the second

dimension (plate) in each case. He again arrives at an equation

exactly similar to Southwell's strut equation. In both the above cases,

the theoretical buckling load for a perfectly straight continuously

elastically supported strut and a perfectly flat plate supported on

three edges was known. These were employed in arriving at the

Southwell's equation' for these cases.

15

The same author next considers the case of a plate which is

supported on all four edges. In this case, the differential

equation is not linear as the surface is not developable and

extensional strains are not negligible. He uses the stress function

together with the energy principle and deflection series as before

and obtains a slightly modified form of Southwell's equation. He

claims that this modified equation would be applicable provided the

deflections were small compared to the thickness of the plate.

From the above review of application of Southwell's method to

various cases, the author felt that it could be applied to the present

problem and the results interpreted as a 'yard-stick' to evaluate

the advantages of multiple point support of the blade.

16

CHAPTER 3

SCOPE OF THE PRESENT INVESTIGATION :

The scope of this investigation is to examine the topographical

pattern of a blade in the virgin condition and after usage of 150

hours of cutting and 300 hours of cutting. The reasons for choosing

the intervals of 150 and 300 hours were based on discussions with

the saw doctor at the saw-mill where the experimental work, as

regards cutting of the timber, was done. The work was done on a

20 inch diameter 12 gauge blade as it was most commonly used by them

for cutting medium hardness timber to size.

It was hoped to study the pattern of deformation of the blade

driven by one pin as usual. The saw doctor was instructed not to

tension the blade when it was removed for sharpening, gulleting and

setting of the teeth. Only when the blade was very distorted and

definitely needed a fair amount of tensioning, was he to stop using

the blade. It was at this stage, after about 150 hours of use, that

the author wanted to examine the topographical pattern. The blade

was thereafter deliberately used for a further 150 hours to cut timber,

without being tensioned.

A similar study was carried out for a differently supported

blade (driven by three pins at intervals of 120 degrees). Topographical

patterns of the two cases were obtained. These patterns were also

plotted linearly for the purpose of studying the regions of deformations

and their magnitudes. In order to show the improvement and

superiority of one method of support over another, Southwell's plots

were constructed for both types of support. To this end, some static

17

elastic tests for both types of support were carried out in the

laboratory in addition to the work done at the saw-mill.

To further show the superiority of a three pin support over a

one pin support, a photo-elastic analysis was also carried out in the

laboratory, under static conditions similar to those of the elastic

test.

The elastic and photo-elastic tests carried out in the

laboratory on new blades and the plastic deformation patterns

obtained from the blades used in the saw mill conclusively showed

that a definite improvement was made by merely changing the method

of support of a cutting blade. The results, incidently, also proved

false the belief that a blade is being driven by friction between the

collars and itself and that the pin only comes into use when a very

hard knot is encountered in the timber.

18

CHAPTER 4

EXPERIMENTAL WORK :

4.1 Experimental Programme :

The experimental work was divided basically into three parts.

The first part was to obtain topographical patterns for a one pin

driven blade after 150 hours and 300 hours of use and for a three pin

driven blade after 150 hours of use. The cutting of the timber was

done, at the saw-milling firm referred to earlier, under the super

vision of the author and a saw doctor, by a saw operator. The second

part was to carry out static elastic tests and the third part was

to carry out photoelastic analysis, both of which were done in the

laboratories of the department of mechanical engineering.

4.2 General Description of Apparatus :

4.2.1 Description of saw-mill equipment :

The saw at the mill was a Wadkin circular saw-bench, type

B.S.W. 20" fig. £l(a).J The blade was driven through a system of

pulleys and V belts by a 5.5 H.P., 3 P.H., 50 c/s motor running at

1420 R.P.M. The pulley attached to the motor shaft was a multiple_ , '/belt 5 ̂ .diameter pulley and that attached to the blade shaft was also

. aa multiple belt -4- diameter pulley. This gave the blade a fixed

speed of 2200 R.P.M. The entire system of blade shaft, motor and

pulleys was linked to a linkage system. By turning the handwheel,

this system of linkages was activated and the blade protrusion increased

or decreased above the surface of the table. The guide is adjustable

and can be utilised to cut timber to various sizes.

19

4.2.2 Description of Laboratory equipment :

The rig had been designed so as to be useful for carrying out

tests on blades ranging from 6 inches in diameter to 4 feet in diameter.

It is basically a box-like structure and is shown in fig. £l(b) .J

It is made up of 2” x 2” x l/4" angle iron welded together to form

a rigid structure. At one end of the rig is welded on a frame (A)

for holding the blade. Every precaution was taken to weld this part

rigidly to the box-like structure so that it would not bend upwards

when the blade was loaded and hence give a false reading. On top

of the rig are fixed three supports. Two of these (B) are merely

simple supports for carrying the loading bars (c) and the loading shackle (D). The third support (e ), which is stronger in design,

is for supporting the load. It is capable of taking a safe working

load of upto 2,000 lbf. The supports (b) and (E) are fixed to the

rig by means of bolts in slotted holes so that the loading bars can

be aligned in line with the blade before applying a load to the blade.

The turnbuckle : A cast 5/8” turnbuckle was purchased locally

and was used to apply the load. The end of the turnbuckle, with the

loop, was retained by a plate on the load bearing support (E) of

the rig, fig. f"l(b)J . A groove on the retaining plate prevented the

loop from rotating whilst applying load. The other end of the

turnbuckle was fixed into the clamp of the proving ring.

The proving ring : If the turnbuckle alone were used, a few

turns would cause a large deflection in the tangential direction and

hence buckle the blade before any substantial readings could be taken.

To overcome this difficulty, a proving ring was incorporated to allow

for some latitude. The ring was made of mild steel and has a mean

20

diameter of 6 inches. The width of the ring was made 3/4" and the

thickness .183 inches. This allowed for a deflection along the mean

diameter of the ring of about 0.7 inches. Two mild-steel grooved

clamps connected the ring to the turnbuckle and the loading bars

respectively. This can be clearly seen in fig. [Kb)].

The shackle : A tool steel shackle was used for measuring load,

details of which are shown in fig. P 5 J . The shackle satisfies the

following conditions (12) :

(a) maximum sensitivity to axial loads

(b) minimum sensitivity to bending moments

(c) loaded well below the elastic limit

(d) uniform stress distribution at gauge area

(e) free from drift.

A uniaxial wire resistance strain gauge was fixed to each flat

side of the shackle. In order to obtain maximum sensitivity, the

gauges were connected to a strain-indicator in the usual manner so

as to form a bridge circuit which would cancel any bending effects.

[Appendix 2 (i)J . The shackle was fixed to the loading bars by means

of clamps and pins. These clamps were designed so as to nullify any

bending moments that might be caused in the loading bars due to any

slight misalignment. A PEEKEL T 630 strain indicator was used for

measuring the readings of the gauges on the shackle.

The tooth clamp : This part was made of two flat mild steel

plates 3/l6 inch thick x l£ inch wide and about 6 inches long. The

plates are screwed onto a mild steel block which has a 5/8" B.S.W.

thread in it for the loading bar to be screwed in.

21

The blade clamp : This vas made out of mild steel and was

basically a boss welded onto flat plate which had holes in it for

bolting onto the frame (a ) of the rig. Provision had been made on

the boss for either one or three pins to hold the blade. The shaft

collars and nuts which were used for fixing the blade into the clamp

are shown in fig. ̂ J .

Magnetic base dial gauge : A magnetic base dial gauge was used

for measuring deflections during the elastic tests. The magnetic

base was attached to the framework of the rig and adjusted so as to

enable the dial gauge to take readings at various stations fig.

The radial arm for obtaining the topographical pattern of the

(a ) in which a square groove is made. A threaded block carrying a

dial gauge (b) slides in the groove. The aluminium bar is mounted

onto a 2" x "2 x l/4" piece of bright drawn mild steel angle. The

whole is mounted onto the blade shaft by means of a block bracket (C)

which fits exactly on a reduced portion of the shaft holding the

blade. It can be clamped in any angular position by means of a 1/4”

B.S.W. bolt, on the block bracket, which screws onto the reduced

portion of the shaft. The radial movement of the dial carrying block

is achieved by means of a lead screw and knob (d ). This arm was

tested to be accurate within .05 mm from one end to the other.

has an ordinary light source from a Phillips bulb. The light is

emitted from a l/4" diameter hole in order to have a point source.

