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come into existence. They tend to give the rivet a slight bend, and this deformation will cause an irregularity in their distribution, the
pressure being proportional to the deformation. This is evidently a similar case to the one which presented itself when dealing with the influence of end plates on the distribution of stresses in cylindrical shells, and which, as was there shown, leads to the most complicated formulæ. Here it will be assumed that as long as the elastic limit has not been reached, p, is uniformly distributed over the thickness t of the shell plate, and p, is distributed over the thickness of the butt straps in
2.Q the shape of a triangle. Then in the above case p = 2p2
t.d" Stress in Rivets.--These pressures produce bending and shearing stresses which, when reduced to right-angled resultants (see p. 173), act in directions which are indicated in figs. 185 and 186. In both the intensities are indicated by the shading of the various zones, and if placed
over each other it would be found that for any particular point the two sets of stresses cross each other at right angles. Fig. 185 shows only compression stresses, and fig. 186 shows tension stresses at the left edge, which are gradually reduced and change into compression stresses at the right-hand centre. The stresses and their angles have been calculated on the assumption that the rivet diameter is equal to the thickness of the shell plate and twice as thick as the butt straps, and that the pressure is distributed as in fig. 184, but rounded off at the corners.
Then the mean stress along the line of shear, which is the mean of the tension and compression stresses, is 1:13 ton per ton of mean bearing pressure, and the maximum stress in this line is 1.55 ton. On the upper part of the left edge there are some severe compression stresses, the maximum being 2.33 tons; and in the lower right-hand corner shrinking stresses are met with, consisting of two right-angled stresses of 1.2 and 1:45 ton per ton of mean bearing pressure. These diagrams and values might be used for the purpose of calculating the elastic deformation of the rivet, and the bearing pressures would then have to be modified ; but as no very important deductions will be drawn from these results, the more correct curves have not been determined.
Distribution of Bearing Pressure.— Another correction has to be introduced on account of the irregular distribution of the bearing pressure over the rivet diameter. Let the dotted line (fig. 187) represent the imaginary outline of the rivet section if it had been free to shift its position. But having come in contact with the circumference of the hole, which is shown by a black line, a pressure is called into existence, which may be taken to be proportional to the distance between the black and dotted lines. This normal pressure between the rivet
FIG. 187 and the plate is shown in fig. 188. p=P. cos a. Here Po is the maximum bearing pressure. The sum of p cos a for the diameter d of the rivet is 1 1..d.t. This is equal to the load Q on the rivet, and the maximum bearing pressure at the centre line is
In the butt straps, as has been explained, this pressure will perhaps be double as much, depending on the flexure of the rivet and compression of the rivet hole.
Fig. 188 shows that d, d cos a, and that if the rivet were divided into numerous axial laminæ of equal thickness, the thrust on each one
would be proportional to its own width, and if not connected amongst each other they would be deflected by different amounts (see figs. 189 and 190).
As these independent motions cannot take place there must exist a series of cross stresses in the rivets, which prevent the laminæ from slipping. It is also probable that in adjusting themselves to their surrounding conditions the outer laminæ do not offer as much resisttance as the inner ones, and therefore the normal pressure (fig. 188) is more likely to be distributed according to the formula p=p. cos as, which, if true, would lead to the conclusion that the maximum bearing pressure to be found between the rivet and its hole is
2.1.1 which is 50 % greater than the mean.
The Shearing Stress in a Rivet increases from nothing at either of its ends and at the centre of its length to a maximum at the plane of shear. In this plane it is greatest at the centre of the rivet, falling off in a parabolic curve towards the circumference. The maximum shearing stress in the centre is therefore about double the mean, viz.
which coefficient is nearly the same as 1:55 x 1.5 as found above.
Shearing and Bending Stresses.—If the leverage with which a load acts on a beam is equal to one-third of its thickness, then the limits of elasticity, both for tension compression and shear, are reached simultaneously (see p. 197). In a lap joint this condition most likely exists. In butt-strapped joints it will depend upon the thicknesses, and on the distribution of the bearing pressure, whether this is so or not. If the strap is equal to the thickness of the rivet, and the pressure distributed as in fig. 184, or if the thickness of the strap is į of the rivet diameter, and V. is uniformly distributed over the length, then the above condition exists. In other cases the rivet gives way first, either by bending alone or by shearing alone. Of course these remarks only apply up to the elastic limit, and even here they are seriously modified by the various cross stresses which it has only been possible to hint at.
