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Cross Deformation of Plastic Beams.—When the elastic limit has been passed the value of 1, the coefficient of cross contraction, increases from to . For very narrow beams the ratio of p to Pi (fig. 162) would then be as 1 to 2. In wide ones, such as shell plates,

this deformation is of course impossible, and it is evident that a cross stress must have been set up, whose intensity is nearly equal to half the longitudinal one, or, say, + 12 tons per square inch, under the action of the bending rolls. But this condition also demands supplementary longitudinal stresses at the edges of the plate and coinpression stresses near its centre. They are all modified by the spring of the plate as it leaves the rolls ; but it is quite clear that the residue is both

of a severe and of a complicated nature. Stresses of a similarly complex nature also exist in small bent test samples; and it is interesting to note that when the steel is of an unsatisfactory

quality it usually Fig. 162

FIG. 163

cracks at the centre

of the width, showing that it was incapable of withstanding the combination called drum tension (see fig. 163 and p. 166) to be found there.

The complicated nature of these stresses, particularly in wide beams, readily explains that the maximum stresses to be found in them when the plastic limit has once been passed differ very materially from those found by the formula

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dy

case

Shearing Stresses in Plastic Beams.—The previously found do

ds formula,

is independent of the elasticity of the material.

dx' If, therefore, the longitudinal stresses in a rectangular plastic beam are distributed as shown in fig. 164, then the shearing stresses must be distributed as in fig. 165. This illustration represents a where the bending stresses have been produced by means of a light force acting on a long leverage, as, for instance, when shell plates are being passed through the bending rolls. The case is

very different when a strong force with a short leverage produces the same moment. Then the shearing stresses will be nearly uniformly distributed over the thickness (fig. 167), and the longitudinal stresses will have to accommodate themselves accordingly somewhat after the fashion shown by the curves in fig. 166. In the one case the beam gives way at the outside fibres, in the other at the neutral fibre.

An intermediate stage is worthy of mention. Suppose that the plastic limit of tension and compression is 20 tons, while for shearing it is 15 tons; then both outside fibres of a beam would give way to longitudinal stresses, at the same time that the centre fibres commence

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to shear if these stresses are reacted on simultaneously. This happens in a square bar if the bending force is applied at a distance equal to } of its thickness, and doubtless most rivets behave in this way.

Stresses in Cylindrical Shells.—The circumferential stress in a cylindrical shell of the mean diameter D is found by the well-known formula

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where p is the internal pressure and t the thickness of the shell plate. If the ends of the shell plates are joined together by riveted longitudinal seams, the percentage of their strength will have to be taken into account. If the boiler shell is built up of several strakes, and the longitudinal seams break joint, it has been argued that the stresses are proportionally reduced, and that they should be calculated as follows:

S = P.D.2

2.t. (Al)

Here l is the length of the boiler, and E(Al) is the sum of the widths of the plates, including the flanged end plates, from which the rivet holes of only one longitudinal seam have been subtracted. As (11) is often greater than 1, this view would lead to the conclusion that a riveted shell is stronger than a solid one. Clearly, this argument is only true as regards the mean stresses, and that it leads to valueless results is proved by the fact that the same reasoning applied to a beam would lead to a wrong conclusion, for its mean stress is just nothing, being + S at the top and – S at the bottom.

A point where variation of stresses in shell plates may be expected is at A (fig. 168), in the solid plate near the end of a longitudinal seam. Let it be assumed that this seam is more elastic than the plate, i.e. that whereas a stress of one ton would stretch the latter 73000 of its *length, the same stress would cause the joint to spring or stretch, say, four times as much. Then, as these two parts are firmly connected by the circumferential seam, the solid plate at A would be subjected to a four times greater stress than the joint. A solitary and perhaps not very reliable experiment on this subject showed that the stress was

E.

actually eight times greater, and similar results were obtained near the single butt-strap joints on ship's sides (C. E. Stromeyer, ' N. A.,' 1886, vol. xxvii. p. 34).

The remedy which readily suggests itself is to make the longi. tudinal seams more substantial-say, of the double butt-strap type. Now, however, this part may be more rigid than the plate, and will have to bear a proportionally heavier load, and being perforated is less capable of sustaining it. A more correct principle would be to make the longitudinal joints exactly as elastic as the solid plate. A few remarks on the elasticity of a riveted joint will be found further on (p. 222), when discussing their theories, but a true solution can be obtained only by careful experiments.

The order in which seams naturally range themselves as regards rigidity is

1st. Single butt-strap joint.
2nd. Lap joint.
3rd. Butt joint with one wide and one narrow strap.
4th. Butt joint with two equally wide straps.