The Analyser and Polariser are clearly shown in the figure. The two

lenses are employed to get parallel light. A plate camera is

blade It is fabricated from an aluminium bar

The photoelastic bench : The bench is shown It

J

22

incorporated on the bench as shown in the figure. It has a filter

attached to it which can be swung in and out of position. The rig

for holding and loading the model is a very versatile one. It has

provisions for vertical and horizontal movements, and for tension

and compression loading. The rig can be moved without upsetting the

rest of the bench. This was very useful when changing the position

of the model each time during the analysis.

The model : a 5" and 4" diameter disc was cut out of photo

elastic material (c.t. 200) which was l/8" thick. A central hole

in diameter and a pin hole l/4" in diameter was made in the disc.

The cutting of the disc was done with a special high speed cutter

used for photoelastic work in order to avoid residual stresses. The

centre distance between the central and pin holes was made 5/®

Load application holes were made at various positions on the periphery

of the disc (these were l/8" in diameter).

The collars : Two collars thick and 2" in diameter were made

out of perspex. The collars had a central diameter hole and a

series of l/4" diameter holes for the pin. These latter holes were

at 60° intervals. The collars also had a 1" wide l/4" deep slot on

one side of each collar only. These slots were used to hold the

collars stationary with respect to the perspex supports fig. ^2(b)J.

These supports were fixed to the rig on the bench by means of pins

supplied with the rig.

A clamp made from Mecano set parts was fixed to the disc. A

steel wire was fixed to the clamp and passed over a pulley fixed to

the side of the rig fig. H . At the other end of the wire, a

weight carrier was fixed.

23

4.3 Experimental Procedure :

4.3.1 Topographical Pattern for Virgin Blade :

A new 20" diameter Spear & Jackson blade with a 1 l/4" diameter

central shaft hole was used for the purpose of investigation. The

blade was l/8" (12 G) thick and was used with a collar of 4^" diameter.

It had a provision for being driven by one pin. (The centre distance

between pin hole and shaft hole was 1 3/8").

The blade was marked off into a grid pattern. This was done

in the following fashion : Concentric circles were marked in felt

pen ink relative to the centre of the blade. They were marked at

intervals of 1" radially. This gave 8 stations, starting from the

zero station concentric circle at the collar diameter. These concentric

circles were intersected by radial lines emanating from the centre

and marked off at intervals of 10 degrees. This method of marking

gave 288 sub stations. The measurements of deflections of the surface

of the blade was made using the rig and the radial arm described earlier.

The blade and radial arm was mounted as shown in fig. and

the dial gauge was adjusted to give zero reading at station 1 on the

0° radial line. The lead screw was turned by means of the turning

knob. The reading on the dial gauge was noted at station 2 and in

this manner all stations were covered upto and including station 7.

The radial arm was then released from the 0° radial line and positioned

at the 10° radial line. Here again, the dial gauge was adjusted to

zero at station 1 and readings at other stations noted relative to

station 1. The pattern of the entire blade was noted in this manner.

Readings towards the plane of the blade were considered positive

24

deflection and away from the plane of the blade as negative deflection.

This being relative to an observer facing the markings on the blade.

The blade was thereafter mounted between centres on a lathe,

fig. £ 4J . A magnetic base dial gauge was suitably placed on the. 0 apron and the dial gauge was adjusted to read zero at the 0 and zero

station mark. The blade vas then rotated by hand so as to note the 0 0 0readings at 10 , 20 , 30 etc. for the zero station concentric circle

until 360° were covered. After this, the magnetic base dial gauge

was moved from zero degree, zero station, to zero degree station 1

by using the cross traverse. The reading at station 1 was noted and

the blade rotated by hand so as to obtain the readings for various

10° intervals at station 1. These were the new readings for the

various first stations (the initial readings were all zero). The

readings originally taken by the radial arm for the various stations

were then adjusted relative to these new readings. In this manner,

the entire pattern of the blade was related to an absolute reference

point at 0° and zero station. The topographical pattern of the virgin

blade obtained as described above is shown in fig. I 9(a)!.

4.3.2 Topographical pattern after 150 and 300 hours of use :

The blade was sent to the local firm of saw-millers who used

it on their saw for cutting medium hardness timber with as few knots

as possible and returned it after 150 hours of use. The topographical

pattern of the blade was obtained in the same manner and for the same

stations as explained in 4.3.1 . The pattern obtained is shown in

fig. ^9(b)J • The blade was again returned to the firm for use of

another 150 hours bringing the total usage to 300 hours. The pattern

obtained at the end of 300 hours of use is shown in fig. j^9(c)J .

25

4.3.3 Topographical pattern for a three pin driven_blade :

A similar blade as for the one pin case was used and holes

made for 3 pins at 120° intervals. A similar pattern as explained

in 4.3.1 was obtained for the virgin blade and is shown in fig.

|ll(a)J . The blade was used for 150 hours but this time with three

pins driving it. On return, a similar topographical pattern as

before was obtained and this is as in fig. |jLl(b)J .

4.3.4 Calibration of Shackle :

Prior to carrying out elastic tests, the shackle had to be

calibrated. For this purpose, it was mounted in a bench model Dillon

tensile machine, which had a dynamometer of capacity 500 lb. fig.r •l> ] . The leads from the two strain gauges mentioned earlier were

connected to a "Peekel T 630" strain indicator. Dummies were also

connected to the strain indicator. The connections formed a double

active bridge circuit on the indicator as explained earlier.

The shackle was loaded in pure tension in increments of 50 lbf,

from zero upto 500 lbp# Values of strain were read for each increment

of load (table l) and plotted against the load to give a calibration

curve as shown in fig. ^8^.

4.3.5 Elastic analysis :

The blade was mounted between the two arms of the fork in the

rig as shown in fig. H . Also fig. ^6^ shows the detail of the

shaft, collars and nuts used for holding the blade. The radial line

passing through the centre of the shaft avid the pin hole on the blade

was regarded as the zero degree radial line. The supports (b ) and (E)

were adjusted so as to get the loading bars, shackle, proving ring

and turnbuckle in line with the blade.

26

From the Dynamic test results (explained later), the

plastically damaged region under practical conditions was found to

be between 60° and 300°. This region was chosen for elastic analysis

and to get representative ideas of the start of buckling in this

region, the directions of 60°, 190° and 300° were chosen. (Part of

the framework was in the way and therefore the 180° direction could

not be chosen). Load was applied at the tooth which correspondedo . .with the 0 position. Before applying load, the dial gauge was

positioned at 60° and station 7. The dial gauge was adjusted to give

a zero reading. A load was applied by turning the turnbuckle and

the reading on the strain indicator and dial gauge noted. The load

was increased in stages and the readings on the dial gauge and strain

indicator noted. This procedure was repeated for other stations on

the 60° radial line. A similar procedure was also repeated for the

stations on the 190° and 300° radial lines. In some cases, it was

difficult to take readings below the 4th station as the rig was in

the way and the stations were inaccessible. These readings are given

in table ^2(a), (b) and (c)^ . Southwell plots of versus A for

these stations are shown in figs. £l3( a) to (g)J . The buckling loads

for these stations are in table (3).

In order to simulate the application of the load at various

points on the periphery of the blade (the practical condition of

timber cutting) the position of the pin was moved through (a) 120°

and (b) 240°. This was done by turning the blade through 120° and

240° respectively so that the 0° radial line (where the pin hole was

situated) was fixed at 120° and 240° respectively to its original

position. The load then being applied at 120 and 240 respectively.

The experiment of measuring the deflections at 60 , 190 and 300 for

27

the above two cases was repeated as for the pin at 0°. The readings

are given in tables ^4(a), (b) and (c)^ and ̂ 5(a), (b) and (c)^ .

Southwell plots for these cases are shown in figs. £l4(a) to (gjj

and f*15( ») to (h)] . The minimum buckling loads for these cases are

shown in tables (6) and (7).

The above experiment was repeated with the blade held by three

pins at 0°, 120° and 240° simultaneously. In this case, the load

was applied at the 0° radial line as before. However, the blade was

not re-orientated for application of load at 120° and 240° as this

would not alter the situation as there were three pins supporting the

blade. The readings for the 60°, 190° and 300° directions are shown

in tables ^8(a), (b) and (c)^ . The Southwell plots for this case

are shown in figs. £l6(a) to (f)]. The buckling load values are as

s hown in t able (9).

4.3.6 Photoelastic analysis :

The 5" diameter disc was clamped between the perspex collars and supports with a 7/l6" bolt. The disc (model) was positioned so

that the pin hole was in the 0° position and a l/4" diameter mild steel

pin was inserted into the hole passing through the disc and the collars.

The whole was fixed to the rig which was then put into position along

the photoelastic bench.

