Hydraulic Tests of Purves' Ribbed Furnaces Length Diameter Thickness Collapsing Remarks ins. ins. ins. ins. ins. ins. lbs. tons tons Tests made at Sir John Brown & Co.'s Works, Sheffield Curved Plates. In the above case the radius of curvature of the furnace, instead of being constant (18 ins.), varied from 17 ins. at the crown to 19 ins. at the sides, and this irregularity was evidently sufficient to overstrain the material. What, it may be asked, will be the effect if the curvature changes still more rapidly, viz. from, say, 24 ins. to a straight Tension FIG. 178 Com ron FIG. 179 Tension line, as often happens in combustion chamber tops (fig. 178)? or if it changes from one side to the other, as in the case of combustion chamber bottoms (fig. 179)? When the pressure is on the convex side, the circumferential stresses are all compression, while when it acts on the concave side they are tension. This is shown in fig. 179, which represents part of the bottom of a combustion chamber with two furnaces. As sketched, the conditions are more unfavourable than when building an arched bridge without abutments and without any support, for the ends are in this case actually being pulled at. The stresses which are necessarily set up are of the most complicated nature. Thus there will be circumferential shearing stresses, which increase very seriously towards the forward and back ends of these plates, particularly along the line of change of flexure. Horizontal seams should not be fitted here. These shearing stresses are also S.dr S.dr FIG. 180 P2 very severe on the circumferential seams of the furnace back ends and of the combustion chamber backs; and the extra circumferential compression stresses which are thereby set up in the furnace saddles may have been the cause of occasional mishaps. The staying of these parts to the boiler shell will necessarily afford local relief, but very often leads to excessive bending strains in the screwed stay near the water level. Thick Cylindrical Shells. This subject would be out of place here were it not that some of the mathematical deductions will assist in estimating the distribution of stresses in riveted joints. Let the left and right-hand side of fig. 180 represent half the tube, respectively with and without the internal and external pressures P1 and p2. Then, dealing with the thin-walled ring dr of the radius r, we find that if p is the radial pressure at this distance from the centre, and if S is the circumferential stress, S being circumferential tension, we have S.dr = (p + dp) (r + dr) − p.r. Neglecting dp. dr, we have The thickness dr of the ring under consideration has decreased by If there are no longitudinal stresses in the cylinder, then its length. increases = But d (r) (dr), and combining this with (3) It follows that p + S is constant for all values of r, which shows that dl (4) is also constant, and need not be taken into account. Differentiating (1) leads to the equation Integrating this and introducing the various constants leads to the following equations: (8) If r2 = ∞, S1 - P1; the combination of S, and Pi is a pure shearing stress of the intensity σ = p, = -S1. 2 From this it will be seen that p + So when p2r22 = pir12. In that case too the metal is exposed to uniformly distributed spiral shearing stresses, σ, as shown in fig. 181: its effects are often seen around punched holes. For This condition is also approached, at least near the internal circumference, when r2 is very large. practical illustrations see C. E., L2 1893, vol. cxi. p. 212, and fig. 182, which represents a piece of a steel shell burst by a high explosive. sheared and inclined at an angle of 45°. Ꭱ P2 All the fractured surfaces are Other conclusions to be drawn are 1st. The maximum stresses, either tension or compression, are always found at the internal circumference. 2nd. If the internal pressure p1 = 0, then the maximum circumferential compression stress is An approximately correct formula for cylindrical tubes would therefore be where D is the external diameter and t the thickness of plate. 3rd. If the external pressure P2 = o, then the maximum circumferential tension stress is For boiler shell plates an approximately correct formula is therefore where D, is the internal and D, + t the mean diameter of the shell. In both these cases, therefore, the maximum stress has to be estimated as if the pressure were acting on an imaginary tube whose diameter is a little larger than the actual one. These corrections are too unimportant to be taken into serious count in practice, except, as will shortly be explained, in the case of rivet holes. The deformations of thick-walled tubes are found by inserting the values of S, and p, in equation (2): Plastic Tubes. As regards the distribution of stresses in thickwalled cylinders when the plastic limit has been reached, little can be learnt until the plastic properties of steel subjected to compound stresses are better known. However, as the value of changes when this limit is passed from about to, it is evident that it ought to be introduced into the various formulæ as a function of r, and then μ δι is not any more a constant value, nor is S + p; and what complicates matters still more is that longitudinal stresses are set up which increase with increasing length of tube. The stresses in a gun barrel which is on the point of bursting are therefore not at all as simple as is usually assumed. A thorough analysis of the shape of the swelling of the material round a carefully drifted hole (fig. 183), similar to that carried out in the case of torsion tests, could perhaps be made to throw light on this subject. FIG. 183 Riveted Joints.-At first sight it would appear that there is no simpler problem than to find the stresses in a riveted joint. Given the sectional area of the perforated plate, and the sectional areas of the rivets, the average stresses ought to be easily calculated. But these are only the average stresses. To find the positions and intensities of the maximum stresses is far more complicated, but worth examining, not only on account of the importance of the subject, but also because the difficulty of dealing with it more in detail will make it clear how little is yet known about other problems of boiler mechanics which, even at first sight, strike one as more complicated than this one. Deformation of Rivets.-Fig. 184 represents part of a butt-strapped joint. A force Q is being transmitted from the central (shell) plate to the two butt straps, and in doing so the bearing pressures p, and p2 |