Of course the number of rows of rivets, their pitch and diameter, and the thickness of the cover plates, will affect the results.

S.1 The Longitudinal Contraction of Cylindrical Shells is where

μ' l is the length and S the circumferential stress. In a boiler of 17 ft. length this will amount to about 0·025 in. at the ordinary working pressure. If no stays are fitted to take up the longitudinal stress, which is one-half of the above, it will be found that the elongation,

S
Al= .7

.1

This can be utilised for the E

T

1 determination of the coefficient of cross contraction. By riveting

H ongitudinal strips on a boiler shell, the longitudinal contraction is

reduced, at least locally, and this is exactly what all riveted longitudinal seams do. Consequently, longitudinal compression stresses are set up in them, which must produce

tension stresses in some other part of the A

shell, probably along either side of the seam. A longitudinal compression stress c will also be set up at A (fig. 168), which, combining with the circumferential stress s, constitutes a shearing stress, and that, as has been

shown by experiments (see fig. 123, p. 169), Fig. 168

is far more injurious to iron and steel than

a simple tension. Therefore, whether elastic lap joints or the more solid butt-strapped joints are used, very severe stresses will be found in the adjoining plates.

Influence of End Plates on the Stresses in Boiler Shells.— In fig. 169 let the line CD represent the shape of half the length l of a boiler shell of the diameter D, while subjected to an internal pressure pı: The shell plate will acquire this curvature if secured to the end plate

P.D.

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at C, as shown; if unsecured it would have remained cylindrical and have occupied the position at the line AB. Under that condition the increase of half the diameter is

where t is the thick

4. E. ť ness of the plate and E the modulus of elasticity. The stress will be

D s

; but, as will be seen in the diagram, if the ends are held 2.t down there is no circumferential straining of the shell at C, while at D it is reduced in the ratio of -y. to . The object of this investigation is to ascertain what this value is.

PI

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A little reflection will show that the circumferential stress S at any point x or z is proportional to the distance of that point of the shell from its original position.

1-= P1.D (91-y).
Y 1

2.t Y1

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But as this stress is capable of balancing only part of the internal pressure, there remains

y P=P

yi'

which has to be transmitted to the end plate, and in doing so longitudinal bending stresses are set up in the shell, which have to be ascertained. This can be done by examining a long strip of the boiler shell-say, 1 in. wide--loaded irregularly with a pressure P, and supported at its ends by loads Q. A bending moment m, will also be found there.

Evidently the load Q is proportional to the area ABDC.

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This leads to the equation

Y.
y = (eux + e -ar) cos ax + A (ell - e -ax) sin ax.
2

Y. By integration and differentiation the values of 2, A, and a can be found, and numerical values obtained, when the conditions of the external forces are known. Thus, when very thick shell plates are attached to thin end plates, or when these have a very weak and wellrounded flange, m, = 0 or nearly so. In the latter of these two cases y, is also reduced, on account of the spring of the flange, which is proportional to Q, and this value has therefore also to be reduced, and the relief afforded to the shell plate is small. The calculations are too complicated to be reproduced here. The problem has been discussed on somewhat different lines by Dr. F. Grashoff, 1878, p. 316, and by J. T. Nicolson, N.-E. C. I.,' 1821, vol. vii. p. 205. Adapting some of his results to a boiler of 15 ft. diameter, with 1-in. shell plates at 100 lbs. pressure, we find the circumferential stress in the centre of the length to be as follows:

Length between end plates (feet)
Circumferential stress at centre (tons)

15 3:35 13:35

10 3:34

1 2 1 2-96 70

3.15

But this is only true if the end plates are rigid while m, = 0. In practice this is never the case, and a very considerable deformation must take place in the end plates. The back end, which is practically a flat plate, will be strained uniformly. If made exceptionally thin, it would have to expand as much as the shell, but, not being able to contract crossways, the consequent drum tension would be equal to

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or about 50 % more than the circumferential tension in the shell.

With the front plate, which is perforated by furnaces, manholes, and tube holes, it would at first sight appear as if locally the stresses might grow to be excessive, but it is clear that only under exceptional circumstances could they exceed the circumferential stresses in the shell plates. The weakest points are undoubtedly to be found between the furnace front holes; but, as no fractures have ever taken place there, it is but reasonable to suppose that the stresses in the end plates, due to their attachment to the shell, are small, the roundness of the flange providing the necessary springiness.

Stresses near the Dome Holes.—A problem which is often met with in boiler designing is the efficient staying of the corners of two

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