The light source was switched on and the weights gradually

increased on the weight carrier. At 100 lb£> a reasonable stress pattern

was obtained with three to four fringes. The filter was put into

position in front of the camera lens and a photograph taken. The

load was removed and the blade re-positioned so that the pin passed

through the 60° pin hole (i.e. the blade had moved through 60 against

28

the load). The blade was again loaded to a maximum of 100 lb£ and

a photograph taken. The above procedure was repeated for the pin at

120°, 180°, 240° and 300°. The photographs for the various positions

are shown in figs. [l7( .) to {£■)] .

Two further pin holes were made on the 5" diameter disc

equally spaced with the first hole and on the same pitch circle

diameter. The disc was then supported with three pins; one at 0°o oand the other two at 120 and 240 respectively. The disc was loaded

owith the load application point at the 0 position and a photograph

taken of the fringe pattern. The disc was then re-positioned so. . o O . .that the load application point was at 60 from the 0 pin position,

i.e. the point of application of the load was midway between 0° and

120° pin positions. A photograph was taken for the fringe pattern

with the load at this point of application. The photographs for the

above two positions of a three pin supported disc are shown in figs.

^18(a) and (b)j .

The entire above procedure was repeated for the 4 " diameter

disc of the same photoelastic material as the 5" disc. However, the

central hole diameter and pin hole diameter were kept the same. The

same collars were also used. The resulting fringe patterns for the

one pin supported and three pin supported discs for various positions

of loading are shown in figs. [l9(a) to (f)J and [20(a) and (b)]

respectively

THE WADKIN SAW

FIG. 1 (a)

to oth clamp

THE

support (E)supports (B)

\

COo

fig t(b)

E X P E R I M E N T A L RIG

POLARISER

LIGHT SOURCEANALYSER

THE PHOTO ELASTIC BENCH

FIG. 1 (c)

PLATE CAMERA

THE RADIAL

A R M

33

PULLEY

CLAMP

WEIGHT CARRIER

DISC

SUPPORTS

COLLARS

THE PERSPEX COLLARS AND SUPPORTS FIG. 2 (b)

fig 3

B L A D E

oco

mCM

f'9 5

LO A D I N G S H A C K L E

LOOs

fig 6

\nut

collar

S H A F T , C O L L A R S

shaft

AND N U T S .

38

DILLON MACHINE FOR CALI BRATI ON OF

S H A C K L E

fig 7

■

39

fig 8

40

CHAPTER 5

EXPERIMENTAL RESULTS :

5.1 Linear Plots Prom Topographical Pattern :

5.1.1 Linear Plots_for one pin driven Blade :

A description of how the topographical pattern was obtained

is in 4.3.1 The results for the virgin blade and after use of 150

these topographical patterns, linear plots were obtained. Degrees

were plotted on a horizontal scale and the deformations, which are

indicated by figures in mm. on the topographical pattern, were plotted

( c ) J . The virgin condition, the condition after 150 hours and 300

hours are all shown on the same graph of the linear plot. The virgin

condition is indicated by 'zero line', the condition after 150 hours'

use by 'plastic deformation' and the condition after 300 hours' use

by 'further plastic deformation'. The sets of lines for the various

stations ranging from zero station at the collar diameter are clearly

marked on the figures.

5.1.2 Linear plots for three_pin driven Blade :

Just as in the case of the one pin driven blade, linear plots

were also drawn for the three pin driven blade. These are shown in

deformation' line sure clearly shown. The sets of lines for the various

hours and 300 hours is shown From

vertically. These linear plots are as shown

figs The 'zero line' and the 'plastic

stations are also marked on the figures.

41

5.2 Southwell's Plot

5.2.1 One pin supported blade

The results of the Elastic Analysis of the one pin driven

blade for the load at 0° are shown in tables ^2(a), (b) and (c)N .

The Southwell plots for these are as shown in figs. £l3(a) to (»)] .

and the buckling loads obtained from these figures are tabulated in

table (3). For the load at 120° the results are shown in tables

^4(a), (b) and (c)^ , the Southwell plots in figs. £l4(a) to (g)]

and the buckling loads obtained from them in table (6). In the case

of the load at 240°, the results are in tables < * a), (b) and (c)^ ,

the Southwell plots in figs. £l5(a) to <>>)] , and the buckling loads

in table (7).

5.2.2 Three pin supported blade :

In this case, the load was applied only at the 0° position.

The results of the elastic analysis are shown in tables <8(a), (b)

and (c)^ . The Southwell plots are in figs. [l6(a) to (fj| and the

buckling loads obtained from them in table (9).

5.3 Photoelastic Stress Patterns :

5.3.1 One pin and_three_pin supported disc (5" disc) :

The stress patterns for the various positions of the one pin

supported disc are shown in figs.£l7(a) to (f)] and for the three

pin supported case in figs. |l8(a) and (b)J .

5.3.2 One_and threepin supported disc (4” disc) :

The corresponding stress patterns in this case are shown in

figs.£l9(a) to (f)J and figs. ^20(a) and (b)J .

42

Topography of one pin driven Blade

Virgin Blade

43


After 150 hours' use

r44


After further 150 hours* use

station 3

Linear Plot of one pin driven Blade

(uiiu) (ujuj)

fig ioc b>

o’ 20* 40* 60* 80* 1OO* 120* 140* 160* 180* 200* 220* 240* 260* 280* 300* 320 340 360■I I I | ■ i-------- 1-------- l-------- t---------T-------- i-------- t T 1 I 1 ’ T >


fig loco

>• to 40* e«0* ao‘ ioo‘ iao‘ v*o' ifo‘ mo* 200' ago* 7*0' 2*0' **o‘___ *» * » f° ‘__ 2«£l__ 2 *°'


48

Topography of three pin driven Blade

Virgin Blade

49

After 150 hours’ use

(UIUI) (UiUi >

(UJUl)

figl2 (a)

Linear Plot of three pin driven Blade

(mm)

(mm)

station 4

U ------------ L— ---------- L _ ---------- L _-----------L_----------- L _ ---------- 1-------------- 1------------- 1— ----------- 1 I_________ I_________ I-------------- 1-------------- 1-------------- 1-------------- 1-------------- 1. - I0 20 40 SO 80 100 120 140 160 180* 200* 220* 240* 260* 260* 300* 320* 340* 360*

fig I2 (b)


(own)

* fT'rn

) 2 station 7

0* 20’ 40* SO* 80* 100* t20* 140* 160* 180* 200* 220* 240* 260* 280* 300* 320* 340* 360*

fig 12 (C)


f i g 1 3 ( a ) S O U T H W E L L P L O T ( O N E P I N D R I V E )

P I N A T 0 ° P O S I T I O N

S T A T I O N S 6 S 7 F O R 8 0 ° R A D I A L L I N E

A

A in mmP in 1

v-nLO

0 0*5 A

f i g 13( b) S O U T H W E L L P L O T ( O N E P I N D R I V E )

P I N A T 0 ° P O S I T I O N_ A i n mm

S T A T I O N S 3 , 4 6 5 F O R 6 0 R A D I A L L I N EP in I bf

A/ p

PIN A T 0 ° P O S I T I O N

S T A T I O N S 6 8 7 F O R 1 9 0 ° R A D I A L L I NE ^ «" m

' P i n I b f

viVI

P I N A T 0 ° P O S I T I O N A in m m

S T A T I O N S 3 , 4 8 5 F O R 1 9 0 ° R A D I A L L I N EP i n l b f

2*5

2*4

2*3

2*2

2>I

2-0

I .9

I - S

1*7

1*6

1*5

1*4

fig 13(e) SOUTHWELL PLOT (ONE PIN DRIVE)

P I N A T 0 ° P O S I T I O N

S T A T I O N 7 F OR 3 0 0 ° R A D I A L L I N E

VJ1"J

S T A T I O N 7

J ___________1*5

^ in m m

i n I b fP

PIN AT 0 P O S I T I O N

FOR 3 0 0 ° R A D I A L L I N ES T A T I O N 6

VJ1oo

S T A T I O N 6 J_______________

Z ^ in mm

p ini b f

1*5

f1

f i g 1 3 ( g ) S O U T H W E L L P L O T ( O N E PI N D R I V E )

OPI N A T 0 P O S I T I O N

A i n m m

P i n I b f

VJ1y£>

ST AT I ONS 4 a 5 FOR 3 0 0 R A D I A L LINE

r

' p

0 0 " 5 1 * 0 1*5

fig 1 4 ( a ) S O U T H W E L L P L O T ( O N E PI N DR I V E )

P I N A T 1 2 0 ° P O S I T I O N

S T A T I O N S 6 8 7 F O R 6 0 ° R A D I A L L I N E

A in mmP in lb.

