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A new theory of plate springs
A N E W T H E O R Y OF P L A T E SPRINGS.* BY DAVID
LANDAU
and P E R C Y PAPER
H. P A R R .
2.
IN our first paper 1 we made a liberal effort to show the fundamental principles underlying our new theory of plate springs, and to indicate briefly, thoug,h succinctly, the irrationality of the old theory. No direct mathematical proof was offered to substantiate our statements regarding the er.rors of the old theory for it seemed to us .that such proof was not demanded; the facts themselves, as marshalled into evidence, being deemed to be, as flae lawyers say, prima facie evidence of its inaccuracy. The reader who has followed our exposition with sufficient care will, however, doubtless find some objections and be inclined to take exception .to some of the statements as not having been proved; this may be the case, to an even greater degree, with the engineer who has had much experience in the use of plate springs and especially to him who has kept a record, or compiled statistics, of the breakages of spring plates. Our intention in the first paper was to avoid tediousness in the exposition of this apparently simple but really very complex sttbjeot, hence, it ~was, in some ways, but a foreword to this more complete and prolix expo.sition. Much, necessarily, had to be left • to the present paper; even this one will not include all of the many elements that have to be .considered, certain of which must be left for the third and (for the present) concluding paper. We shall preamble the present exposition by a few remarks which have been verified by tests and experience. Most experienced spring makers have noticed that the short plate is the one which breaks most often on ordinary plate springs ; so far then our theory has offered an intelligible--and as we hope to show, ultimately, a true--explanation of the observed facts. On the other hand, there are many exceptions to this breakage of the short leaf. It is a fact beyond contradiction, that with very many springs, other plates break with equal or, in some * Communicated by the Authors. 1 This JOURNAL,VOI. X85, April, I918, p. 48x. 699
700
DAVID LANDAU AND PERCY I-I. PARR.
[J. F. L
rare cases, with even greater frequency than do the short plates. F o r instance, in a series of endurance tests of several hundreds of springs, made a few years ago, the .majority of the breakages occurred in the master leaves; in several cases the intermediate leaves broke, but only in a comparatively small number of cases was the fracture confined to the short leaf. W h y should this be so? Indeed, from the exposition of our theory, as. given so far, it might be reasonably asked, " I f your theory is correct, how is it possible that any plate other than the sho.rt one can break first? ~' It may also be mentioned as an established fact that many commercial springs, constructed from the point of view of low cost of production combined with reasonable safety in use, are made of ordinary qualities of "carbon spring steel" except for the master leaves, which are made of high-grade "alloy spring steels." W h y this practice if the theory we have so. far expounded be correct? And how can we, on the basis of our theory, account for the many apparent exceptions to it? The answers to these questions are fairly simple, but the formal proofs are tedious and long. In order to allay the desire of the reader to. understand the reasons of the apparent--and they are only apparent--discrepancies between our theory and the facts of practice, we shall give a brief "word-picture" first and afterwards proceed to the formal mathenlatical proof. There are two principal causes which operate to induce a leaf, in a leaf spring, other than the short one to break first: for the moment we shall name only one of these and that is the taper-
ing of the ends of the leaves. This one cause--tapering--produces the most astonishing modifications in the reactions and, of course, the stresses in the leaves. T,he "life" of the spring is increased or decreased in proportion as this seemingly minor, but really most important, de c tail is given proper consideration in the design and in the manufacture. Our theoretical investigations and long practical experience justifies our making immediately the definite statement that, next to the homogeneity of the molecular structure of the
l~nished product, the tapering of the ends of the leaves is of the greatest importance, and in order to obtain the best results in the
Dec., 1918.]
A NEW THEORY OF PLATE SPRINGS.
7Ol
longevity of ~ plate spring the tapering should be carried out with math.ematiccd exactitude. W e believe that we have, in a very great measure, solved this most important question of tapering on scientific principles, and our researches appear to prove that we have been the first to define the precise effects of tapering and its resultant effect on the strength and life of leaf springs. Before the publication of the present paper, if anyone had made an inquiry of the engineer, or even of the experienced spring maker, as to what was the effect of tapering the ends of the leaves on the "st1"ength," the "life" or on the "endurance" of a spring, the average answer would have been about as follows: "The tapered-end leaves look better, but are of no particular advantage;" and as confirmatory evidence, if that be needed, "the railroad springs are scarcely ever tapered." A more astute engineer might "guess" the,t, "the tapered-end leaf is better," 'but as to why, how, and ho.w much better? he could give no answer. W e sllall, in fact, show later on that the effect of tapering may be either for the better or for the worse--the good effects produced have limitations, although, in general, tapering is beneficial as compared to no tapering. The mathematical proof is rigorous but not too easy, and an attempt will first be made to give a mentM concept of the physical effects. Let us consider, for example, Fig. IO; this is the same as Fig. 3, with the exception that the short leaf is tapered in the plane of the width ( f o r illustrative convenience only we have shown the most simple taper). Now, it is clear that with any given load placed on the end of the short leaf, it will deflect more than it would if it were not tapered. This is a simple and obvious fact which does not seem to require a formal proof but which is of the greatest importance. In other words, the effect of tapering the end of a leaf is to make that leaf, as a whole, more flexible; also, since the leaf still has the same section at the point of maximum stress, namely at the point of encastr~ment, the safe load that may be placed thereon still remains W1, although this load will cause a~ greater deflection of the short leaf than in the case of a non-tapered leaf. This being granted, we may next consider the effects on the leaf above. Since the load, 1/71, on the end of the short leaf (for convenience it may be taken as the load producing the maximum
702
DAVID LANDAU
AND P E R C Y H .
PARR.
[J. F. I.
allowed stress in that leaf) can only come from the pressure of the plate above on it, it follows .therefore that the plate above must also deflect a greater distance when the bottom plate is ~apered than w~en it is =ot. But, in order to obtain this. greater deflection of the t~late above 'we must place 'a load, W2, on it which is greater *han it was in the cases of the non-tapered bottom leaf shown in Figs. 8 and 9. Some readers may take exception to the statement that the load I/F'2 becomes greater in this .case; their judgment is not to be questioned here, but we ask them to withhold any contrary opinion until we give the more rigid proof; we ask them to acFIG. Io.
2
[
j
I
i
mr,S
cept, provisionally, our statement that I4/2 is greater when the bottom plate, No. I, is tapered than when it is not. On this basis we will continue further. Assuming ~hen that W2 has now become greater than 4/5 W1 (the value found in our first paper) we see that we have the apparently curious result that when we taper the end of the leaf below we can increase the safe load on the leaf above, and, of course, on the spring as a whole--a result that would 'hardly be expected, much less looked for. This concludes, briefly, the "word-picture" of the principles in question and of which the formal proofs will be given shortly. To avoid misunderstanding in future expositions concerning tapered-end leaf springs it is to be understood that we use
Dec., I918.]
A NEW THEORY OF PLATE SPRINGS.
703
the word taper in its most general sense, except in cases where it is specifically given a limited meaning. A taper, straight or curved, in width only, as in Fig. IO, or a taper in thickness, or any combination of the two, may be assumed to apply with equal force to the discussion. It seems to us, as the result of our experience of teaching some of these principles to others, that clarity of the mathenaatical exposition may be gained, and greater confidence placed in the formal reasoning if we place some additional physical illustrations before the reader. Realizing, as we do, the difficulties FIG.
I I.
/:
I
N
I
we had in the earlier years of this investigation, we are prone to expand somewhat our illustrations in order to avoid later dilation. T~h.e reader who is so fortunale as to have already obtained a clear mental grasp of our general ideas may skip the next few paragraphs. Consider again Fig. IO; we have stated that W2, in this case, is greater than the corresponding 1//2 of Fig. 3 for the same value of Wa; suppose then that W2 instead of being equal to 4/5V/1, as was the case for Fig. 3, is now equal to, say 9/IOW1, and let us see what happens to the stress distribution ir~ the two plates. The result of this assumption is shown in Fig. I I, which
704
DAVID LANDAU AND PERCY H. PARR.
[J. F. I.
is to the same scale as Fig. 8, in which we showed the stress distribution for the two-leaf spring of Fig. 3. The most important result is that the stress in the main, or master lea.f, has increased, by about 6 per cent. on an average, and so the metal in this leaf, which is still not stressed as high as that in the short leaf, is, however, being ~se.d to much greater advantage. The stress in the short leaf has been increased slightly, in the tapered part, but/che difference here is no/c of great account; at the same time, it is all in the direction of greater load capacity per unit weight of sprihg, the weig~ht of metal has been reduced, and that which has "been left is utilized to greater a dvaniage. The whole effect then may be said to be: If, in any spring, the "plc~te below" is tapered in any of the well-known ways, the stress in the "plate above'" is increased and, therefore, within certain limits (to be defined later), the safe load on the spring as a whole may be increased. Our inves/cigations of the history of the theory of plate springs given in our first paper indicate that this very important effect (tapering) has never seriously been considered by any prev!ous investigators. W e have used the terms "plate below" and "plate above"; it seems opportune to define here the sense in which we always use these terms before proceeding further. Any plate in a spring may be taken as the datum plate for particular purposes, in which case any shorter plate may be called the "plate below" and any longer plate, the "plate above" : thus in a two-plate spring, the short plate is the "plate below" and the long or master plate, the "plate above." We have already seen that with any plate spring composed of non-tapered plates of equal cross section, the stresses in t h e upper plates are less than those in the lower ones, and we have just shown that the effect of tapering the ends of the lower leaves is to increase ~h,e s/cresses in the upper ones. The question naturally follows--is it possible so to arrange the tapers that the reactions and stresses shall all be equal? The answer is in the affirmative. A spring having these characteristics can be constructed, and is shown in Fig. 12, which really is none other than that of Fig. A: of our Historical Irttroduction. TIh.is is the special spring regarding which all of the old and
Dec., I918.]