Plastic Rivets. As soon as the stresses increase to such an extent that either at one point or another the elastic, or even the plastic, limit has been reached, then the conditions are altogether changed. All the pressures are more evenly distributed. The shearing stress is similarly affected, and is probably uniformly distributed, as shown in fig. 167, p. 197. The ultimate strength of a rivet may, therefore, be estimated by those formulæ which give the mean stresses, while its working strength up to the elastic limit has to be found as explained above. It is, therefore, very misleading to assume that the elastic limit and ultimate strength of a rivet stand in the same relation as the elastic limit and ultimate strength of a simple bar. In the latter case the ratio is about as one to two, in the former it is more nearly as one to ive. A nominal factor of safety five leads to a construction in which the limit of elasticity of a rivet is just reached with the ordinary working pressure.
When the dimensions of a rivet have been so arranged that the limit of elasticity of the metal is reached simultaneously at various parts, which is of course the most advantageous condition, then it does not at all follow that, on increasing the load, rupture will also take place simultaneously at all these points. In fact, there can be no doubt but that the bending moments increase at a relatively greater rate than the load, while the axial stresses due to them increase irregularly. While this is going on, the shearing stresses distribute themselves more uniformly and sink into relative insignificance, so that a rivet designed on the best principles for working conditions would rupture primarily through bending, and at a lower load than another which is so designed that all the ultimate stresses are reached simultaneously. Of course in this case the rivet would not be an equally efficient one under working conditions. That unexplained actions of this sort do exist is proved by nearly all experiments on this subject, for it will be found that it is not always the part which is apparently most strained that gives way. In fact, the stronger part-either the rivet or the plate-seems as if it were always endowed with extra strength. Thus a joint may tear through the plate when its stress is, say, 28 tons, while that in the rivets is 14 tons. But by reducing the sectional area of the latter it is possible to construct a joint which will give way at, say, 20 tons, while the plate remains intact, though in this case it has been strained up to 32 tons or more.
Rivet Holes.-Circumferentially the bearing pressure in a rivet hole is of course distributed exactly in the same way as on the rivet. The stresses which are thereby produced in the surrounding material might be determined with considerable accuracy by a careful and exhaustive analysis; but this cannot be attempted here, and it will only be possible to recapitulate some of the views on this subject, and which are intended to be approximately correct.
Firstly, the stresses between the rivet holes are supposed to be uniformly distributed. This is evidently incorrect.
Secondly, the shell plates, or the straps, are supposed to be built up of a number of tapes or links, each one being of a sufficient section for carrying the load which is placed on it (figs. 191, 192). The depths of these links over the rivets must be made 50 % greater than their widths. Although this view rather begs the question, it seems
to give results which are well supported by practice, and if oftener applied would lead to modifications in some of the complicated riveted joints. Thus in fig. 191 it will be noticed that the inner row of rivets is irregularly pitched. This is necessary, as otherwise the
tapes from the outer row would have to be split in two and spread out at an angle.
Thirdly, in order to estimate
Pa the amount of metal required be
tween the rivet hole and the edge L.
of the plate, this part is looked
upon as part of a continuous FIG. 193
Fourthly, the circumferential shearing stresses are used as a basis.
Fifthly, a more thorough method is to deal with the metal surrounding the rivet as if it were part of a thick-walled cylinder.
Distribution of Stresses near Rivet Holes.-In fig. 193 the curves A and B represent the distribution of the circumferential stresses at these two points, due to the internal pressure po, which in this case is supposed to be uniformly distributed. In the formula (8, p. 213)
d for the curve A we have ri =
4u2 + d2
4u? – d2
d And for the curve B we have ri = El
2 4.12 - 4.1.0 +2.02
2.12 Si = - Po:
S, =-P 4.12 4.1.d
4.12 – 41.d Of course, as the adjoining rivet also produces stresses, S, has to be added to S,, and we have
12 - 1.d + (12 S.= S2 + S, - Po:
12 - 1.d