ONo

P I N A T 1 2 0 ° P O S I T I ON

STATIONS 4 8 5 FOR 60° RADIAL LINE

i n m m

p in l b f

4*0

0 0 - 5

fifl 1 4 ( C) S O U T H W E L L P L O T ( O N E PI N D R I V E )

PI N A T 1 2 0 ° P O S I T I O N

S T A T I O N S 6 8 7 F O R 1 9 0 ° R A D I A L L I N E

in mm

P i n l b f


STATI ONS 4 8 5 FOR 19 0 ° R A D I A L L I NE

0

3-0 -

-V

2 - 0 -

1*0

________________________ I________________________I0 0 1 0*2

f i g 1 4 ( e ) S O U T H W E L L P L O T ( O N E PI N D R I V E )

P I N A T 1 2 0 ° P O S I T I O N

S T A T I O N 7 F O R 3 0 0 ° R A D I A L L I N E

©STATION 7

_J_________________________0*3

-<£5k. in mmP i n Ib j

ON

0 - 20 0 1

f i g 1 4 ( f ) S O U T H W E L L P L O T ( O N E PI N D R I V E )

P I M A T 1 2 0 ° P O S I T I O N


S T A T I ON 6

x

o>

i n mm

P in l b f

STATION 5

f i g 1 4 ( g ) S O U T H W E L L P L O T ( O N E P I N D R I V E )


STATION 5 FOR 300° RADI AL LINE

in mm

P in l b f

PIN A T 2 4 0 ° P O S I T I O N P m l b f

S T A T I O N 7 F O R 6 0 ° R A D I A L L I N E

P I N A T 2 4 0 ° P O S I T I O N

S T A T I O N 6 F O R 6 0 ° R A D I A L L I N E

ONOD

STATION 6

A in mm

P in l b f

1

S T A T I O N S 5 3 6 F O R 1 9 0 ° R A D I A L L I N E

\o-

f i «3 1 5 ( e ) S O U T H W E L L P L O T ( O N E P I N D R I V E )

PIN A T 2 4 0 ° P O S I T I O N


%

P I N A T 2 4 0 ° P O S I T I O N Z ^ m m r

S T A T I O N 6 F O R 3 0 0 ° R A D I A L L I N E P 10 l b f

-jro

M

1-0

09

0-8

0 * 7

0*6 A

£» £>

0 * 5

f i g 1 5 ( g ) S O U T H W E L L P L O T ( O N E PI N D R I V E )



'

CO

2 ^

in mm

P in I b j

S T A T I O N 5

0' -----------------------------------------------------------0 0 * 5

f i g 1 5 ( h ) S O U T H W E L L P L O T ( O N E P I N D R I V E )

P I N A T 2 4 0 ° P O S I T I O N


1

STATION 4

<A. in m m

P in I b f

f i g 1 6 ( a ) S O U T H W E L L P L O T (

S T A T I O N 7 F O R 6 0 ° R A D I A L

0T H R E E

L I N E

1

P I N D R I V E )

i n m m

p i n l b f

/ _____________________________0 0 f i g 1 6 ( b ) S O U T H W E L L P L O T ( T H R E E

S T A T I O N 6 F OR 6 0 ° R A D I A L L I NE

J___•5

P IN

1

in mmp in

F i

f i g 1 6 ( c ) S O U T W E L L P L O T ( T H R E E P I N D R I V E )

S T A T I O N S 4 6 5 F O R 6 0 ° R A D I A L L I N Ei n mm

P in I b j

r

. / \ in m m

p in Ib f

-ooo

S T A T I O N S 5 , 6 a 7 F O R 1 9 0 ° R A D I A L L I N E

• zx

S T A T I ON S 6 S 7 FOR 3 0 0 * in mm

P in I bf

VO

R A D I A L L I NE

1

A

S T A T I O N S 4 & 5 F O R 3 0 0 R A D I A L L I N E^ in mm

P in lb

ooO

f

fig. 17(a) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)

PIN AT 0° POSITION

fig. 17(b) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)

PIN AT 60° POSITION

fig. 17(c) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)


oo

fig. 17(d) FRINGE PATTERN FOR 5"


DIA. DISC. (ONE PIN DRIVE)

-

fig. 17(e) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)


1

4

fig. 17(f) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)


■ v

fig. 18(a) FRINGE PATTERN FOR 5" DIA. DISC. (THREE PIN DRIVE)

PINS AT 0°, 120° AND 240°

fig. 18(b) FRINGE PATTERN FOR 5" DIA. DISC. (THREE PIN DRIVE)

PINS AT 6C°, 180° AND 300°

ooVO

fig. 19(a) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)

PIN AT 0° POSITION

fig. 19(b) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)


\

v£>

fig. 19(c) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)


f

fig. 19(d) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)


r

fig. 19(e) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)


jfig. 19(f) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)


&

fig. 20(a) FRINGE PATTERN FOR 4" DIA. DISC. (THREE PIN DRIVE)

PINS AT 0°, 120° AND 240°

fig. 20(b) FRINGE PATTERN FOR 4" DIA. DISC. (THREE PIN DRIVE)

PINS AT 60°, 180° AND 300°

97

CHAPTER 6

ANALYSIS AND DISCUSSION :

6.1 Analysis of the Linear plots for one pin driven blade :

6.1.1 Region of deformation s

A linear plot of the topographical pattern of the one pin

driven blade for the virgin condition and use of 150 hours and 300

hours is shown in figs. [l0(a), (b) and (c)J . From these, it can

be seen that the region of appreciable deflection lies between 60°

and 300° and it occurs from zero station onwards. The region around

the pin between 300° and 60° is virtually undeformed.

6.1.2 Region of maximum deformation :

oThe region of maximum deformation seems to be at 180

see figs. £l0(a), (b) and (c)] . At this region the deformation

curves for the third and higher stations show a characteristic 'hump'.

This only occurs in the case of the curves after 150 hours of use,

the curves after 300 hours' use do not show this characteristic.

The curves after 300 hours of use (denoted further plastic deformation)

do however have a maximum deformation in the same region of 180 and

occur on the opposite side of the zero line (which is the linear plot

of the virgin blade). It is worthy of mention that the magnitude of

the maximum deflection, with reference to the zero line, is less in

the case of 300 hours' use than 150 hours' use.

98

6.2 Analysis of the Linear plots for three pin driven blade :

6.2.1 Region of deformation :

A linear plot of the topographical pattern in this case is

shown in figs. £l2(a), (b) and (c)J . In this case, the deformation

appears to be more uniformly distributed over the entire surface of

the blade as compared to the case of the one pin driven blade.

However, for the third and higher stations, there appears to be a

deformation region between 60 and 220 , the maximum being in the

region of 140°. The magnitudes of the maximum deformations in this

case are not as large as in the case of the one pin driven blade

after 150 hours of use.

6.3 Discussion of Linear plots :

It was mentioned above that for the one pin driven blade, the

region of deformation was between 60° and 300 , the region between

300° and 60° being virtually undeformed. The pin at 0 seemed to have

had a stabilising effect on the blade for a region of 60° on either

side of zero. This in fact tempted the author to use three pins at

intervals of 120° on the blade. This, from the above reasoning,

have a stabilising effect on the whole blade.

In the case of the one pin driven blade there is a deformation

in the region 60° to 300° from the zero station onwards; vhereas in

the case of three pins, the deformations are in a smaller r g

are only worthy of mention after the third station. I*

(at the periphery of which is zero station) was driving the blade and

as such was a source of rigid support, the deformation

stations should have been negligible uniformly around

99

This, however, is not the case as can be seen in fig. I 1 0 ( a ) J .

However, for the three pin driven blade, the deformations at the

initial stations are virtually negligible as can be seen in fig.

[l2(a)] . Unfortunately, the use of 3 pins does not completely remove

the region of deformation; it however does decrease the region so

as to exist between 60° and 220°. The reason why this particular

region is still affected must be due to the complex stress structure

the blade is subjected to during cutting of timber. Not much can be

said about this until further experimental work is done. For bothothe one pin and three pin driven blade, the region between 300 and

60°, and 220° and 60° respectively has not been affected. This could

lead one to assume that the blade must be made stronger by the

manufacturers in the region of the hole for the pin, since markings

for the hole were clearly visible. The hole for the one pin driveno . .blade was made at this marking and also the hole at 0 position for

the 3 pin driven blade. Again one cannot be absolutely sure of this

"metulurgically stronger” region without further experimental analysis.