A NEW THEORY OF PLATE SPRINGS.
705
ordinary formulae are based, but it is never, or at least very rarely, found in practice. It will .be shown later that such a spring made of equal ,thickness plates and in which the tapers are so made that t,he momenlt of inertia is everywhere proportional to the bending moment in the overlap or step, has equal reactio.ns and the stresses are everywhere the same. T~heoretically this should be ,12he ideal spring, as it would then carry a load directly propo.rtional to the number of plates ; there are other modifying circumstances, however, w~hich will be duly discussed, that prohibit suc,h springs from carrying loads directly p.roportional .to the number of leaves, and such springs, irt consequence, are not to be found in general use. FIG. I2.
1
I
I
Before closing the general description of the physical effects of tapering the ends o.f the leaves we must pus,h the question a little further and ask: what is the effect o[ tapering the plates ~oo much? The answer is, as we shall prove in due course, that the reaotio.ns go on. increasing and, after a certain limiting amount of taper, any further taperirtg will cause the stresses to be greater in the upper leaves than in the lower ones. Since it is usually, from a practical point of view, most important that the master leaf of a spring should not be the first one to fracture, it follows that an excessive a m o u n t of tap,erir~g is most objectionable. It appears also t.hat the exact details of the tapering are of the utmost importance, and while this is a question which has been altogether neglected from the scientific point of view up to the present, it is necessary to study it With the greatest care in order to. obtain the best springs, and it is mainly through the VOL. I86, No. III6--56
706
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
study of this major detail that we have been able to. obtain some of the results indicated in our first paper. The physical illustrations of the effects of tapering the ends of the leaves can no longer be dealt with by exemplifications and analogy; they would soon grow so complex as to become unintelligible. T'he following mathematical exposition will clarify the situation. T H E G E N E R A L T H E O R Y OF L E A F S P R I N G S .
H a d we followed the usual sequence, the general theory of leaf springs, of whick the larger part of the present paper is art exposition, would have been treated in our first paper, but, for b reasons already stated the actual complexity of this apparently simple subject indicated that it would be advisable for us to discard custom in order to gain clarity, hence our first paper was written so as to give the reader greater confidence in the mathematical generalizations which follow. If a leaf spring be so constructed that each leaf is in contact with its adjacent one along its entire length, but without pressur'e--when in the unloaded, or "free" condition--then, on the application of a load, one of three things may happen. First, the leaves may separa,te everywhere except at the tips and at the centre point o.f encas.tr6ment ; second, l~he leaves may continue in contact everywhere but without pressure except at the tips; and third, the leaves may tend to foul one into another, so that there will be pressure acting between the leaves for a greater or lesser part o.f their length. Our first paper dealt with the first condition, which is usually of the greatest importance, and a hint was given as to the occasional existence of the other conditions. The second condition may be considered as the limit of either the first or the third, and so. forming the line of demarcation between them. The ,first case, carefully expounded by the theory given in our first paper, considered the leaves when subjected to load to remain in contact only at the centre point of encas,tr~ment and at the tips of the leaves: most of the commercial springs, especially those of the cheaper grades, and of the heavy springs used by the railways, fall into this class. The second case will be examined from the point of view
Dee., I918.]
A NEW TIIEORY OF 1°LATE SPRINGS.
707
of it being t~he limiting case of the first--it is not of much actual importance, but still requires to be considered in order to make the study complete. The third case is not of common occurrence in practice, nevertheless it is becoming more so, and may occur either through design or through accidents of manufacture. It is of considerable importance, and will, therefore, be considered in due course. With these preliminary remarks we now proceed to establish the generalization of our theory in as simple a manner as FIG. 13.
5
2 )/3
J
Yl 1
[L possible. F o r convenience we will first consider the two lower plates of a spring. Referring to Fig. I3, the load on the end of plate No. I, or the reaction between plates Nos. I and 2, is W1. The load on the . end of plate No. 2 is W2. Considering the bottom plate, No. I, the load W1 acting on it will produce a downward deflection which we will denote by yl, and we may write: yl =A1WI
where N1 is a coefficient which may be calculated when we know the section everywhere of plate No. I. Now, considering the second plate, the reaction WI acting at the distance l~ will produce an upward deflection, say Ya, at the distance l 1, and we may put: y3=AsW1
708
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
where A3 is a coefficient which may be calculated when we know the section everywhere of plate No. 2. Similarly; the load //K2 acting on plate No. 2 at the distance l2 will I),roduce a downward deflection at 11 which we may denote by Y6 and we can ~hus write: y6=A6W2
where, again, A 6 is a coefficient 'which may be calculated when we know the particulars of the second plate. Next, considering the equilibrium of the two plates, it is at once seen~ that: y, = y, -- y3 or :
A iWI = A 6 W 2 - - A *W,
from which. : W, W2 -
A6 A1 + A3
giving ,the ratios of the loads or reactions //V1 and VV2. W e must next note t h a t for the second leaf, the reactioll H/-~ acting at the distance lj will produce an upward deflection Y4 at 12, and we may write: 3'4 =A4WI
where N4 may be calculated. Similarly, the load //V2 acting at 12 will produce a downward deflection of 3'~ at 12, and we have: y~ = A ~W2
where again A~ is a coefficient which may be calculated. The total actual deflection then of the two-plate spring at l2 due to the load I/V2, which we shall denote by Y2, is evidently Y2 = y s - y 4 or : Y2 = A s W 2 - - A 4 W 1
The reasoning, unfortunately, is not too easy, although a careful study of the foregoing, with the aid of the diagram, will show that the question is merely one of equating to equality the • upward and downward deflections of the plates, which are produced by the various forces. Having indicated the method of procedure as applied to the two bottom leaves, we proceed to the general case, and for
Dec., I918. ]
A NEW
TItEORY
OF P L A T E
SPRINGS.
70 9
this we refer to Fig. 14. It is to be understood that the y's (small y's) refer to the deflections of the individual plates, considered apart from the spring as a whole, and the Y's (capital Y's) refer to the deflection of the "partial springs," thus: Y4~ is t h e upward deflection of t h e n + I t h leaf at l~+1 due to t h e reaction W~ a t ln; y4n--1 is t h e upward deflection of the n -t- I t h leaf a t l~ due to the reaction Wn at ln; y 4 n - 2 is the d o w n w a r d deflection of the nth leaf at l~--~ due to the reaction W~ a t l~; y4,,-3 is t h e d o w n w a r d deflection of t h e nth leaf a t In d u e to the reaction W,~ at ln; Y,~ is the d o w n w a r d deflection of the partial spring of n plates at l,~ due to the reaction Wn a t l , .
N o w let : y4n-a = Av~-aW ........................................................
Y.
= B~ W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(9) (Io)
and W , ~ - I = C,~-~ W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(x I)
then a consideration of the equality of the deflection of the plates at the points of contact shows that: y4n--2 --y4n--S = Y , ~ - t or : A 4n-2 W n -
A 4,,-s W n - , = B , ~ - , W n - ,
from which we obtain: W~-I --W,~
= C~--~ = -
A~.-2 . ..................................... B n - - t+A 4,~-- 6
(I2)
We also note that: y4n-- 3 -- y,,~-- 4 = Y,~ or : Av~--sW,~ --A4n--,Wn-
t = BnWn
from whi.ch : B , , =Av~--3--A4,~--*C,~--1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(I3)
and these equations give the fundamental relations between the deflections of the individual leaves, those of the partial (and complete) springs, and the ratios of the various reactions. It must be noticed that A~ (the deflection coefficient for the bottom plate at the eneastr6ment) is always zero, and that B,
7io
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
(the deflection coefficient o.f the partial spring of one plate only) is always equal to A 1. The complete relations of a spring can now be calculated, starting at the short plate, called No. I, and working upwards, as is explained more fully in the succeeding paragraphs. The A's can all be calculated independently, for they depend solely or~ t~he individual details of the various plates: these calFIG. 14.
ln+l
....
1"1+1
~y~n
,l
~
(n+l
y 4 n-3
In-i ~4rt ~7
n-1
xVn.]
3
S Y 2 ~yl
culations for various cross-sectional variations of ih¢ plates will be dealt wifl~ fully in a later section of this paper, and for the moment we will assume that all the A's have been determined. Note that BI=A 1. Next put n = 2 in equation (12) and we obtain : •
C1
=
-
A6
B1 + A3
from which C1 can be calculated. Then put n = 2 in equation ( I 3) and we obtain : BI=Aa=AiCI
from which B2 can be calculated.
Dec., 1918.]
A NEW' T H E O R Y OF P L A T E SPRINGS.
