section joint stress pressure surfaces pull bending parallel surface tension
JOINTS, in the sense in which engineers use the word, may be classed either (a) according to their material, as in stone or brick, wood, or metal ; or (b) according to their object, to prevent leakage of air, steam, or water, or to transmit force, which may be thrust, pull, or shear ; or (c) according as they are stationary or moving (" working " in technical language). Many joints, like those of ship-plates and boiler-plates, have simultaneously to fulfil both objects mentioned under (b).
All stone joints of any consequence are stationary. It being uneconomical to dress the surfaces of the stones resting on each other smoothly and so as to be accurately flat, a layer of mortar or other cementing material is laid between them. This hardens and serves to transmit the pressure from stone to stone without its being concentrated at the "high places." If the ingredients of the cement are chosen so that when hard the cement has about the same coefficient of compressibility as that of the stone or brick, the pressure will be nearly uniformly distributed. The cement also adheres to the surfaces of the stone or brick, and allows a certain amount of tension to be borne by the joint. It likewise prevents the stones slipping one on the other, i.e., it gives the joint very considerable shearing strength. The composition of the cement is chosen according as it has to " set " in air or water. The joints are made impervious to air or water by "pointing " their outer edges with a superior quality of cement.
Wood joints are also nearly all stationary. Lignum vitae is still used by engineers for the one half of seine special working joints, but even in these few instances its use is rapidly dying out. Wood joints are made partially fluid-tight by "grooving and tenoning," and by " caulking " with oakum or similar material. If the wood is saturated with water, it swells, the edges of the joints press closer together, and the joints become tighter the greater the water-pressure is which tends to produce leakage.
Relatively- to its weaker general strength, wood is a better material than iron so far as regards the transmission of a thrust past a joint. So soon as a heavy pressure comes on the joint all the small irregularities of the surfaces in contact are crushed up, and there results an approximately uniform distribution of the pressure over the whole area (i.e., if there be no bending forces), so that no part of the material is unduly stressed. To attain this result the abutting surfaces should bo well fitted together, and the bolts binding the pieces together should be arranged so as to ensure that they will not interfere with the timber surfaces coming into this close contact.
Owing to its weak shearing strength on sections parallel to the fibre, timber is peculiarly unfitted for tension joints. If the pieces exerting the pull are simply bolted together with wooden or iron bolts, the joint cannot be trusted to transmit any considerable force with safety. The stresses become intensely localized in the immediate neighbourhood of the bolts. A tolerably strong timber tension-joint call, however, be made by making the two pieces abut, and connecting them by means of iron plates covering the joint and bolted to the sides of the timbers by bolts passing through the wood. These plates should have their surfaces which lie against the wood ribbed in a direction transverse to the pull. The bolts should fit their holes slackly, and should be well tightened up so as to make the ribs sink into the surface of the timber. There will then be very little localized shearing stress brought upon the interior portions of the wood.
Metal Joints. - Iron and the other commonly used metals possess in variously high degrees the qualities desirable in substances out of which joints are to be made. The joint ends of metal pieces can easily be fashioned to any advantageous form and size without waste of material. Also these metals offer peculiar facilities for the cutting of their surfaces at a comparatively small cost so smoothly and evenly as to ensure the close contact over their whole areas of surfaces placed against each -other. This is of the highest importance, especially in joints designed to transmit force.
Wrought iron and mild steel are above all other metals suitable for tension joints where there is not continuous rapid motion. Where such motion occurs, a layer, or, as it is technically termed, a "bush," of brass is inserted underneath the iron. The joint then possesses the high strength of a wrought iron one and at the same time the good frictional qualities of a brass surface.
Where the running speed is high and the intensity of pressure can be made small by adopting large bearing surfaces, cast iron is now increasingly preferred for pressure joints. But when, owing to want of space or for other reasons, the bearing surface cannot be made large in proportion to the thrust to be transmitted, gun-metal, i.e., the toughest quality of brass, should be used if the speed be high, and steel if the speed be small.
Leakage past moving metal joints can be prevented by cutting the surfaces very accurately to fit each other. Steam-engine slide-valves and their seats, and piston " packing-rings " and the cylinders they work to and fro in, may be cited as examples. A subsidiary compressible " packing" is in other situations employed, an instance of which may be seen in the "stuffing boxes" which prevent the escape of steam from steam-engine cylinders through the piston-rod hole in the cylinder cover.