It was mentioned earlier that for the one pin driven blade, a

characteristic 'hump' existed for the third and higher stations.

It seems as if the hump started at some earlier stage and at 150 hours

probably progressed from this stage and at some time before 300 hours

'flipped' in the opposite direction. This is similar to a strut which

would proceed to the second and higher modes of deformation once the

first critical load was exceeded. Exactly at how many hours this

'flipping' in the opposite direction took place, is difficult to assess

unless topographical patterns are obtained at smaller intervals of

say 50 hours. In this way, one could possibly define a "buckling time

had reached the stage shown It

100

for the blade in question.

As mentioned earlier, the curves in figs. [10(a), (b) and (c)J

for the one pin case after 300 hours of cutting have a maximum

deflection which is less than that for 150 hours' use. Fran this,

it seems that the blade goes through a process of strain-hardening

after the first buckling which presumably exists at or after 150

hours of use.

6.4 Analysis and Discussion of Southwell plots :

It is seen from the tables 3, 6, 7 and 9 mentioned in 4.3.5

of buckling loads that in the case of the blade being supported by

one pin the magnitudes of the buckling loads at various stations are

rather irregular. The minimum of all these values is about 205 lbf.

In the case of three pins, the blade is in a more uniform state of

buckling and the minimum of all the concerned buckling loads is a

higher value of about 600 lbf#

One has to mention here that the elastic analysis made in

4.3.5 was under static loading conditions and the thermal conditions

of the blade were at room temperature. In actual practice, there

are the added effects of temperature variation, vibrations, intermittent

action of load etc... Nonetheless, as mentioned in 2.2, the above

analysis does give a fairly representative picture of what could be

happening to the blade in actual operation as regards forces.

6.5 Analysis of Photoelastic patterns & discussions :

6.5.1 Large disc “_Single_pin_drive :

The patterns, figs. [1 7 (a) to (f)], at 60° intervals of the pin

101

positions, were obtained in order to simulate the actual working

conditions of the blade rotating whilst the work piece (timber)

remained fixed. Certain characteristic features are clearly depicted

by these patterns and are as follows :

1) If one neglects the influence of the point of application of the

load, the fringe patterns are almost symmetrical about the

diameter of the blade passing through the pin for all positions

of the pin.

2) The zero fringe order is on the boundary of the disc. The fringe

orders 1, 2 and 3 start from very near the pin and terminate on tv,|“

boundary of the collar on either side of the diameter mentioned

above. The angle between the diameter of the blade through the

pin and the radial line from the centre of the disc passing

through the intersection of the second fringe with the collar

is about 60°. This is supprisingly true for all positions of

the pin in a rotation of the disc.

3) It is also noticeable that the number of fringes is constant

(and is 3) on either side of the diameter mentioned above, as

the disc is rotated.

From the above remarks, one can conclude that as the blade

rotates, values of the difference in principal stress ( <r"|—

in the region of roughly 60°, on either side of the diameter throuah

the pin,-is greater than the rest of the disc (blade).

6.5.2 Small disc - Single pin drive

The characteristics mentioned above for the large disc are true

for the small disc (and the same load) in every respect except that

102

the fringe orders in the more stressed areas are reduced (2 in this

case) figs. ^19( a) to (f)] . This clearly indicates the advantages

of reducing the area of the blade between the collar and the periphery

by increasing the collar size. However, this would be impractical.

6.5.3. Three_pin drive for large and small discs :

The patterns, figs. Tl8(a) and (b)J and [20( a) and (b)]

clearly show the evening out of stressed areas and also reducing the

magnitudes of stresses. This explains the advantages of multiple pin

support for large or small blades.

6.6 Conclusions :

(1) If the blade was driven by friction between itself and the

tightened collars, the pattern of the plastic deformation after use

of any number of hours would have been uniform i.e. it would have

varied only radially and not circumferentially. This does not preclude

it from deforming in a higher harmonic mode, but symmetrically with

respect to the axis through the hole. The deformation pattern as it

is, shows that the pin in fact does the driving of the blade. The

linear plots of the three pin driven blade clearly show a unifying

effect of the deformation pattern. This phenomenon shows that the

blade is in fact driven by the pin.

(2) In actual use, it was found that irregular cutting of timber

(indication of severe plastic buckling) using one pin drive occured

in about half of the time of using a three pin drive. This can be

noticed from figs. ^10(a), (b) and (c)J and [l2(a), (b) and (c)J

where the deformations for the one pin drive after 150 hours of use

are larger than for the three pin drive after 150 hours of use.

103

Sharpening frequency in the case of three pin drive was less than

that of one pin drive. Smoother running was also a characteristic

°f the three pin drive (as reported by the technicians operating the

saw). This, together with the foregoing conclusion definitely show

the delaying of the onset of buckling by changing the method of support.

(3) The minimum buckling load obtained by harmonising the various

values over the critical region of the blade (for the one pin case)

was about ten times the maximum cutting force encountered in practice,

[as calculated by using parameters of the saw, such as horse power

of the motor, speed and size of the blade]- see Appendix 2 (ii).

Bearing in mind that the static elastic laboratory test does not

exactly simulate the actual working conditions such as (a) the applica

tion of the load continuously along the periphery, (b) the temperature

variation over the blade, (c) vibration of blade or shaft, (d) jerking

action of the cutting force etc..., the test results indicate that

application of Southwell's method to predicting critical load of the

blade, under the devised peripheral loading conditions, can at least

lead to a better understanding of the problem of circular saw blades.

The elastic analysis also showed that the change of method of support

(three pins) would delay the onset of buckling, this since the minimum

buckling load for the three pin case was almost thrice that for the

one pin case.

(4) The photoelastic analysis has given us some important clues

regarding the nature of stresses induced in the blade. The photoelasti

pattern for the one pin driven case predicts a highly stressed area

exactly opposite to that observed under actual working conditions (the

topographical pattern). The Southwell plots indicate buckling again

104

ln re<Jions different from those expected from photoelastic stress

patterns. Since compressive stresses cause buckling at any point,

the main conclusion that can be arrived at is that due to working

conditions other than those assimilated in the laboratory, compressive

stresses are produced in regions different from those shown by

photoelastic analysis. This is understandable since cutting of timber

produces temperature gradients which undoubtedly produce thermal

stresses, which combine with the stresses seen by photoelastic analysis.

This combination produces sufficient compressive stresses to start

buckling in the regions indicated by the linear plots of the topo

graphical patterns. Thus the main fact revealed by the photoelastic

analysis is that in any theoretical treatment of this blade problem,

temperature stress calculations is very important to predict areas

of buckling.

(5) All the above analysis indicate the advantages of using a

multiple pin drive. The photoelastic analysis indicates the advantage

of using a large collar. Hence dependant upon the size of timber to

cut, the size of collar could play an important part in the prolonga

tion of the onset of buckling.

(6) The three fields of analysis has given sufficient guide lines

for the advanced theoretical and experimental investigation of the

problem. From the literature review, one can see that no previous

attempt has been made to show experimentally the possible influence

of temperature stresses on the stresses due to load. There are also

no previous conclusions about the size of collar or multiple pin

drives.

105

The conclusions arrived at in this investigation have at least

given an indication as to where more theoretical work should be done

to understand the problem of the buckling of a saw blade and perhaps

to devise a scientific method of tensioning, if not a prevention

altogether.

106

CHAPTER 7

RECOMMENDATIONS FOR FUTURE tfORI :

The problem of circular saw tensioning is a complicated one

and, to the author's knowledge, the least investigated one compared

to other problems occuring in industry. The present investigation

has opened up at least one or two fields where further work would

prove fruit full.

Since temperature seems to play an important part in the actual

buckling and distortion of the blade, an experimental determination

of the temperature gradient of a blade in actual use would be useful.

From such an investigation, the temperature stress pattern could

possibly be obtained.

A stress pattern of the blade due to cutting forces is essential

in the study of the problem. A dynamic stress pattern obtained by

measuring dynamic strains of a blade in actual use would have the

effects of temperature stresses intrinsically involved. From this

and the temperature stress referred to in the previous paragraph, the

effect of cutting forces on the blade can be studied.