711
Having now calculated C~ and B2, put n = 3 in equations ( i 2 ) and ( I 3 ) , giving: C~ -
Al0 Bi + A7
and B3 =A9-AsC~
from which C2 and B3 are obtained. Proceedinff in a similar manner will give the deflection and reaction coefficients for the complete spring. The work is somewhat tedious'but not difficult and, after a few springs have been calculated out, it will easily be seen that the work of arithmetical computation can be systematized so as to minimize the actual amount of labor. It must be admitted, however, that the labor of working out completely, say, a ten-plate " g r a d e d " * spring is very considerable, and in practice, where many springs have to be calculated out, a calculating machine is certainly a desirability-almost a necessity. The above exposition contains the complete theory of the deflection and reaction relations for all leaf springs which are such that the leaves do not tend to foul into one another, and therefore covers, as previously mentioned, the great majority of commercial springs. W e now proceed to details and show how the values for the A's for the various plates are to be derived. In general terms, if y is the deflection at the distance x from the point of encastr~ment, then using the well-known relation: dZy
M
dx 2
EI
(14)
we have : E
=
£
M
dx.
E y -~
.(I5)
dx dx
.................................
(I6)
and it is seen at once that: /~l,~ ; l ~ E W~ y ~,,-s =
1,,.--X dx dx. I,~ ................................. jo
. (17)
Jo
* A " g r a d e d " spring is one whose leaves are not all of the same thickness.
712
DAVID L A N D A U AND PERCY H .
E
fi 1 , , . ~
[l,~.~
W n Y 4n-,z =
,j o E
way 4n-1
fin ,_., o
flnlnT_ ,J o
[J. F. I.
(18)
dx dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In
,),o
=
I,
PARR.
(I9)"
x dx dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~[n+l
and also : dy . " . . . . . . . . . . . Y 4,~ = y 4na + (ln+l -- In) dxl,,
~ ..........
(20)
In -- x dx . . . . . . . . . . ln-~-a
(21)
. ................
or : E W n y 4n =
do
,Jo
l n - - X dx dx + (ln+l-- In) In+~l
dy
No constants are added to the integrals, since both ~-x and y vanish together with x. These general relations, as here given, are of course in a somewhat intangible form for practical use; they will be reduced to definite formulm for particular cases later on, but before doing so, we will prove a very importaht theorem, which shows l~hat A4,-2=A4a-4, so that when A4,-4 has been calculated, A4a-2 may at once be written down as the same, thus saving twentyfive per cent. of the preliminary work. FIG.
5 ¢ 5 •
ll
15.
W
'IW
5
5 5/ 5 /
Theorem." With any cantilever (see Fig. _r5) the de/~ection at la due to a load Vd at 12 is equat to the deflection at 12 due to a load W at ll. In general, if the load V/ is distant l from the point of encastr6ment, the bending moment M is V / ( l - x ) , and if we put 7 = f ( x ) , where f(,v) is any function of x, and apply the standard
Dec., 1918.]
equation
A NEW THEORY OF PLATE SPRINGS.
713
14), we at once obtain; E d2y = l f ( x ) - x f W dx 2
(x)
Zd--Y=lff(x) d x W d d xz- - f zf~x) say = I F(x)--G(x)
y. = l
Y ( x ) d x --
G(x) dx
Now integrating the first term by parts, using unity as one factor, and writing H(x) for the second term, t'here results: -~.y = l
x F (x)--
~c
F (x) d x
- H (x)
=I { z E (z) - f x f (x) dx } --H (x) = z {x F (x) -- G (x) }--.H (x) Making~ use of these relations, the deflection at I1 due to I4/" at /2 is :
l={ l~F (I1)--G (I~))--H (I,) The deflection at 12 due to W at 11 is:
l,{11F(Ix)--a(l,) }--H(h)+(h-ll){llF(h)-G(h)} =12{l,F (l,)--G ( / , ) } - - H (I,) the same as before; this then proves the theorem. We may now proceed with the evaluation of the fundamental constants, or A's, for plates with various types of ends. or points. The most usual types of points are shown by Fig. I6, and the common names of them are as follows: No. I. Square, or plain ; No. 2. Trimmed, or trapezoidal (trap') ; No. 3. Round; No. 4. Circular;
714
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
No. 5. Parabolic, oval, o.r egg-shaped; No. 6. Square-tapered, or plain-tapered; No. 7. Trimmed-tapered, or trapezoidal-tapered; No. 8. Round-tapered ; No. 9- Circular-tapered ; No. Io. Parabolic-tapered, oval-tapered, or egg-shapetapered. There are various other shapes of points in use but those, shown in the Fig. I6 are the most common and also the most impoxhant. In many cases the leaves are finished with a slight PIG. I6.
]
N°- I.
I
N°~2. ~.~ ]
N94-.
I
)
I
No-5.
N°-3. I
>
No-6.1 I
NO_lO,' ~ bevelling of the ends, but this is of no importance from the present point of view as it is merely a matter of appearance. N o w proceeding to the calculations the first and simplest case is of course that of: No. I or Squa,re Leaf Point. For this point the leaves are simply there is no tapering either in the width or For this case, w.hich is simply that of uniform section~ and loaded at the end, if
cut off "square" and in the thickness. a simple cantilever of l is the lengt'h and y
Dec., 1918.]
i
715
N E W T H E O R Y OF P L A T E SPRINGS.
the deflection at the distance x from the point of encastr~ment, we have : . (22)
d2Y E 1 ~x 2 = W ( l - x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E l d " = W ( lx -- ~ EI
y=w(d2
'
"z*) 6
(23)
.............................................. .............
(24)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and at the end, when x = l, we have E 1 dy = VVfl~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dx 2
(23a)
W--l s , ..................................................... 3
(24a)
E1 y=
and these relations are all that are necessary in order to make the calculations for a spring with the No. I or square points to the leaves. FIG. 17 . .
2 1
4"
I
6 i,
I 7"
Considering the somewhat extensive abstract work of the preceding, it may now be well to give a prac~cal example, showing the application of the foregoing to a particular concrete case. The spring for which the calculations will be worked out is shown in Fig. 17, for which we will take plate No. I as 2 " × 3 / 1 6 " with 11=.OOlO99 and Z 1 = . O l I 7 2 ; plate No. 2 as 2 " × 7/32'' with I2 = .0o1745 and Z2 = .01595 ; and plate No. 3 as 2" × 1/4" with In = .002604 and Z~ = .02084. E will be taken as 28 × lO3
716
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
Now, in order to calculate the A's we have: F o r Plate N o . r. li=4, It =.001o99,
and yt
WtX43 3 X 2 8 X IoOX .oo I o99
.0006933 W~
or : A 1 = .oo06933.
A2 is zero always, as mentioned previously, being only introducdd for the sake of symmetry of the mathematical expressions. F o r Plate No.
2. 11=4, 12 = 6 , I 2 = . o o i 7 4 5 ,
and WD<43 = .0004366 W1 3 X 2 8 X IO6X .001745
y3 or
A :, = .ooo4366 y4 = y 3 + = .ooo4366 W ~ +
(12--/1)
dy dxa
2 )< W ! X 4 ! 2X28X 1o6X.ooI745
= .ooo4366 W l + . o o o 3 2 7 5 W 1 =.ooo7641 Wt or A 4 = ooo764 I . y5 =
14~X63 - - - = . o o i 4 7 3 W2 3 X 2 8 X IO°X .0OI 745
or A5 = .0o1473.
From t~he theorem given above: Aa = A 4 = .0007641.
F o r Plate No. , l.., = 6, l:~= 7, Is = .002604,
and yT= W'2X6 '~ = .ooo9875 I V 3 X 28 × I o nX .oo26o 4 or ,47 =
.ooo9875.
3's=3'7 + (l~--l~) ddYxl2 = .00o9875
W2 +
W~X62
2 X 2 8 X lO6X .002604
= .o0o9875 W * + .ooo2469 W , = .oo1234 rU,
D e c . , 1918.1
A
NEW
THEORY
OF PLATE
SPRINGS.
717
or A
= .0o1234
W 3 X 7~ YP - 3 X 28 X 106 X . o o 2 6 o 4 =
. o o i 5 6 8 14~ ,
Or Ag = .oo1568
F r o m the theorem: Al0 = As = .ooi234,
H a v i n g ncnv determined the values of all the A's, the B's and C's m a y readily be found, thus: B I = A 1 = ,0o06933 C~ = - A ~ = .oo07641 B I + A3 .0006933+.0004366
-.6763
B~ = ' s - - A 4 C~ = . o o i 4 7 3 - . o o o 7 6 4 I X . 6 7 6 3 = . 0 0 0 9 5 6 2 C~=
A~o _ B2+ A
.001234 - .6349 .0009562+.0009875
B3 = A 9 - - A ~ C2 = . o o i 5 6 8 - - .o01234 X .6349 =
.00o7845
It may be also noted that" Wl
Wl
W',=(;I = .6763 =
1.479 W 1
and W~
W~ _ C.,
1.479W1 .6349
2 . 3 3 0 W~
T h e s t r e n g t h modulus for bending of plate No. I being Z = . o 1 1 7 2 , then if we allow, for convenience of calculation, a m a x i m u m stress of IOO,OOO lbs. per square inch, the safe load ~Va will be iooooox.oiI72 4 -- 293 lbs. //V2 = 293 x 1.479 = 433 lbs. and l/Va, which is the safe load on the three-leaf cantilever spring, will be 293 x 2.330 = 683 lbs. Also, as the deflection I
is B 3 inches per lb. of //Va, the stiffness of the spring is B = 1275 lbs. load per inch deflection. T h e bending m o m e n t in plate No. 2, directly over the tip of plate No. 1, is 433 × 2 = 8 6 6 in. lbs. and as Z2 = .01595 the stress is 8 6 6 / . o i 5 9 5 = 54,290 lbs. per sq. in. At the centre the bending moment is 433 x 6 - 293'x 4 = 1426 in. lbs. with a stress of 89,400 lbs. per sq, in.