Fixed metal joints are made fluid tight - .(a) by caulking a rivetted joint, i.e., by hammering in the edge of the metal with a square-edged chisel (the tighter the joint requires to be against leakage the closer must be the spacing of the rivets - compare the rivet-spacing in bridge, ship, and boiler-plate joints) ; (6) by the insertion between the surfaces of a layer of one or other of various kinds of cement, the layer being thick or thin according to circumstances; (c) by the insertion of a layer of soft solid substance called " packing " or "insertion." A. special kind of indiarubber and canvas sheet is prepared for this purpose. A very effective species of " insertion " is thin copper gauze. Sometimes a single round of thick copper wire laid in opposite grooves cut on joint-surfaces serves the purpose.
The Principles of the Strength of Joints. - The conditions of strength of cemented and glued joints are too obvious to require description. It may, however, be mentioned that in most cases the joint is stronger the thinner the layer of cementing material interposed between the surfaces.
Nearly all other joints are formed by cutting one or more holes in the ends of the pieces to be joined, and inserting in these holes a corresponding number of pins. The word "pin" is technically restricted to mean a cylindrical pin in a movable joint. The word "bolt" is used when the cylindrical pin is screwed up tight with a nut so as to be immovable. When the pin is not screwed, but is fastened by being beaten down on either end, it is called a "rivet." The pin is sometimes rectangular in section, and tapered or parallel lengthwise. " Gibs " and "cottars " are examples of the latter. It is very rarely the case that fixed joints have their pins subject to simple compression in the direction of their length. They are, however, frequently subject to simple tension in that direction. A good example is the joint between a steam cylinder and its cover. Here the bolts have to resist the whole thrust of the steam, and at the same time to keep the joint steam-tight.
If D be the cylinder diameter, t the thickness of the flange of the cover, and n the number of bolts used, it can be shown that the amount the flange rises between the bolts by bending is proportional to , where p is the steam pressure per unit area. If the same pressure, it would follow that the diameter of bolt a philD1. Again, if t were reckoned in accordance with the shearing force of the steam on the circular section of the cover at the circumference of the cylinder, i.e., t ccpD, we would have and bolt diam. a pipk For reasons connected with technical difficulties in the foundry, t is made larger in proportion to 1) than this rule indicates for the smaller sizes of cylinders ; and, therefore, the spacing and the bolt diameter are not made to increase quite so rapidly as the a and I powers of D.
No moving joints have their pins exposed to simple stress on sections transverse to the pins' axes. The pins of such joints have these transverse sections subjected to shearing and bending stresses, and the sections parallel to the pin axes to compressive stress.
The simplest case by which the subject can be illustrated is that in which a cylindrical pin passes through the ends of two links - one forked, and the other simple and lying between the branches of the fork of the other.
Let the accompanying diagram represent the end of the unforked link. The width of the link parallel to CC' is taken as unity, and the letters on the figure indicate the ratios of the respective dimensions to this width. Let b represent the ratio of the thickness, perpendicular to the paper, of the "eye " to the thickness in the same direction of the main body of the link at 1). Let also/ be the intensity of uniform tensive stress on the section at D. Evidently no pressure comes on the under side of the pin below CC'. The whole pull at D is passed round half on each side of the pin, and is delivered to the upper side of the pin, on which it produces compression. Since the side sections t, through which the pull passes, lie out of the direct line of that pull, the stress is much higher on the parts of these sections towards the centre line DD' than on those further off. The linos of force crowd as close as possible together near the surface of the pin, i.e., towards the main line DD' of the pull. In other words, the inequality of stress is occasioned by the bending moments due to the centre of force not passing through the centres of gravity of area of the sections. The inequality begins at the root of the widening out of the link to form the eye, and reaches its maximum at CC'.
The bending moment at CC' and the stress caused by it at the edge of the section can be found by the help of the ordinary theory of elasticity. The best method of doing so is to calculate the amount by which the portion of the eye below CC' is bent by the forces applied to it. In the equations the bending moment at CC' is inserted as an unknown quantity. The section. on DD' remaining unmoved, each element of the linear deflexion is resolved parallel to CC', and the integral from DD' up to CC' of all these components parallel to CC' is equated to zero, the resultant deflexion at C in the direction of CC' being evidently nil. This equation gives value of the bending moment at CC', and from it the correspond stress is obtained.