107

A P P E N D I X 1

T A B L E S 0 F O B S E R V A T I O N S A N D R E S U L T S

108

LOADING UNLOADING

Me p .i u e

50 12 50 ii

100 21 100 21150 32 150 30200 42 200 41250 53 250 51300 63 300 62350 74 350 72400 85 400 83450 96 450 95500 106 500 106

Table 1

Shackle Calibration Data

60° - S7 60° - S6 60° - S5 60° - S4 60° - S3u c R > , Arnm A/p -3 » 10 Me P ibf Am m % - a

X 10 JJLC P lbf Am m t y p - 3 x 10 JULC R w Avnm */£> JJLC Amm A/ p _3X 10

15.5 72 .05 14.0 67 +.12 13.5 67 + .11 12.0 57 + .09 12.5 57 + .0828.0 132 .01 26.0 125 +.07 29.0 137 + .06 25.5 120 +.05 22.5 105 +.0742.0 200 .11 .55 47.5 225 - .2 2 .98 42.0 200 - .1 8 .90 39.0 190 - .1 4 .74 38.5 182 - .1 4 .7753.0 252 -.2 6 1.03 62.5 297 - .3 4 1.14 56.5 267 - .2 8 1.05 55.0 262 - .2 7 1.03 52.5 250 - .2 4 .9664.5 310 -.3 1 1.0 79.0 377 -.4 8 1.27 68.0 325 - .3 5 1.08 69.0 330 - .3 5 1.06 68.5 325 -.3 3 1.0280.0 382 -.3 8 .99 88.5 422 - .5 3 1.26 83.0 397 - .4 4 1.11 82.5 392 - .4 2 1.07 79.0 377 - .4 0 1.0690.0 432 - .4 4 1.02 99.5 475 -.5 5 1.16 96.0 460 - .5 0 1.09 95.0 455 -.4 8 1.06 94.0 450 - .4 7 1.04

103.0 492 - .4 8 .97 110.5 527 - .5 9 1.12 108.5 517 - .5 4 1.04 107.5 562 - .5 4 .96 105.0 502 - .5 2 1.04112.0 537 - .5 1 .95 141 675 - .6 5 .96 120.0 575 -.6 1 1.06 119.5 570 - .6 2 1.09 117.5 560 - .5 8 1.04121.0 580 -.5 8 1.00 124.0 595 - .6 5 1.09

_________________ 1.

PIN TRAVELLED THROUGH 0° TABLE 2(a)

Southwell Plot Data

190° -• S7 190° - S6 190° - S5 190° - S4M e P Ibf A mm A/ P -3 X 10 J J i t P Ib f Am m A / P -3X 10 3 u e P Ib f Amm * / P -3

X 10 M e %Amm A/ p _3 x lo

20.0 95 + .10 +1.05 15.0 72 +.06 .833 20.5 95 +.09 .948 14.0 67 + .03 .4535.5 167 + .24 +1.44 25.5 120 +.14 1.17 30.0 142 +.16 1.13 23.5 110 +.09 .8249.0 235 + .37 +1.575 38.0 182 +.23 1.265 43.0 205 +.17 .83 39.5 190 +.16 .8469.5 335 +.49 +1.46 51.0 245 +.30 1.22 58.0 277 +.25 .9 54.0 258 +.21 .8180.0 382 +.59 +1.55 62.0 297 +.37 1.25 72.0 345 +.33 .96 10.5 335 +.25 .7589.5 432 +.69 +1.6 79.0 377 +.45 1.19 82.0 392 +.38 .97 84.5 402 +.29 .72

100.5 480 +.75 +1.56 91.0 435 +.56 1.29 95.0 455 +.44 .97 98.5 470 +.35 .75114.0 545 + .87 +1.6 103.0 492 +.65 1.32 108.0 517 +.49 .95 112.5 537 +.40 .75118.0 565 +.94 +1.66 112.5 537 +.72 1.34 119.0 570 +.58 1.02 124.0 595 +.52 • /

119.0 | 570 +.79 1.38 123.0 590 +.71 1 .2

PIN TRAVELLED THROUGH 0°

TABLE 2(b)

Southwell Plot Data

1

300° - S7 300° - S6 300° - S5 300° - S4

Jdc P IbfA m m A / p .3 x to3 U e P | b f

Am m

A /p _3 * to3 J l i Pi b# A

m m A/P -3X 10 Jdc P lb fAm m

Ay/ p -3** 1014.0 67 - . 1 1 13.5 67 -.06 1 1 .0 52 - .1 0 13.0 62 -.13

23.0 120 -.09 22.0 105 -.07 21.5 100 - .1 2 23.5 110 - . 1 2

39.5 190 +.28 1.47 36.0 172 +.10 .58 35.0 167 -.07 35.5 167 -.05

53.5 252 +.64 2.54 52.0 250 +.37 1.48 49.0 235 +.28 1.19 47.5 225 + .2 1 .93

64.0 310 +.69 2.22 64.0 310 +.50 1.61 60.5 287 +.55 1.93 60.5 287 +.40 1.4

77.5 370 +.92 2.48 76.5 365 +.63 1.72 73.5 350 +.56 1 .6 75.5 360 +.40 1 . 1 1

91.5 435 +1.13 2.6 87.0 417 +.77 1.85 85.5 407 +.61 1.5 93.5 445 +.58 1.3

10 1.0 485 +1.14 2.36 101.5 485 +1 .0 1 2.08 95.0 455 +.72 1.5 8 105.0 502 +.63 1.25

1 1 1 . 0 530 +1.24 2.34 113.5 542 +1 .1 2 2.06 105.0 502 +.77 1.54 115.0 552 + .65 1.18

121.5 580 +1.14 1.97 115.5 552 +.89 1.61 124.0 595 +.60 1 .0 1

1__________125.0 600 +.87 1.45


TABLE 2(c)

Southwell Plot Data

111

PCR 60° 190° 300°

S7 5750 lbf# 1900 lbf# 633 lbf.

S6 475 " 3100 " 1100 '•

S5 1900 " 1000 " 1600 "

S4 5200 " 1700 " 950 "

S3 1400 " 1750 "


TABLE 3

Buckling Loads from Southwell Plot

112

60° - S6 60° - S7 60° - 5 60° - S4

>ue P Ibf Awfn

A /7 R, ,5* juc P lb* Am m % - » x 10 J d C P i i f

Am m

A y' P - 3

x 10 U c % Am m A / P -3 x io13.0 62 -.38 6.15 13.0 62 -.45 7.25 11.5 52 -.25 4.8 14.0 67 -.24 3.58

21.5 100 -.55 5.5 20.0 95 -.66 6.95 19.5 90 -.37 4.1 22.5 105 -.35 3.24

36.5 172 -.77 4.5 28.5 132 -.81 6.14 29.0 137 -.53 3.86 32.5 152 -.47 3.1

47.0 225 -.99 4.4 34.0 162 -.95 5.87 42.0 200 -.75 3.75 40.5 192 -.64 3.34

61.5 292 -1.06 3.64 47.5 225 -1.2 5.3 51.0 245 -.88 3.6 56.5 267 -.76 2.85

71.0 340 -1.22 3.6 59.0 277 -1.43 5.1 60.5 287 -1.02 3.56 69.5 330 -.81 2.46

67.0 320 -1.49 4.66 72.0 345 -1.09 3.16 82.5 392 -.96 2.45

80.5 382 -1.73 4.5 90.5 432 -1.07 2.48


TABLE 4(a)

outhvell Plot Data

113


TABLE 4(b)

Southwell Plot Data

114

300° - 07 300° - ?6 300° - c5

M e p ,bf A m m JUe A m m A/ P -3 xIO JUL Plbf Am m

V p ,x \o9.5 42 -.15 3.57 10.0 50 -.16 3.2 14.0 67 -.14 2.09

18.0 85 -.30 3.54 19.5 90 -.26 2.9 25.5 120 -.21 1.75

29.0 137 -.32 2.34 53.0 252 -.27 1.07 40.5 192 -.22 1.14

62.3 297 -.35 1.18 67.0 320 -.31 .97 57.0 272 -.24 .88

76.0 365 -.37 1.015 82.5 392 -.34 .87 74.5 355 -.29 .82

92.5 440 -.42 .95 97.5 265 -.35 1.32 93.5 445 -.32 .72

100.5 480 -.42 .875 104.5 500 -.37 .74


TABLE 4(c)

Southwell Plot Data

115

O' 0 o 1 CO 60° - S6

M e Plb* Arty m A/ P _9X 10 u e P lb* Am rr\ A/ p -*X 1011.3 52 -.07 1.34 15.0 72 -oo.• .834

22.0 105 -.14 1.33 24.5 115 0r-i•1 .87

32.5 152 -.24 1.58 35.5 167 -.20 1.19

45.5 215 -.38 1.77 51.0 245 -.32 1.3

59.0 277 1 • C0 1.73 66.5 315 -.43 1.37

71.5 340 -.63 1.85 83.0 397 -.64 1.61

86.0 412 -.79 1.92 97.5 465 -.76 1.64

98.0 470 -.95 2.02

PIN TRAVELLED THRJUGH 240°

TABLE 5(«)