718
DAVID LANDAU AND PERCY H. PARR.
[J. F. I.
For plate No. 3, the bending moment directly over the tip of plate No. 2 is 683 × I = 683 in. lbs. with a stress of 32.77 o lbs. per sq. in. and at the centre the bending moment is 683 x 7 433 x 6 = 2 1 8 3 in. 1,bs. with a stress of Io4,75 o lbs. per sq. in. This last result is interesting, as it shows a slightly higher stress in the master leaf than in the short one; as previously mentioned, this is not of common occurrence and cannot occur with a non-graded non-tapered spring, but it may~ and sometimes does occtlr with graded and tapered springs. The new theo.ry thus accounts for the observed facts that with some types o.f springs the short leaf nearly always breaks first, while with other springs - - a s in the particular case just given--it is the master leaf which is usually the first to fracture. There are, of course, intermediate conditions, when one of the intermediate leaves is most apt to break, 'but these are few in comparison with the cases of fracture of either the short or the master leaf. The square-point leaf is in such general use, especially for the heavier grades o.f springs, that it is desirable to simplify, as much as possible, the calculations for the reactions and deflections of this style of spring. This can be done to a considerable extent in the fo.llowing manner: Referring back to the fundamentaI equation (7), it will be found that this can be rewritten as follows:
i(",:, ?
,
3 ~+1
1,)
~
In .
.
.
.
.
.
.
.
.
.
I
and if we write Pn for ~
'
Qn for ln+l, and Rn--1 for 3Q,~-l-r Q,3 - ,
~
we obtain : Wn-~-I
2 Wn (Pn + I) -- W n - =
3 O,)-x
]PnR,~--I
..............................
(26)
which is a reasonably simple formula for practical manipulation. In order to facilitate the practical use of this equation (26) ~'e have calculated the values of the function of 0,-1 denoted by ~,_,, and these are given in Table No. VII.
Dec., 1918.1
A N E W T H E O R Y OF P L A T E SPRINGS.
719
TABLE VII. On-x
g,,.,
Dif.
On-x
R,z.t
D~f.
i.oo 1.o1 1.o2 1.03 I.O4 1.o5 1.o6 1.o7 1.o8 1.o9 i.io i.ii 1.12 1.13 1.14 I.I 5 I.I6 1.I7 I.I8 1.19 1.2o 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 ° 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4o 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5o 1.51 1.52 1.53 1.54 1.55 1.56
2.0000 1.97o 3 1.9412 1.9127 1.8847 1.8573 1.83o4 1.8o4o 1.7782 1.7529 1.728o 1.7o37 1.6798 1.6564 1.6334 1.61o 9 1.5888 1.5672 1.5459 1.525I 1.5o46 1.4846 1.4649 1.4456 1.4266 1.4o8o 1.3897 1.3718 1.3542 1.3369 1.32oo 1.3o33 1.287o 1.27o9 1.2551 1.2396 1.2244 1.2095 1.1948 1.18o4 1.1662 1.1522 1.1386 1.1251 1.1119 1.o989 1.o861 1.o735 1.o611 1.o49o 1.o37o 1.o253 I.OI37 1.oo24 .9912 .9802 .9693
297 291 285 280 274 269 264 258 253 249 243 239 234 230 225
1.57 1.58 1.59 1.0o 1.61 1.62 1.63 1"64 1.65 1.66 1.67 1.68 1.69 1.7 ° 1.7I 1.72 1.73 1.74 1.75 1"76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9o 1.91 1.92 1.93 1.94 1.95 1"96 1.97 1'98 1-99 2.00 2.02 2.04 2.o6 2.08 2.I0 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26
.9587 .9482 -9379 .9277 .9177 .9079 .8982 .8887 .8793 .87Ol .861o .8520 .8432 .8345 .8260 .8175 .8092 .8Oll .7930 "7851 .7772 .7695 .7619 .7545 .7471 -7398 .7326 .7256 .7186 .7117 .7o5o .6983 .6917 .6852 .6788 .6725 .6662 .66Ol .6541 "6481 .6422 .6364 -63°7 .6250 .6139 .6o31 .5926 .5823 .5723 .5625 .553o .5438 .5347 .5259 .5173 .5089 .5oo 7
lo5 lO3 lO2 ioo 98 97 95 94 92 9I 9° 88 87 85 85 83 81 ~1 79 79 77 76 74 74 73 72 7° 7° 69 67 67 66 65 64 63 63 61 60 60 59 58 57 57 III :08 1o5 lO3 1oo 98 95 94 oI 88 86~ 84, ~ 82 8O
22I
216 213 208 205 200 197 193 I9 ° 186 183 179 176 173 169 167 I63 16I 158 155 152 149 147 144 142 14° 136 135 132 I3 ° 128 126 124 121 12o 117 116 113 112 IiO lO9 lO6
720
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
TABLE V I I . - - C o n t i n u e d . Dif.
On.x
R ....
Dif.
.4927 .4849 .4773 .4698 .4626 .4554 .4485 .4417 .4351 .4286
78 76 75 72 72 69 68 66 65 64
2.66 2.68 2.70 2.72 2.74 2.76 2.78 2.80
.3709 .3657 .36o7 .3558 .35Io .3463 .3416 .3371
.4222
62
.4160 .4099 .404 ° .3982 .3925 .3869 .3814 .3761
61 59 58 57 56 55 53 52
52 50 49 48 47 47 45 44 44 43 42 41 4° 4° 39 38 37
Q ....
Rn-t
2,28 2.3 ° 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46, 2.48 2.5 o 2-52 2.54 2.56 2.58 2.60 2.62 2.64
2.82
.3327
2.84 2.86 2.88 2.90 2.92 2.94 2.96 2.98 3.00
.3283 .3240 .3198 .3157 .3117 .3077 .3038 .3000 .2963
In order to show the utility of this equatiort and table, we will now work out t'he same spring (Fig. 17) that we have solved above, by this method. First find the values of the P's and Q's, thus : P : = .oo1754/.OOlO99 = 1.588 P2 = .0o2604/.oo1745 = 1.492 Qt = 6 / 4 = 1.5oo • Q2 = 7 / 6 = 1.167
R1 is to be taken from Table vii, opposite the value of Q1, or 1.5oo , where we find R1 = I.o37o. For R2, corresponding to Q2, we find 1.5888 as the tabular value corresponding to 1.16, with a first difference of 216 and 216 × .7 = 151 which: must be subtracted from the 1.5888 since R decreases as Q increases, thus giving 1.5737 as the value of R 2. Next inserting these values in equation (26) and putting W1 = I we have. since/;Vo = o : W2 W_2 =
2 ( P 1 + 1 ) _ 2 (1.588 + I) = 1.479 3QI - I 3 X 1.5 - I W 2 ( P 2 q 1) - 14"1P~RI 3Q2 - I
2 X 1.478 (1.492 + I) - 1.492 >( 1.o37 3 X 1.167 - I
= 2.327
these values being almost exactly the same as before--if more figures were taken they would be exactly the same, and in any case the differences are practically negligible.
Dec., I918. ]
A
NEW
THEORY
OF P L A T E
SPRINGS.
721
The same table (VII) can be used' for simplifying the deflection calcuhtions, for the deflection coefficient B . may be written in the form: B. . . . K. . . . . .l*,~ .................................................
(27)
6 EIn
where : K.
= 2
W,,_~
W.
R,,_~
...........................................
(28)
To apply this to the above case, we have n = 3 for the threeplate spring, 13 = .002604, 13 = 7, W, = 1.479, W3 = 2.327, K3 = 2
W3R3 _ W3
2--
1.479)< 1 . 5 7 3 7 = . 9 9 9 5 2.327
and Ba =
.9995 × 7'
6 X 28 X lO 6 X .002604
= .ooo7837
which is sensibly the same as obtained :before. It will easily be seen that this method of calculation greatly reduces the labor of computation in cases where square-point leaves are concerned. F o r most practical cases where the leaf ends are "trap" points ~he present formulre may be u s e d - - w e shall see that this is the case in the later portion of the present paper. (To be concluded.)
Jerusalem Has New Water-supply. ANON. (Engineering News-Record, vol. 8I, No. 2o, p. 89o, November I4, I918. Through the London Surveyor.)--Following closely upon the occupation of Jerusalem by the British, investigations for a new water-supply were begun by the Royal Engineers. Four days later a scheme was outlined. This was on February I4th but shortage of transportation facilities and bad weather put off the beginning of construction until about the second week in April. On June I8th water was being delivered. The supply is pumped from springs to the city. Many miles of pipe have been laid, and the supply is being delivered directly to hospitals, to "stand-pipes in every quarter of the city" and also to cisterns, the latter on condition that the cisterns be cleaned out to the satisfaction of the water authorities before they are filled. The daily yield of the springs is about 4o0,0o0 U. S. gallons. The water consumption of the city is said to be about ten times what it was formerly. It is reported that the "beneficial effects upon the health of the city has been instant and widespread." VOL. 186, NO. 1116---57
LANDAU
and P E R C Y PAPER
H. P A R R .
2.