If the section at D be rectangular, as also that at CC', then the average tensive stress on t is ,r1 and the extra stress caused at the edge of the section by the bending moment is 4t2b i+ d The total maximum stress is, therefore, fl +f"frer6) 5t + 3d– 4t÷d • This gives the ratio of the maximum tension at the side of the eye to the uniform tension (f) on the main body of the link at D. If the section at D be circular while t remains rectangular, the corresponding ratio is a little more than 7, or about I of the above. If it is desired that this maximum should not exceed!, we obtain a relation between the ratios d, t, and b by putting f +f'=f. The following table exhibits the results of this calculation for rectangular section at D: - 1= „ 5= 4 3! 2'1 2 1.7 d=1 „ 4•9 3'3 2.8 2.3 1•9 For circular section at D, b is about i of these values.
Although the values of t and d that are commonly used all fall considerably within the limits of the above tables, the values of b usually found in practice are much less than those shown above. This means that the eyes of links as commonly proportioned are much more severely stressed than is the main body of the link.
In working joints the frictional resistance to rotation throws more than half the main pull on one side of the eye, and this side is therefore still more severely stressed than is indicated by the above equations.
The stresses on the portion of the eye lying above CC' are complicated by the combination with the direct pull already mentioned of the pressure of the upper surface of the pin. This latter is normal at each point if the surface be smooth and the joint a motionless one. It increases from zero at CC' to a maximum at the line DD'. At this point the intensity of the surface pressure is, according to an approximate theoretic estimate, about - 4 , or lk times greater than if the whole pull were evenly distributed over the projection on CC' of the upper half surface of the pin. It has often been fallaciously imagined that the central section t1 is exposed to severe shearing stress. From the symmetry of the case, however, it is evident that on this section the shear is zero. The maximum shear occurs on a section nearly parallel to DD', and somewhat less than f,d distant from DD'. The exact position of this section of maximum shear depends upon the dimension-ratio t„ which is usually made considerably greater than t.
The pin surface pressure has transverse components parallel to CC', which produce tension and a bending moment on the section A theoretical approximation to this bursting pressure or about A, of the whole pull exerted by the link, and the line of the resultant (parallel to CC') is situated Ad distant from the centre of the pin. A small portion of this is borne by the central section on DD' of the main part of the link below CC', but by far the larger part is borne by the section marked t,. If it were wholly borne by that section, the average tension on t1 would, for a circular section at D, would be 44 f 4 t + --(1)• Other bending moments, however, are thrown on this section due to - first, the resultant of the pin-surfacepressure-components parallel to DD', which lies at 3!-d, or about id, from the line DD'; and, second, the stress at the section CC'. Adding all these together, there is obtained an approximation to the actual tension parallel to CC' on the lower edge of the section t.„ namely, 1 tab 5t,.(4t+ d)l• The shearing and bending stresses upon the pin itself depend upon whether one of the links is forked or both are simple ; and also greatly upon the exactitude with which the pin fits the holes.
When the link exerts a thrust instead of a pull through the joint, a similar investigation of the state of stress may be made.
A couple of plates joined together by a single row of rivets may, so far as concerns the sections lying between the rivets, be looked upon as a number of flat links laid side by side with their eyes of equal width with the body of the link.
We may therefore apply the first of the above equations for!' -bf" to find the stress close to the rivets on the section coinciding with the line of the rivets. To adapt the formula to this case, it is only necessary to put 4=1 and I= (1 -d). The formula thus derived, however, gives results probably considerably higher than those actually occurring, because of the strips into which the plate has been supposed to be divided, acting on each other in such a way as to produce bending moments partly neutralizing the above increase of stress.
2t, the maximum stress on a section perpendicular to the plate edge. To make!' =f, it is necessary to make 11= N/0.5-0.7. The edge of the plate will then be amply strong enough to resist the greatest shear to which it is anywhere exposed.
When there are two or more rows of rivets the investigation of the stress is quite similar to the above.
In joints where the movement is rapid and continuous, the size of the pin is determined by considerations of durability against wear. The metal wears rapidly if the bearing surfaces are not well lubricated. The lubricant is pressed out from between these surfaces if the intensity of pressure exceeds a limit determined by the character of the lubricant. The size of the pin must be sufficient to prevent this limiting pressure being reached. Even before the oil is wholly squeezed out the friction becomes so great qs to heat the metal surfaces to a high temperature, which hastens the evaporation of the remaining oil. In order to ensure that the temperature may be kept low by the continuous dissipation of the heat generated, some engineers design the bearing surface in proportion to the product of the pressure and the speed, so as to allow a certain area of " conducting surface" for each unit of heat generated per second. (n. 11. s.*)