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 ...........
Linear plot of three pin driven blade ...........
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 (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 0° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 120° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (one pin drive, pin at 240° position)
Southwell Plot (three pin drive)....................
Southwell Plot (three pin drive) ..................
Southwell Plot (three pin drive) ..................
Southwell Plot (three pin drive) ..................
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 (one pin drive, pin at 300 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 60 position) ...................................
Fringe pattern for 4" disc (one pin drive, pin at 120 position) ..................................
Fringe pattern for 4" disc (one pin drive, pin at 180 position) ...................................
Frigge pattern for 4" disc (one pin drive, pin at240 position) ...................................
Fringe pattern for 4" disc (one pin drive, pin at 300 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 ................................ 4(a) 113
Southwell plot d a t a ................................ 4(b) 114
Southwell plot d a t a ................................ 4(c) 115
Southwell plot d a t a ................................ 5(a) 116
Southwell plot d a t a ................................ 5(b) 117
Southwell plot d a t a ................................ 5(c) 118
Buckling loads from Southwell plots ............... 6 119
Buckling loads from Southwell plots ............... 7 120
Southwell plot d a t a ................................ 8(a) 121
Southwell plot d a t a ............................... 8(b) 122
Southwell plot d a t a ............................... 8(c) 123
Buckling loads from Southwell plots ............... 9 124
(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
Topography of one pin driven Blade
After 150 hours' use
r44
Topography of one pin driven Blade
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 >
Linear Plot of one pin driven Blade
fig loco
>• to 40* e«0* ao‘ ioo‘ iao‘ v*o' ifo‘ mo* 200' ago* 7*0' 2*0' **o‘___ *» * » f° ‘__ 2«£l__ 2 *°'
Linear Plot of one pin driven Blade
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)
Linear Plot of three pin driven Blade
(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)
Linear Plot of three pin driven Blade
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
PI N A T 1 2 0 ° P O S I T I O N
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 O N 6 F O R 3 0 0 ° R A D I A L L I N E
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 )
PI N A T 1 2 0 ° P O S I T I O N
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
S T A T I O N 7 F O R 3 0 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 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 )
PI N A T 2 4 0 ° P O S I T I O N
S T A T I O N 5 F O R 3 0 0 ° R A D I A L L I N 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
S T A T I O N 4 F O R 3 0 0 ° R A D I A L L I N E
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)
PIN AT 120° POSITION
oo
fig. 17(d) FRINGE PATTERN FOR 5"
PIN AT 180° POSITION
DIA. DISC. (ONE PIN DRIVE)
-
fig. 17(e) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)
PIN AT 240° POSITION
1
4
fig. 17(f) FRINGE PATTERN FOR 5" DIA. DISC. (ONE PIN DRIVE)
PIN AT 300° POSITION
■ 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)
PIN AT 60° POSITION
\
v£>
fig. 19(c) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)
PIN AT 120° POSITION
f
fig. 19(d) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)
PIN AT 180° POSITION
r
fig. 19(e) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)
PIN AT 240° POSITION
jfig. 19(f) FRINGE PATTERN FOR 4" DIA. DISC. (ONE PIN DRIVE)
PIN AT 300° POSITION
&
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
PIN TRAVELLED THROUGH 0°
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 "
PIN TRAVELLED THROUGH 0°
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
PIN TRAVELLED THROUGH 120°
TABLE 4(a)
outhvell Plot Data
113
PIN TRAVELLED THROUGH 120°
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
PIN TRAVELLED THROUGH 120°
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
PIN TRAVELLED THROUGH 240°
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 "
PIN TRAVELLED THROUGH 120°
TABLE 6
Buckling Loads from Southwell Plot
119
PCR 60° 190° 300°
S7 1850 lbf> 700 lbf. 825 lbf>
S6 1116 " 1700 " 1300 '•
S5 1550 " 425 "
S4 1000 "
PIN TRAVELLED THROUGH 240°
TABLE 7
Buckling Loads from Southwell Plot
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
Buckling Loads from Southwell Plot
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.
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C a n a d a , Ottawa, O n t a r io .
134
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P h i la d e lp h ia 42 P .A . V o l . 5 N o . 1 J a n u a r y , F e b r u a r y ,
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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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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.
1 9 . " T h e S t a b i l i t y o f a f l a t s t r i p c la m p e d a t e a c h
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 .
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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 .