IN our first paper 1 we made a liberal effort to show the fundamental principles underlying our new theory of plate springs, and to indicate briefly, thoug,h succinctly, the irrationality of the old theory. No direct mathematical proof was offered to substantiate our statements regarding the er.rors of the old theory for it seemed to us .that such proof was not demanded; the facts themselves, as marshalled into evidence, being deemed to be, as flae lawyers say, prima facie evidence of its inaccuracy. The reader who has followed our exposition with sufficient care will, however, doubtless find some objections and be inclined to take exception .to some of the statements as not having been proved; this may be the case, to an even greater degree, with the engineer who has had much experience in the use of plate springs and especially to him who has kept a record, or compiled statistics, of the breakages of spring plates. Our intention in the first paper was to avoid tediousness in the exposition of this apparently simple but really very complex sttbjeot, hence, it ~was, in some ways, but a foreword to this more complete and prolix expo.sition. Much, necessarily, had to be left • to the present paper; even this one will not include all of the many elements that have to be .considered, certain of which must be left for the third and (for the present) concluding paper. We shall preamble the present exposition by a few remarks which have been verified by tests and experience. Most experienced spring makers have noticed that the short plate is the one which breaks most often on ordinary plate springs ; so far then our theory has offered an intelligible--and as we hope to show, ultimately, a true--explanation of the observed facts. On the other hand, there are many exceptions to this breakage of the short leaf. It is a fact beyond contradiction, that with very many springs, other plates break with equal or, in some * Communicated by the Authors. 1 This JOURNAL,VOI. X85, April, I918, p. 48x. 699
700
DAVID LANDAU AND PERCY I-I. PARR.
[J. F. L
rare cases, with even greater frequency than do the short plates. F o r instance, in a series of endurance tests of several hundreds of springs, made a few years ago, the .majority of the breakages occurred in the master leaves; in several cases the intermediate leaves broke, but only in a comparatively small number of cases was the fracture confined to the short leaf. W h y should this be so? Indeed, from the exposition of our theory, as. given so far, it might be reasonably asked, " I f your theory is correct, how is it possible that any plate other than the sho.rt one can break first? ~' It may also be mentioned as an established fact that many commercial springs, constructed from the point of view of low cost of production combined with reasonable safety in use, are made of ordinary qualities of "carbon spring steel" except for the master leaves, which are made of high-grade "alloy spring steels." W h y this practice if the theory we have so. far expounded be correct? And how can we, on the basis of our theory, account for the many apparent exceptions to it? The answers to these questions are fairly simple, but the formal proofs are tedious and long. In order to allay the desire of the reader to. understand the reasons of the apparent--and they are only apparent--discrepancies between our theory and the facts of practice, we shall give a brief "word-picture" first and afterwards proceed to the formal mathenlatical proof. There are two principal causes which operate to induce a leaf, in a leaf spring, other than the short one to break first: for the moment we shall name only one of these and that is the taper-
ing of the ends of the leaves. This one cause--tapering--produces the most astonishing modifications in the reactions and, of course, the stresses in the leaves. T,he "life" of the spring is increased or decreased in proportion as this seemingly minor, but really most important, de c tail is given proper consideration in the design and in the manufacture. Our theoretical investigations and long practical experience justifies our making immediately the definite statement that, next to the homogeneity of the molecular structure of the
l~nished product, the tapering of the ends of the leaves is of the greatest importance, and in order to obtain the best results in the
Dec., 1918.]
A NEW THEORY OF PLATE SPRINGS.
7Ol
longevity of ~ plate spring the tapering should be carried out with math.ematiccd exactitude. W e believe that we have, in a very great measure, solved this most important question of tapering on scientific principles, and our researches appear to prove that we have been the first to define the precise effects of tapering and its resultant effect on the strength and life of leaf springs. Before the publication of the present paper, if anyone had made an inquiry of the engineer, or even of the experienced spring maker, as to what was the effect of tapering the ends of the leaves on the "st1"ength," the "life" or on the "endurance" of a spring, the average answer would have been about as follows: "The tapered-end leaves look better, but are of no particular advantage;" and as confirmatory evidence, if that be needed, "the railroad springs are scarcely ever tapered." A more astute engineer might "guess" the,t, "the tapered-end leaf is better," 'but as to why, how, and ho.w much better? he could give no answer. W e sllall, in fact, show later on that the effect of tapering may be either for the better or for the worse--the good effects produced have limitations, although, in general, tapering is beneficial as compared to no tapering. The mathematical proof is rigorous but not too easy, and an attempt will first be made to give a mentM concept of the physical effects. Let us consider, for example, Fig. IO; this is the same as Fig. 3, with the exception that the short leaf is tapered in the plane of the width ( f o r illustrative convenience only we have shown the most simple taper). Now, it is clear that with any given load placed on the end of the short leaf, it will deflect more than it would if it were not tapered. This is a simple and obvious fact which does not seem to require a formal proof but which is of the greatest importance. In other words, the effect of tapering the end of a leaf is to make that leaf, as a whole, more flexible; also, since the leaf still has the same section at the point of maximum stress, namely at the point of encastr~ment, the safe load that may be placed thereon still remains W1, although this load will cause a~ greater deflection of the short leaf than in the case of a non-tapered leaf. This being granted, we may next consider the effects on the leaf above. Since the load, 1/71, on the end of the short leaf (for convenience it may be taken as the load producing the maximum
702
DAVID LANDAU
AND P E R C Y H .
PARR.
[J. F. I.
allowed stress in that leaf) can only come from the pressure of the plate above on it, it follows .therefore that the plate above must also deflect a greater distance when the bottom plate is ~apered than w~en it is =ot. But, in order to obtain this. greater deflection of the t~late above 'we must place 'a load, W2, on it which is greater *han it was in the cases of the non-tapered bottom leaf shown in Figs. 8 and 9. Some readers may take exception to the statement that the load I/F'2 becomes greater in this .case; their judgment is not to be questioned here, but we ask them to withhold any contrary opinion until we give the more rigid proof; we ask them to acFIG. Io.
2
[
j
I
i
mr,S
cept, provisionally, our statement that I4/2 is greater when the bottom plate, No. I, is tapered than when it is not. On this basis we will continue further. Assuming ~hen that W2 has now become greater than 4/5 W1 (the value found in our first paper) we see that we have the apparently curious result that when we taper the end of the leaf below we can increase the safe load on the leaf above, and, of course, on the spring as a whole--a result that would 'hardly be expected, much less looked for. This concludes, briefly, the "word-picture" of the principles in question and of which the formal proofs will be given shortly. To avoid misunderstanding in future expositions concerning tapered-end leaf springs it is to be understood that we use
Dec., I918.]
A NEW THEORY OF PLATE SPRINGS.
703
the word taper in its most general sense, except in cases where it is specifically given a limited meaning. A taper, straight or curved, in width only, as in Fig. IO, or a taper in thickness, or any combination of the two, may be assumed to apply with equal force to the discussion. It seems to us, as the result of our experience of teaching some of these principles to others, that clarity of the mathenaatical exposition may be gained, and greater confidence placed in the formal reasoning if we place some additional physical illustrations before the reader. Realizing, as we do, the difficulties FIG.
I I.
/:
I
N
I
we had in the earlier years of this investigation, we are prone to expand somewhat our illustrations in order to avoid later dilation. T~h.e reader who is so fortunale as to have already obtained a clear mental grasp of our general ideas may skip the next few paragraphs. Consider again Fig. IO; we have stated that W2, in this case, is greater than the corresponding 1//2 of Fig. 3 for the same value of Wa; suppose then that W2 instead of being equal to 4/5V/1, as was the case for Fig. 3, is now equal to, say 9/IOW1, and let us see what happens to the stress distribution ir~ the two plates. The result of this assumption is shown in Fig. I I, which
704
DAVID LANDAU AND PERCY H. PARR.
[J. F. I.
is to the same scale as Fig. 8, in which we showed the stress distribution for the two-leaf spring of Fig. 3. The most important result is that the stress in the main, or master lea.f, has increased, by about 6 per cent. on an average, and so the metal in this leaf, which is still not stressed as high as that in the short leaf, is, however, being ~se.d to much greater advantage. The stress in the short leaf has been increased slightly, in the tapered part, but/che difference here is no/c of great account; at the same time, it is all in the direction of greater load capacity per unit weight of sprihg, the weig~ht of metal has been reduced, and that which has "been left is utilized to greater a dvaniage. The whole effect then may be said to be: If, in any spring, the "plc~te below" is tapered in any of the well-known ways, the stress in the "plate above'" is increased and, therefore, within certain limits (to be defined later), the safe load on the spring as a whole may be increased. Our inves/cigations of the history of the theory of plate springs given in our first paper indicate that this very important effect (tapering) has never seriously been considered by any prev!ous investigators. W e have used the terms "plate below" and "plate above"; it seems opportune to define here the sense in which we always use these terms before proceeding further. Any plate in a spring may be taken as the datum plate for particular purposes, in which case any shorter plate may be called the "plate below" and any longer plate, the "plate above" : thus in a two-plate spring, the short plate is the "plate below" and the long or master plate, the "plate above." We have already seen that with any plate spring composed of non-tapered plates of equal cross section, the stresses in t h e upper plates are less than those in the lower ones, and we have just shown that the effect of tapering the ends of the lower leaves is to increase ~h,e s/cresses in the upper ones. The question naturally follows--is it possible so to arrange the tapers that the reactions and stresses shall all be equal? The answer is in the affirmative. A spring having these characteristics can be constructed, and is shown in Fig. 12, which really is none other than that of Fig. A: of our Historical Irttroduction. TIh.is is the special spring regarding which all of the old and
Dec., I918.]