Southwell Plot Data

116

190° - C7 190° - 6 190° - S5 190° - A

M e P lb*Am m A/ P - 3

X IO M e F V Am m A / P - 3*10 M e PIbf Anom A/ P , - 3 M e P,k, Ahn idA/P

12.0 57 - . 1 0 .175 17.0 80 1 . o CO 1.0 16.5 77 i . o .91 10.0 50 -.07 1.4

2 1 . 0 1 0 0 -.16 1 . 6 24.5 115 - . 2 1 1.83 25.0 1 2 0 -.16 1.33 20.5 95 -.23 2.42

33.0 157 -.38 2.42 35.5 167 -.31 1 . 8 6 35.5 167 -.17 1 . 0 2 30.5 142 -.28 1.97

50.0 240 -.49 2.04 44.0 2 1 0 -.37 1.76 52.5 250 -.25 1 . 0 41.0 195 -.39 2 . 0 0

63.0 300 -.71 2.37 54.5 258 -.54 2.1 66.5 315 -.39 1.24 55.5 262 -.38 1.4

77.5 370 -.70 1.89 6 8 . 0 325 -.57 1.75 82.5 392 -.44 1 . 1 2 75.0 360 -.48 1.34

93.5 445 -.79 1.78 85.0 407 -.61 1.5 95.5 455 -.56 1.23

1 0 2 . 0 487 - 1 . 0 2 2 . 1 98.5 470 -.74 1.57 105.0 502 -.52 1.04

PIN TRAVELLED THKJUG34 240° TABLE 5(b)

Southwell Plot Data

117

300° - S7 300° - S6 300° - S5 300° - S4

M P ib f Am m A/ P ,-, JJLC P l k f A m m A/P -aM IO U C P lb,. Am m A/pX l» J i e P l b f Anrt m A/p,

30.5 142 + .02 .14 25.5 120 +.05 .42 28.0 132 + .08 .6 41.0 195 +.05 .256

35.5 167 + .09 .54 33.0 157 + .09 .57 36.5 172 +.10 .58 48.5 230 +.10 .435

42.5 200 + .10 .5 39.0 190 + .15 .79 47.5 225 + .21 .93 59.5 • ill +.14 .50

49.0 235 +.15 .64 46.5 220 + .23 1.04 57.0 272 + .30 1.1 70.0 335 + .17 .51

55.5 262 + .19 .72 57.5 272 +.30 1.1 74.5 355 + .33 .93 82.5 392 + .24 .61

60.5 287 + .20 .7 65.0 310 +.35 1.13 87.5 417 + .36 .86 90.5 432 +.28 .65

71.0 340 + .26 .76 73.0 350 + .41 1.17 95.5 455 +.39 .86 100.0 480 + .29 .6

81.0 , 387 + .32 .83 93.5 445 + .43 .97

100.0 480 + .35 .73


TABLE 5(c)

118

PCR 60° 190° 300°

S7 435 lbf> 825 lbf. 217 lbf.

S6 320 " 587 " 215 "

S5 640 " 312 " 205 "

S4 820 " 250 "


TABLE 6


119

PCR 60° 190° 300°

S7 1850 lbf> 700 lbf. 825 lbf>

S6 1116 " 1700 " 1300 '•

S5 1550 " 425 "

S4 1000 "


TABLE 7


120

60° - S7 60° - Si 60° - 8 5 60° - 84

jx e P l b * Am m A /P -3 j x c P l b fAm m A/v P u f

A m m A /P.3K 10 3 u e FWAm m * V p . -3 *|o

15.5 72 +.09 1.25 13.5 67 ♦ .08 1.19 13.5 67 ♦ .07 1.04 13.0 67 +.07 1.04

30.5 142 ♦.17 1.2 21.5 100 ♦ .20 2.00 22.5 105 + .18 1.72 25.0 120 + .19 1.58

48.5 230 +.24 1.04 32.0 152 ♦.29 1.91 31.5 147 ♦ .22 1.5 40.0 192 ♦ .31 1.62

56.5 267 + .31 1.16 46.0 220 ♦ .34 1.55 44.0 210 ♦.31 1.48 52.0 252 ♦ .40 1.59

72.0 345 ♦.41 1.19 59.5 277 ♦ .48 1.73 57.5 272 ♦.38 1.4 67.5 320 +.30 1.56

82.0 392 ♦ .50 1.28 69.0 330 ♦.51 1.55 70.0 335 ♦ .49 1.46 85.5 407 +.57 1.4

89.0 432 ♦ .66 1.53 86.0 412 ♦ .67 1.63 85.5 407 ♦ .63 1.55 93.5 445 +.64 1.44

TABLE 8(a)

routhvell Plot Data

121

190° - S7 190° - S6 190° - S5

J d i Plb t Am m A/PX /o J Plb, Am m A/ p -3 * 10 Plbf Amrn A / e.»39.0 42 + .01 .238 10.0 50 + .01 .2 15.0 72 + .05 .69

21.0 100 +.10 1.00 19.0 90 +.03 .33 27.0 132 +.08 .6

35.0 147 +.11 .66 29.0 137 +.05 .366 39.0 190 +.10 .527

50.5 240 +.19 .79 42.0 200 + .11 .55 51.5 245 +.13 .53

65.0 310 +.24 .77 59.0 277 +.13 .47 64.5 310 +.14 .45

TABLE 8(b)

Southwell Plot Data

122

1In k

300° - S7 300° - S6 300° - S5 300° - S4

U C P-kf Am m A/p . 3 x lo * Jd€ F5 ib* Am rr» A/ px lo U e %

A m m J U e P lbf A m m A /r,->11.0 52 -.16 3.08 12.5 57 -.14 2.46 10.5 50 -.10 2.00 12.0 57 -.10 1.75

23.0 110 -.50 4.55 23.5 110 -.31 2.82 19.5 90 -.23 2.56 22.0 105 -.24 2.28

29.5 137 -.63 4.6 40.0 192 -.65 3.38 32.0 152 -.51 3.36 32.0 152 -.43 2.82

38.5 182 -.93 5.1 57.5 272 -.91 3.35 46.5 220 -.78 3.54 42.5 200 -.65 3.25

45.5 215 -1.02 4.75 72.5 345 -1.18 3.42 61.0 292 -.85 2.91 59.0 277 -.71 2.56

57.5 272 -1.17 4.3 86.0 412 -1.39 3.38 79.5 377 -1.13 3.00 74.0 355 -1.07 3.02

68.5 325 -1.39 4.28 96.0 460 -1.59 3.46 91.0 435 -1.25 2.88

85.5 407 -1.86 4.57

94.01-----------

450 -2.1 4.67

TABLE 8(c)

Southwell Plot Data

123

* L

PCR 60° 190° 300°

S7 900 lbf> 1650 lbf. 1600 lbf.

S6 750 " 3000 " 1200 "

S5 1550 " 1900 " 600 "

S4 2000 " 760 "

TABLE 9


124

125

APPENDIX 2(i)

Connection of bridge for cancelling effectt>

R1 x Vrab = R1 + r2

bd R1 R3= ^ + R2 * S, * R. }

ad

R 1 (R 3 ♦ V ■ r 3 ( R l + r 2 ) '

L (Ri + R2 > ( R3 + v .

r v R3+ R.. .R

1 4 — iv ̂ . R 2 — R^.S

+ + R4 )

K - R4 R3 * K2 v+

R3 x V

R3 + R4

For one active gauge R^ changes to R̂ +AR

Substitute 2 into 1 :

bd{(R1 + AR) (R4) - R3.R2} x V

= <̂ (R1 +AR) + ™R 2 J ( R 3 + V

126

(Rr R4 + R4 *AR - R 3.Rg) x v = (Rx + R2 + ^ R ) (R3 + R4 )

( R 1 # R4 + R ^ . A R - R 3 . R 2 ) x V

^ 1 • ^ 3 + R2 • S 3 + A R , R3 + f t . i<4 + R J T F J + A R • R 4

I f R ^ , R2 , R^ and R^ a r e e q u a l t o s a y R :

.AR .V

bd 4 R + 2 A R .R

4 R . V4R + 2AR

A R . V2 (2R + A"RT I f A R « R Vbd =

A R.V4R

For 2 a c t i v e gauges t o g e t h e r w i t h b e n d in g :

c h a n g e s t o R^ + A R + A Rb

R4 c h a n g e s t o R4 + A R - A R b )

3

Where -t-AR.^ i s s t r a i n dae t o b e n d in g and i s t e n s i l e

- A i<d i s s t r a i n due t o b e n d in g and i s c o m p re s s iv e

S u b s t i t u t e 3 i n t o 1 :

{ ( R x + AR +ARb ) (R4 +AR -AI?b ) - R g . R , } V

'b d = £ (R X + A R +ARb ) + R2 H r3 + r 4 ^ - i R b }

I f R, = R0 = R = R = R th e n :1 2 3 4

v = { ( R +AR +ARb̂ (R + AR " ^ ' V " r2} V { 2 R + AR + A Rb | ^ 2R + AR - A ? b J.

