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conditions for the end supports, and then there is sufficient material available for finding, firstly, the deflections, y, at each stay; secondly, the bending moments, m, at each stay, both in the girder and in the plate; and thirdly, the pulls, Q, in each stay.

It will then be found that much depends on the rigidity of the girder and of the flanges, and also on the distance of the first stays from the flanges, as to whether these stays are in tension or in compression. The present rules of 'Lloyd's Register' and of the Board of Trade encourage the practice of placing the end stays near the flange seams, thereby throwing an undue stress on the curved parts. This is also bad practice, for the proximity of the stay nut to the flange makes caulking difficult. Assuming, however, that the end stays are so placed that they carry the same load as the others, the approximate maximum bending moment in the girder can easily be worked out. Let L be the total length of the girder, while 1 and n are the pitch and number of stays; then L - (n-1).1 is double the distance of the end stays from the girder ends. If w is the distance apart of the girders, and p the pressure in the boiler, then the maximum bending moment in the girder is m as follows:

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h2.t

s
6. m
3 p.W

Here h and t are respectively the 2' h2.t height and thickness of the girder.

In the above formulæ the product n.l is the length of the plate whose load is borne by the girder. For even numbers of stays the formulæ of the table are reduced to a simpler one.

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This formula will also serve approximately for uneven numbers of stays, but, as can easily be seen by trial, the remainder of the fraction of is not 0-5, but always a little less, viz. 3, 3, , 6, , 1's, &c.

Stayed Flat Plates.—The formulæ which were worked out for the continuous beam are not directly applicable to stayed flat plates; they only give average results for the whole width of such plates. They also do not take into account the cross bending. Some idea of the stresses in these plates will be gained by the following view. Let it be assumed that the boiler pressure is halved, and that one

half is active in producing bending along the pitch line B, fig. 146, while the other half produces bending along the pitch line A. The mean bending moment in either direction is then

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On account of the easy elastic curvature of stayed plates along lines which pass half

way between the stays, these bending moments

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are probably reduced to half of what they would be in a continuous beam, viz. :

P.B. A2
ma =

P. A.B2
i me =
2.2.24

2.2.24

Now, according to Mr. Guest's experiments (p. 168), the presence of a cross tension stress does not affect the strength of the material, so that on the tension side of the plate we could allow two cross proof stresses of, say,

M 25,000 = 6.

; also 25,000 B.t? 16.t2

16.t?

P. B2

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This is a very high result, and would indicate, as would be imagined, that stayed plates are not weakest in the spaces between the stays. If we take the values deduced from Professor C. Bach's experiments (p. 169) as being correct, we must admit that plastic bending due to double compression on one side will take place if the stress exceeds say 6,000 lbs. per square

inch :

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This is also a very high value.

If now we look at the plate close to the stays, we may safely assume that, because of the concentration of forces at these points, the values of m at the supports as found above would not be reduced by half, but probably increased twofold.

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As the plates are perforated, and as the screwed stays cannot be assumed to be part of the plate, there can be no radial stresses at the circumference of the holes, and we have only simple stresses, which, up to the proof pressure, might be allowed to reach 25,000 lbs. per square inch.

25,000 =
B.t? 2.t2

2.72

6 6.ma

P. A2

; also 25,000 = p.B?

This indicates that the stresses near the stays are about eight times. as severe as between the stays, though, if the deductions drawn from Professor Bach's experiments are correct, the material, being more plastic under the conditions near the centres of the plates, would only be about 100 per cent. stronger than near the stays. If these views are correct the working pressure of a stayed flat plate would be

25,000. t2 p=

A?

all dimensions being expressed in inches. If they be expressed in sixteenths of an inch, then

t2 p = 97.5.

A2

and the first indication of giving way would be at about double this. pressure.

That even this result is not quite correct will be admitted when we consider the question of size of stay. Assuming that the screwed plate is not supported by screwed stays, but is resting on supports, then we may distinguish three cases. Firstly, the plate over the supports is not perforated. We have severe drum tension on the one side and severe strangling stress on the other. Secondly, if we drill a very small hole just over the support the compound stresses give way to single tension and compression stresses of great intensity. Thirdly, if the holes are made large, the diameters of the supports being of course proportionately increased, these simple stresses will be much reduced. Therefore the size of the stay is an important factor, all the more because of the shearing stress set up in the hole. This is fully borne out by experiments, and published rules have followed this up in so far as that they only allow a low working stress for screwed stays, and that they allow higher working pressures to plates supported by stays with nuts and washers. In such cases the working stress on the stays might also be increased, and, on the other hand, allowances might be made in the thickness of the plate when the screwed stays without washers are of very large diameter—as, for instance, when these stays are tubes.

Irregularly Stayed Plates.-By applying the formulæ of continuous beams to irregularly stayed plates-for instance, such as occur in boiler backs (fig. 147)-it can be shown that the stresses near

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FIG, 147

the stays on either side of the pitch b are greater than the usually accepted formulæ would lead one to expect. The value of m, has to be increased from

p.53 12

1.12

m =

P.12

to mi

It is exceedingly difficult to arrive at a satisfactory view of the case illustrated in fig. 148, where the pitches A and B are not equal. The stresses set up parallel to the pitches A will certainly be affected by the pitch B, for the more this is reduced the less metal will remain

between the stays, and in the extreme case 6

when they nearly touch each other, i.e.

when B=d, the stress S,, due to the A--* А

curvature at the line of holes, would be 6*- increased to

2. p.78

: and comparing this

d. t2
B with the above value S, for such cases

when A and B are equal, we find that
S 4 A
S

3. ď
FIG. 148

On the other hand, when A and B ar

nearly equal, a slight reduction of B will tend towards a more uniform distribution of the stresses between the stays, thereby reducing the maximum values of S; and it is possible that an alteration of B may in the one case reduce, and in the other case increase, the maximum stresses.

Diagonal Pitches.-The Board of Trade has adopted the view that the maximum stresses are proportional to the product of A into B, but certain allowances are made for large variations (see p. 324). Formerly · Lloyd's Register ' did not take into account the smaller of the two pitches, but since the first edition of this work was issued the view expressed in it, that the stresses in flat plates are probably proportional to the sum of the squares of the two pitches, has been accepted. This amounts to the same thing as measuring the

are

diagonal pitch and doubling the accepted constants. For irregularly stayed plates the practice commonly followed on land boilers is to inscribe circles which will touch the rivet centres or stays: the diameters of these circles are taken to be the diagonals of squares. It should also be noted that on land considerably higher pressures than at sea are permitted on flat plates of the same dimensions. This is due to the experience that thick plates frequently crack, owing to

panting; yet there is no indication that these practices lead to weak boilers.

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m =

Manhole Flanges.-A problem which presents itself with

reference to the fitting of manFIG. 149

holes in flat plates is the follow

ing (see fig. 149) Let Qu=p.lv, where p is the working pressure and l, is half the width of the manhole door; then the bending moment at the position of the stay is

piliola +

2 If experience has shown that for this particular thickness of plate a pitch of stays equal to L is permissible, then evidently the following equation must be true:

12
=p. 1 l2 +
12

2 So that the maximum value for l = li + la is

1 L2 1+

6° 2,2 There L is the pitch for stays in flat plates as found by accepted rules. The manhole doors must be estimated independently. Valuable experiments have been made as to their strengths by the German Admiralty, Danzig.

Effects of Boiler Deformations.--A little familiarity with the working of the equations for a and y of continuous beams (see p. 179) will soon lead to the conviction that the changes of form to be expected in boilers often produce stresses which far exceed any

that may

be due

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