A NEW THEORY OF PLATE SPRINGS.
705
ordinary formulae are based, but it is never, or at least very rarely, found in practice. It will .be shown later that such a spring made of equal ,thickness plates and in which the tapers are so made that t,he momenlt of inertia is everywhere proportional to the bending moment in the overlap or step, has equal reactio.ns and the stresses are everywhere the same. T~heoretically this should be ,12he ideal spring, as it would then carry a load directly propo.rtional to the number of plates ; there are other modifying circumstances, however, w~hich will be duly discussed, that prohibit suc,h springs from carrying loads directly p.roportional .to the number of leaves, and such springs, irt consequence, are not to be found in general use. FIG. I2.
1
I
I
Before closing the general description of the physical effects of tapering the ends o.f the leaves we must pus,h the question a little further and ask: what is the effect o[ tapering the plates ~oo much? The answer is, as we shall prove in due course, that the reaotio.ns go on. increasing and, after a certain limiting amount of taper, any further taperirtg will cause the stresses to be greater in the upper leaves than in the lower ones. Since it is usually, from a practical point of view, most important that the master leaf of a spring should not be the first one to fracture, it follows that an excessive a m o u n t of tap,erir~g is most objectionable. It appears also t.hat the exact details of the tapering are of the utmost importance, and while this is a question which has been altogether neglected from the scientific point of view up to the present, it is necessary to study it With the greatest care in order to. obtain the best springs, and it is mainly through the VOL. I86, No. III6--56
706
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
study of this major detail that we have been able to. obtain some of the results indicated in our first paper. The physical illustrations of the effects of tapering the ends of the leaves can no longer be dealt with by exemplifications and analogy; they would soon grow so complex as to become unintelligible. T'he following mathematical exposition will clarify the situation. T H E G E N E R A L T H E O R Y OF L E A F S P R I N G S .
H a d we followed the usual sequence, the general theory of leaf springs, of whick the larger part of the present paper is art exposition, would have been treated in our first paper, but, for b reasons already stated the actual complexity of this apparently simple subject indicated that it would be advisable for us to discard custom in order to gain clarity, hence our first paper was written so as to give the reader greater confidence in the mathematical generalizations which follow. If a leaf spring be so constructed that each leaf is in contact with its adjacent one along its entire length, but without pressur'e--when in the unloaded, or "free" condition--then, on the application of a load, one of three things may happen. First, the leaves may separa,te everywhere except at the tips and at the centre point o.f encas.tr6ment ; second, l~he leaves may continue in contact everywhere but without pressure except at the tips; and third, the leaves may tend to foul one into another, so that there will be pressure acting between the leaves for a greater or lesser part o.f their length. Our first paper dealt with the first condition, which is usually of the greatest importance, and a hint was given as to the occasional existence of the other conditions. The second condition may be considered as the limit of either the first or the third, and so. forming the line of demarcation between them. The ,first case, carefully expounded by the theory given in our first paper, considered the leaves when subjected to load to remain in contact only at the centre point of encas,tr~ment and at the tips of the leaves: most of the commercial springs, especially those of the cheaper grades, and of the heavy springs used by the railways, fall into this class. The second case will be examined from the point of view
Dee., I918.]
A NEW TIIEORY OF 1°LATE SPRINGS.
707
of it being t~he limiting case of the first--it is not of much actual importance, but still requires to be considered in order to make the study complete. The third case is not of common occurrence in practice, nevertheless it is becoming more so, and may occur either through design or through accidents of manufacture. It is of considerable importance, and will, therefore, be considered in due course. With these preliminary remarks we now proceed to establish the generalization of our theory in as simple a manner as FIG. 13.
5
2 )/3
J
Yl 1
[L possible. F o r convenience we will first consider the two lower plates of a spring. Referring to Fig. I3, the load on the end of plate No. I, or the reaction between plates Nos. I and 2, is W1. The load on the . end of plate No. 2 is W2. Considering the bottom plate, No. I, the load W1 acting on it will produce a downward deflection which we will denote by yl, and we may write: yl =A1WI
where N1 is a coefficient which may be calculated when we know the section everywhere of plate No. I. Now, considering the second plate, the reaction WI acting at the distance l~ will produce an upward deflection, say Ya, at the distance l 1, and we may put: y3=AsW1
708
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
where A3 is a coefficient which may be calculated when we know the section everywhere of plate No. 2. Similarly; the load //K2 acting on plate No. 2 at the distance l2 will I),roduce a downward deflection at 11 which we may denote by Y6 and we can ~hus write: y6=A6W2
where, again, A 6 is a coefficient 'which may be calculated when we know the particulars of the second plate. Next, considering the equilibrium of the two plates, it is at once seen~ that: y, = y, -- y3 or :
A iWI = A 6 W 2 - - A *W,
from which. : W, W2 -
A6 A1 + A3
giving ,the ratios of the loads or reactions //V1 and VV2. W e must next note t h a t for the second leaf, the reactioll H/-~ acting at the distance lj will produce an upward deflection Y4 at 12, and we may write: 3'4 =A4WI
where N4 may be calculated. Similarly, the load //V2 acting at 12 will produce a downward deflection of 3'~ at 12, and we have: y~ = A ~W2
where again A~ is a coefficient which may be calculated. The total actual deflection then of the two-plate spring at l2 due to the load I/V2, which we shall denote by Y2, is evidently Y2 = y s - y 4 or : Y2 = A s W 2 - - A 4 W 1
The reasoning, unfortunately, is not too easy, although a careful study of the foregoing, with the aid of the diagram, will show that the question is merely one of equating to equality the • upward and downward deflections of the plates, which are produced by the various forces. Having indicated the method of procedure as applied to the two bottom leaves, we proceed to the general case, and for
Dec., I918. ]
A NEW
TItEORY
OF P L A T E
SPRINGS.
70 9
this we refer to Fig. 14. It is to be understood that the y's (small y's) refer to the deflections of the individual plates, considered apart from the spring as a whole, and the Y's (capital Y's) refer to the deflection of the "partial springs," thus: Y4~ is t h e upward deflection of t h e n + I t h leaf at l~+1 due to t h e reaction W~ a t ln; y4n--1 is t h e upward deflection of the n -t- I t h leaf a t l~ due to the reaction Wn at ln; y 4 n - 2 is the d o w n w a r d deflection of the nth leaf at l~--~ due to the reaction W~ a t l~; y4,,-3 is t h e d o w n w a r d deflection of t h e nth leaf a t In d u e to the reaction W,~ at ln; Y,~ is the d o w n w a r d deflection of the partial spring of n plates at l,~ due to the reaction Wn a t l , .
N o w let : y4n-a = Av~-aW ........................................................
Y.
= B~ W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(9) (Io)
and W , ~ - I = C,~-~ W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(x I)
then a consideration of the equality of the deflection of the plates at the points of contact shows that: y4n--2 --y4n--S = Y , ~ - t or : A 4n-2 W n -
A 4,,-s W n - , = B , ~ - , W n - ,
from which we obtain: W~-I --W,~
= C~--~ = -
A~.-2 . ..................................... B n - - t+A 4,~-- 6
(I2)
We also note that: y4n-- 3 -- y,,~-- 4 = Y,~ or : Av~--sW,~ --A4n--,Wn-
t = BnWn
from whi.ch : B , , =Av~--3--A4,~--*C,~--1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(I3)
and these equations give the fundamental relations between the deflections of the individual leaves, those of the partial (and complete) springs, and the ratios of the various reactions. It must be noticed that A~ (the deflection coefficient for the bottom plate at the eneastr6ment) is always zero, and that B,
7io
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
(the deflection coefficient o.f the partial spring of one plate only) is always equal to A 1. The complete relations of a spring can now be calculated, starting at the short plate, called No. I, and working upwards, as is explained more fully in the succeeding paragraphs. The A's can all be calculated independently, for they depend solely or~ t~he individual details of the various plates: these calFIG. 14.
ln+l
....
1"1+1
~y~n
,l
~
(n+l
y 4 n-3
In-i ~4rt ~7
n-1
xVn.]
3
S Y 2 ~yl
culations for various cross-sectional variations of ih¢ plates will be dealt wifl~ fully in a later section of this paper, and for the moment we will assume that all the A's have been determined. Note that BI=A 1. Next put n = 2 in equation (12) and we obtain : •
C1
=
-
A6
B1 + A3
from which C1 can be calculated. Then put n = 2 in equation ( I 3) and we obtain : BI=Aa=AiCI
from which B2 can be calculated.
Dec., 1918.]
A NEW' T H E O R Y OF P L A T E SPRINGS.