127

T p 2 + A R .R - ARb .R + AR. R + (A R )2 - A R .A R K + R . A R ,

+ A R K.AR - (ARk ) 2 - R2r _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ L|"4R 2 + A R .2 R - 2R .A R b + 2R.AR + (AR ) 2 - A k , A R b

1

-X V

♦ 2R.ARb + A k .AS. - (ARb )

pAx^.R - (ARb):[ i _ __________________|_4R2 + 4R.AR + (A R ) 2 (ARb ) 2

2A R .R .V 4R (R + ART

A R . R . V" 2R (R + A R)

A R .V2 ( ' +ARJ

A R.V" 2 R A R < < R

T h is g i v e s t w i c e th e v o l t a g e a c r o s s bd com pared to t h e one

a c t i v e gau ge c a s e . T he c o m p r e s s iv e and t e n s i l e s t r a i n s

( -A R b a n d +ARb r e s p e c t i v e l y ) due t o b e n d in g , c a n c e l l o u t .

128

APPENDIX 2 ( ii)

Force at tooth from Saw Parameters :

H.P. of Motor • Speed of Motor Speed of Saw — Size of Blade ■

H.P. x 33,000 27fN

5.5 x 33.000 2 x 2̂ 00 «r7T

= 13.1 lbf# - ft.

F =

157.6 lbf# - ins.T _ 157.6 R " 10 15.8 lbf.

5 .5 H .P .

1420 R.P.M. 2200 R.P.M. 20” Diameter

129

B a c k g ro u n d to S o u th w e l l 's P lo t :

P3

APPENDIX 2(iii)

p D ia g . (a )

When a p e r f e c t l y s t r a i g h t s t r u t i s s u b je c te d to

two e q u a l c o m p re s s iv e f o r c e s , P as show n in d ia g ra m ( a ) ,

the e q u a t io n g o v e rn in g th e fo rm o f th e s t r a in e d c e n t r a l

line can be shown to be ( 1 0 )

d 2B — \ + P = 0 ( F o r s m a ll v a lu e s o f y ) ............. 1

dx y

Where B = E l

E i s th e m o d u lu s o f E l a s t i c i t y

I i s th e secon d moment o f a re a o f th e c r o s s s e c t io n

S i s th e d e f l e c t i o n a t x = >*/2

The f i r s t c r i t i c a l b u c k l in g lo a d can be shown t o be :

When a s t r u t i s n o t p e r f e c t l y s t r a i g h t , b u t has

im p e r f e c t io n s a r i s i n g o u t o f m a n u fa c tu re and w o rk m a n s h ip ,

the c e n t r a l l i n e o f th e u n lo a d e d s t r u t can be r e p r e s e n te d

by yQ = £(*) .

130

A ssu m ing y0 = S0 s i n J p , th e s t r u t can be r e p r e s e n te d

( a f t e r th e a p p l i c a t i o n o f lo a d ) as shown in d ia g ra m ( b ) .

W here So i s th e d e f l e c t i o n o f th e im p e r fe c t s t r u t a t

x = ^ / 2

& i s th e t o t a l d e f l e c t i o n o f th e s t r u t a t

x = / / 2 a f t e r lo a d in g .

The e q u a t io n g o v e rn in g th e fo rm o f th e

s t r a in e d c e n t r a l l i n e ca n be show n to be ( 1 0 ) :

B d2 ( y _ y Q) + p = o (F o r s m a ll v a lu e s o f y) d x 2

I f B i s in d e p e n d a n t o f x :

W herePB

P u t t in gd 2yo

dx= - 6* JL s in 77Vx_

4th e g o v e rn in g

e q u a t io n becomes£ y +#62y . - J ^ i n j E a Ld x “ ^

3

131

T h e c o m p le te s o lu t io n o f t h i s l i n e a r d i f f e r e n t i a l

e q u a t io n i s :

y = c1 c o s o tx + c 2 sincLoc

U s in g th e b o u n d a ry c o n d i t io n s y =

s in

o a t x = o and

x = JL in d ic a te s c i = c 2 = 0

Sf7Tz//l ___Therefore y =s in (2L£_ )

Using th e boundary c o n d i t io n y a t x

f S 0 7T "̂l/ 1 —

8 =

. 2 _ P _ £ _p u t t i n g - b “ 21

r S o 7r 2/ / 2- _ & a J T ± l i M

-

6 0 p c r 6 0_ ______-__—--------

8 = p c r - pl 1 ‘ ^ )

6 — 6 0 6 __Re-arranging p p c r

A d d in g & s u b t r a c t onr . h . s .

A So S6 0 1 - p -

------- — = “ nP F c r p c r ‘a

Re-arranging :

& - <5 0 _ S - & 0 + SoP Pc r Pc r

132

P u t t in g A = 8 — S0 t h i s e q u a t io n becom es

c . f . y = mx + c

P l o t t i n g A / p v e rs u s , th e s lo p e g iv e s p ^ — ( o r th ec r

in v e r s e s lo p e g iv e s P ^ r ) and th e i n t e r c e p t g iv e s ^ ° / p c r »

The p l o t o f A / p v e r s u s A i s know n as 'S o u t h w e l l 's P l o t ' .

Some a u th o r s (1 0 ) assum e a F o u r ie r s e r ie s f o r y 0

in s te a d o f th e s im p le a s s u m p tio n s o f y Q = 80 s in (V x^g .

H o w eve r a f t e r w o r k in g th r o u g h w i t h th e s o lu t io n o f

e q u a t io n 2 an d m a k in g v a r io u s a s s u m p t io n s , th e y a r r i v e

a t th e same r e s u l t as a b o v e .

133

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1 8 0 8 0 , N a ir o b i - E n t e r p r is e Road.

2. 'C i r c u l a r Saw s' M i n i s t r y o f T e c h n o lo g y , U n ite d

K in g d o m , F o re s t P r o d u c t s ' R e s e a rc h L a b o r a to ry .

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E n g in e e r in g S c ie n c e V o l . I pages 8 9 to 100.

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S o l id s 1963 V o l. I I pag es 41 to 47 Pergamon P re s s .

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V o l . IV No. 3

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C a n a d a , Ottawa, O n t a r io .

134

9 . " T h in C i r c u l a r S a vs" by C .P . B e ro lz h e im e r an d

C .H . B e s t . F o r e s t P r o d u c ts ' J o u r n a l I X ( i i ) 4 0 4 . 1 9 5 9 .

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U n iv e r s i t y P r e s s . Second E d i t i o n — pages 4 2 4 to 4 2 8 .

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o f th e s id e s o f a S h ip " b y G .H . B ry a n - C a m b rid g e

P h i lo s o p h ic a l P ro c e e d in g s . Decem ber 1 1 th , 1 8 9 0 .

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and A .K . H e a d . J o u r n a l o f M e c h a n ic s and P h y s ic s o f

S o l id s 1961 V o l . 9 pages 13 1 t o 1 3 9 . P ergam on P r e s s .

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Me G raw H i l l Book C o . - s e c o n d e d i t i o n .

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R o ta t in g S a w s ". R e p o r t 12 B S venska

T r a f o r s k n i n g s i t i t n s u t e t T r a t e k n is k a A v d e ln in g e n .

135

1 8 . M a lc o lm F .B . 1 9 5 7 . "W hy b la m e t h e sa w ? "

Southern Lumberman.

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e d g e and s u b je c t e d t o s h e a r in g f o r c e s " b y R .V .

S o u th w e l l and ( M is s ) S .W . S k a n . P r o c e e d in g s o f th e

R o y a l S o c ie t y V o l . 137A 1 9 3 2 .

2 0 . "S om e T e s ts on t h e S t a b i l i t y o f T h in S t r i p m a t e r i a l

u n d e r S h e a r in g f o r c e s i n th e p la n e o f t h e S t r i p " b y

H . J . G ough and H . L . C o x . P r o c e e d in g s o f t h e R o y a l

S o c ie t y V o l . 1 3 7 A 1 9 3 2 .

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