711
Having now calculated C~ and B2, put n = 3 in equations ( i 2 ) and ( I 3 ) , giving: C~ -
Al0 Bi + A7
and B3 =A9-AsC~
from which C2 and B3 are obtained. Proceedinff in a similar manner will give the deflection and reaction coefficients for the complete spring. The work is somewhat tedious'but not difficult and, after a few springs have been calculated out, it will easily be seen that the work of arithmetical computation can be systematized so as to minimize the actual amount of labor. It must be admitted, however, that the labor of working out completely, say, a ten-plate " g r a d e d " * spring is very considerable, and in practice, where many springs have to be calculated out, a calculating machine is certainly a desirability-almost a necessity. The above exposition contains the complete theory of the deflection and reaction relations for all leaf springs which are such that the leaves do not tend to foul into one another, and therefore covers, as previously mentioned, the great majority of commercial springs. W e now proceed to details and show how the values for the A's for the various plates are to be derived. In general terms, if y is the deflection at the distance x from the point of encastr~ment, then using the well-known relation: dZy
M
dx 2
EI
(14)
we have : E
=
£
M
dx.
E y -~
.(I5)
dx dx
.................................
(I6)
and it is seen at once that: /~l,~ ; l ~ E W~ y ~,,-s =
1,,.--X dx dx. I,~ ................................. jo
. (17)
Jo
* A " g r a d e d " spring is one whose leaves are not all of the same thickness.
712
DAVID L A N D A U AND PERCY H .
E
fi 1 , , . ~
[l,~.~
W n Y 4n-,z =
,j o E
way 4n-1
fin ,_., o
flnlnT_ ,J o
[J. F. I.
(18)
dx dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In
,),o
=
I,
PARR.
(I9)"
x dx dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~[n+l
and also : dy . " . . . . . . . . . . . Y 4,~ = y 4na + (ln+l -- In) dxl,,
~ ..........
(20)
In -- x dx . . . . . . . . . . ln-~-a
(21)
. ................
or : E W n y 4n =
do
,Jo
l n - - X dx dx + (ln+l-- In) In+~l
dy
No constants are added to the integrals, since both ~-x and y vanish together with x. These general relations, as here given, are of course in a somewhat intangible form for practical use; they will be reduced to definite formulm for particular cases later on, but before doing so, we will prove a very importaht theorem, which shows l~hat A4,-2=A4a-4, so that when A4,-4 has been calculated, A4a-2 may at once be written down as the same, thus saving twentyfive per cent. of the preliminary work. FIG.
5 ¢ 5 •
ll
15.
W
'IW
5
5 5/ 5 /
Theorem." With any cantilever (see Fig. _r5) the de/~ection at la due to a load Vd at 12 is equat to the deflection at 12 due to a load W at ll. In general, if the load V/ is distant l from the point of encastr6ment, the bending moment M is V / ( l - x ) , and if we put 7 = f ( x ) , where f(,v) is any function of x, and apply the standard
Dec., 1918.]
equation
A NEW THEORY OF PLATE SPRINGS.
713
14), we at once obtain; E d2y = l f ( x ) - x f W dx 2
(x)
Zd--Y=lff(x) d x W d d xz- - f zf~x) say = I F(x)--G(x)
y. = l
Y ( x ) d x --
G(x) dx
Now integrating the first term by parts, using unity as one factor, and writing H(x) for the second term, t'here results: -~.y = l
x F (x)--
~c
F (x) d x
- H (x)
=I { z E (z) - f x f (x) dx } --H (x) = z {x F (x) -- G (x) }--.H (x) Making~ use of these relations, the deflection at I1 due to I4/" at /2 is :
l={ l~F (I1)--G (I~))--H (I,) The deflection at 12 due to W at 11 is:
l,{11F(Ix)--a(l,) }--H(h)+(h-ll){llF(h)-G(h)} =12{l,F (l,)--G ( / , ) } - - H (I,) the same as before; this then proves the theorem. We may now proceed with the evaluation of the fundamental constants, or A's, for plates with various types of ends. or points. The most usual types of points are shown by Fig. I6, and the common names of them are as follows: No. I. Square, or plain ; No. 2. Trimmed, or trapezoidal (trap') ; No. 3. Round; No. 4. Circular;
714
DAVID LANDAU AND PERCY H . PARR.
[J. F. I.
No. 5. Parabolic, oval, o.r egg-shaped; No. 6. Square-tapered, or plain-tapered; No. 7. Trimmed-tapered, or trapezoidal-tapered; No. 8. Round-tapered ; No. 9- Circular-tapered ; No. Io. Parabolic-tapered, oval-tapered, or egg-shapetapered. There are various other shapes of points in use but those, shown in the Fig. I6 are the most common and also the most impoxhant. In many cases the leaves are finished with a slight PIG. I6.
]
N°- I.
I
N°~2. ~.~ ]
N94-.
I
)
I
No-5.
N°-3. I
>
No-6.1 I
NO_lO,' ~ bevelling of the ends, but this is of no importance from the present point of view as it is merely a matter of appearance. N o w proceeding to the calculations the first and simplest case is of course that of: No. I or Squa,re Leaf Point. For this point the leaves are simply there is no tapering either in the width or For this case, w.hich is simply that of uniform section~ and loaded at the end, if
cut off "square" and in the thickness. a simple cantilever of l is the lengt'h and y
Dec., 1918.]
i
715
N E W T H E O R Y OF P L A T E SPRINGS.
the deflection at the distance x from the point of encastr~ment, we have : . (22)
d2Y E 1 ~x 2 = W ( l - x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E l d " = W ( lx -- ~ EI
y=w(d2
'
"z*) 6
(23)
.............................................. .............
(24)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and at the end, when x = l, we have E 1 dy = VVfl~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dx 2
(23a)
W--l s , ..................................................... 3
(24a)
E1 y=
and these relations are all that are necessary in order to make the calculations for a spring with the No. I or square points to the leaves. FIG. 17 . .
2 1
4"
I
6 i,
I 7"
Considering the somewhat extensive abstract work of the preceding, it may now be well to give a prac~cal example, showing the application of the foregoing to a particular concrete case. The spring for which the calculations will be worked out is shown in Fig. 17, for which we will take plate No. I as 2 " × 3 / 1 6 " with 11=.OOlO99 and Z 1 = . O l I 7 2 ; plate No. 2 as 2 " × 7/32'' with I2 = .0o1745 and Z2 = .01595 ; and plate No. 3 as 2" × 1/4" with In = .002604 and Z~ = .02084. E will be taken as 28 × lO3
716
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
Now, in order to calculate the A's we have: F o r Plate N o . r. li=4, It =.001o99,
and yt
WtX43 3 X 2 8 X IoOX .oo I o99
.0006933 W~
or : A 1 = .oo06933.
A2 is zero always, as mentioned previously, being only introducdd for the sake of symmetry of the mathematical expressions. F o r Plate No.
2. 11=4, 12 = 6 , I 2 = . o o i 7 4 5 ,
and WD<43 = .0004366 W1 3 X 2 8 X IO6X .001745
y3 or
A :, = .ooo4366 y4 = y 3 + = .ooo4366 W ~ +
(12--/1)
dy dxa
2 )< W ! X 4 ! 2X28X 1o6X.ooI745
= .ooo4366 W l + . o o o 3 2 7 5 W 1 =.ooo7641 Wt or A 4 = ooo764 I . y5 =
14~X63 - - - = . o o i 4 7 3 W2 3 X 2 8 X IO°X .0OI 745
or A5 = .0o1473.
From t~he theorem given above: Aa = A 4 = .0007641.
F o r Plate No. , l.., = 6, l:~= 7, Is = .002604,
and yT= W'2X6 '~ = .ooo9875 I V 3 X 28 × I o nX .oo26o 4 or ,47 =
.ooo9875.
3's=3'7 + (l~--l~) ddYxl2 = .00o9875
W2 +
W~X62
2 X 2 8 X lO6X .002604
= .o0o9875 W * + .ooo2469 W , = .oo1234 rU,
D e c . , 1918.1
A
NEW
THEORY
OF PLATE
SPRINGS.
717
or A
= .0o1234
W 3 X 7~ YP - 3 X 28 X 106 X . o o 2 6 o 4 =
. o o i 5 6 8 14~ ,
Or Ag = .oo1568
F r o m the theorem: Al0 = As = .ooi234,
H a v i n g ncnv determined the values of all the A's, the B's and C's m a y readily be found, thus: B I = A 1 = ,0o06933 C~ = - A ~ = .oo07641 B I + A3 .0006933+.0004366
-.6763
B~ = ' s - - A 4 C~ = . o o i 4 7 3 - . o o o 7 6 4 I X . 6 7 6 3 = . 0 0 0 9 5 6 2 C~=
A~o _ B2+ A
.001234 - .6349 .0009562+.0009875
B3 = A 9 - - A ~ C2 = . o o i 5 6 8 - - .o01234 X .6349 =
.00o7845
It may be also noted that" Wl
Wl
W',=(;I = .6763 =
1.479 W 1
and W~
W~ _ C.,
1.479W1 .6349
2 . 3 3 0 W~
T h e s t r e n g t h modulus for bending of plate No. I being Z = . o 1 1 7 2 , then if we allow, for convenience of calculation, a m a x i m u m stress of IOO,OOO lbs. per square inch, the safe load ~Va will be iooooox.oiI72 4 -- 293 lbs. //V2 = 293 x 1.479 = 433 lbs. and l/Va, which is the safe load on the three-leaf cantilever spring, will be 293 x 2.330 = 683 lbs. Also, as the deflection I
is B 3 inches per lb. of //Va, the stiffness of the spring is B = 1275 lbs. load per inch deflection. T h e bending m o m e n t in plate No. 2, directly over the tip of plate No. 1, is 433 × 2 = 8 6 6 in. lbs. and as Z2 = .01595 the stress is 8 6 6 / . o i 5 9 5 = 54,290 lbs. per sq. in. At the centre the bending moment is 433 x 6 - 293'x 4 = 1426 in. lbs. with a stress of 89,400 lbs. per sq, in.
718
DAVID LANDAU AND PERCY H. PARR.
[J. F. I.
For plate No. 3, the bending moment directly over the tip of plate No. 2 is 683 × I = 683 in. lbs. with a stress of 32.77 o lbs. per sq. in. and at the centre the bending moment is 683 x 7 433 x 6 = 2 1 8 3 in. 1,bs. with a stress of Io4,75 o lbs. per sq. in. This last result is interesting, as it shows a slightly higher stress in the master leaf than in the short one; as previously mentioned, this is not of common occurrence and cannot occur with a non-graded non-tapered spring, but it may~ and sometimes does occtlr with graded and tapered springs. The new theo.ry thus accounts for the observed facts that with some types o.f springs the short leaf nearly always breaks first, while with other springs - - a s in the particular case just given--it is the master leaf which is usually the first to fracture. There are, of course, intermediate conditions, when one of the intermediate leaves is most apt to break, 'but these are few in comparison with the cases of fracture of either the short or the master leaf. The square-point leaf is in such general use, especially for the heavier grades o.f springs, that it is desirable to simplify, as much as possible, the calculations for the reactions and deflections of this style of spring. This can be done to a considerable extent in the fo.llowing manner: Referring back to the fundamentaI equation (7), it will be found that this can be rewritten as follows:
i(",:, ?
,
3 ~+1
1,)
~
In .
.
.
.
.
.
.
.
.
.
I
and if we write Pn for ~
'
Qn for ln+l, and Rn--1 for 3Q,~-l-r Q,3 - ,
~
we obtain : Wn-~-I
2 Wn (Pn + I) -- W n - =
3 O,)-x
]PnR,~--I
..............................
(26)
which is a reasonably simple formula for practical manipulation. In order to facilitate the practical use of this equation (26) ~'e have calculated the values of the function of 0,-1 denoted by ~,_,, and these are given in Table No. VII.
Dec., 1918.1
A N E W T H E O R Y OF P L A T E SPRINGS.
719
TABLE VII. On-x
g,,.,
Dif.
On-x
R,z.t
D~f.
i.oo 1.o1 1.o2 1.03 I.O4 1.o5 1.o6 1.o7 1.o8 1.o9 i.io i.ii 1.12 1.13 1.14 I.I 5 I.I6 1.I7 I.I8 1.19 1.2o 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 ° 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4o 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5o 1.51 1.52 1.53 1.54 1.55 1.56
2.0000 1.97o 3 1.9412 1.9127 1.8847 1.8573 1.83o4 1.8o4o 1.7782 1.7529 1.728o 1.7o37 1.6798 1.6564 1.6334 1.61o 9 1.5888 1.5672 1.5459 1.525I 1.5o46 1.4846 1.4649 1.4456 1.4266 1.4o8o 1.3897 1.3718 1.3542 1.3369 1.32oo 1.3o33 1.287o 1.27o9 1.2551 1.2396 1.2244 1.2095 1.1948 1.18o4 1.1662 1.1522 1.1386 1.1251 1.1119 1.o989 1.o861 1.o735 1.o611 1.o49o 1.o37o 1.o253 I.OI37 1.oo24 .9912 .9802 .9693
297 291 285 280 274 269 264 258 253 249 243 239 234 230 225
1.57 1.58 1.59 1.0o 1.61 1.62 1.63 1"64 1.65 1.66 1.67 1.68 1.69 1.7 ° 1.7I 1.72 1.73 1.74 1.75 1"76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9o 1.91 1.92 1.93 1.94 1.95 1"96 1.97 1'98 1-99 2.00 2.02 2.04 2.o6 2.08 2.I0 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26
.9587 .9482 -9379 .9277 .9177 .9079 .8982 .8887 .8793 .87Ol .861o .8520 .8432 .8345 .8260 .8175 .8092 .8Oll .7930 "7851 .7772 .7695 .7619 .7545 .7471 -7398 .7326 .7256 .7186 .7117 .7o5o .6983 .6917 .6852 .6788 .6725 .6662 .66Ol .6541 "6481 .6422 .6364 -63°7 .6250 .6139 .6o31 .5926 .5823 .5723 .5625 .553o .5438 .5347 .5259 .5173 .5089 .5oo 7
lo5 lO3 lO2 ioo 98 97 95 94 92 9I 9° 88 87 85 85 83 81 ~1 79 79 77 76 74 74 73 72 7° 7° 69 67 67 66 65 64 63 63 61 60 60 59 58 57 57 III :08 1o5 lO3 1oo 98 95 94 oI 88 86~ 84, ~ 82 8O
22I
216 213 208 205 200 197 193 I9 ° 186 183 179 176 173 169 167 I63 16I 158 155 152 149 147 144 142 14° 136 135 132 I3 ° 128 126 124 121 12o 117 116 113 112 IiO lO9 lO6
720
DAVID
LANDAU
AND
PERCY
H.
PARR.
[J. F. I.
TABLE V I I . - - C o n t i n u e d . Dif.
On.x
R ....
Dif.
.4927 .4849 .4773 .4698 .4626 .4554 .4485 .4417 .4351 .4286
78 76 75 72 72 69 68 66 65 64
2.66 2.68 2.70 2.72 2.74 2.76 2.78 2.80
.3709 .3657 .36o7 .3558 .35Io .3463 .3416 .3371
.4222
62
.4160 .4099 .404 ° .3982 .3925 .3869 .3814 .3761
61 59 58 57 56 55 53 52
52 50 49 48 47 47 45 44 44 43 42 41 4° 4° 39 38 37
Q ....
Rn-t
2,28 2.3 ° 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46, 2.48 2.5 o 2-52 2.54 2.56 2.58 2.60 2.62 2.64
2.82
.3327
2.84 2.86 2.88 2.90 2.92 2.94 2.96 2.98 3.00
.3283 .3240 .3198 .3157 .3117 .3077 .3038 .3000 .2963
In order to show the utility of this equatiort and table, we will now work out t'he same spring (Fig. 17) that we have solved above, by this method. First find the values of the P's and Q's, thus : P : = .oo1754/.OOlO99 = 1.588 P2 = .0o2604/.oo1745 = 1.492 Qt = 6 / 4 = 1.5oo • Q2 = 7 / 6 = 1.167
R1 is to be taken from Table vii, opposite the value of Q1, or 1.5oo , where we find R1 = I.o37o. For R2, corresponding to Q2, we find 1.5888 as the tabular value corresponding to 1.16, with a first difference of 216 and 216 × .7 = 151 which: must be subtracted from the 1.5888 since R decreases as Q increases, thus giving 1.5737 as the value of R 2. Next inserting these values in equation (26) and putting W1 = I we have. since/;Vo = o : W2 W_2 =
2 ( P 1 + 1 ) _ 2 (1.588 + I) = 1.479 3QI - I 3 X 1.5 - I W 2 ( P 2 q 1) - 14"1P~RI 3Q2 - I
2 X 1.478 (1.492 + I) - 1.492 >( 1.o37 3 X 1.167 - I
= 2.327
these values being almost exactly the same as before--if more figures were taken they would be exactly the same, and in any case the differences are practically negligible.
Dec., I918. ]
A
NEW
THEORY
OF P L A T E
SPRINGS.
721
The same table (VII) can be used' for simplifying the deflection calcuhtions, for the deflection coefficient B . may be written in the form: B. . . . K. . . . . .l*,~ .................................................
(27)
6 EIn
where : K.
= 2
W,,_~
W.
R,,_~
...........................................
(28)
To apply this to the above case, we have n = 3 for the threeplate spring, 13 = .002604, 13 = 7, W, = 1.479, W3 = 2.327, K3 = 2
W3R3 _ W3
2--
1.479)< 1 . 5 7 3 7 = . 9 9 9 5 2.327
and Ba =
.9995 × 7'
6 X 28 X lO 6 X .002604
= .ooo7837
which is sensibly the same as obtained :before. It will easily be seen that this method of calculation greatly reduces the labor of computation in cases where square-point leaves are concerned. F o r most practical cases where the leaf ends are "trap" points ~he present formulre may be u s e d - - w e shall see that this is the case in the later portion of the present paper. (To be concluded.)
Jerusalem Has New Water-supply. ANON. (Engineering News-Record, vol. 8I, No. 2o, p. 89o, November I4, I918. Through the London Surveyor.)--Following closely upon the occupation of Jerusalem by the British, investigations for a new water-supply were begun by the Royal Engineers. Four days later a scheme was outlined. This was on February I4th but shortage of transportation facilities and bad weather put off the beginning of construction until about the second week in April. On June I8th water was being delivered. The supply is pumped from springs to the city. Many miles of pipe have been laid, and the supply is being delivered directly to hospitals, to "stand-pipes in every quarter of the city" and also to cisterns, the latter on condition that the cisterns be cleaned out to the satisfaction of the water authorities before they are filled. The daily yield of the springs is about 4o0,0o0 U. S. gallons. The water consumption of the city is said to be about ten times what it was formerly. It is reported that the "beneficial effects upon the health of the city has been instant and widespread." VOL. 186, NO. 1116---57