The stresses, in the simplest structures are often those we find most difficult to analyze. The most complex condition in mine stresses is found in simple tunnels where the roof, the sides, and the floor are a monolith. The functions of the parts are like the parts themselves not distinct and specialized, and the problems to be solved are like those in a metal structure with riveted joints or a redundancy of bars.

This difficulty explains perhaps why the condition has not been treated. But just because it cannot be discussed in its entirety is no reason why it should be treated as an action of parts with specialized functions as a roof beam with supports and a foundation. The problem cannot be ignored on the ground that it is not of sufficient importance to warrant careful consideration, because conditions of complete monolithism, of which the tunnel is the type, are found materially unchanged in room-and- pillar and in longwall work.

This unity between roof, sides, and floor, which to the coal miner is a difficult conception, really deserves a scientific appellation, and perhaps holoid (from holos, whole, and eidos, form) will serve the purpose as well as any other.

In a simple tunnel the roof, the sides, and the floor form integral parts of one and the same structure, and the distortion of one cannot be conceived without a consequent strain in the others. Thus when the roof of the tunnel droops by reason of its weight, the upper parts of the sides are drawn in because they are integrally connected with the roof and must approach each other whenever, by the sagging of the roof, the distance between any two points in it is diminished. (See Fig. 1.)

The sides in their turn operate on the floor of the holoid structure, producing a tensile stress. The writer has always been impressed with the value of soap as a means of illustrating the action of mine stresses. With that idea in mind a cake of naphtha soap measuring 4 3/8 by 2¼ by 1¾ in. was taken and a tunnel was made through it 1¾ in. long, 2½ in. wide and 1¼ in. high. (See Fig. 2.) A load was then placed at the mid-span of the tunnel. Eventually, the upper bar, or “roof,” broke at the center line and along both “ribs” of the tunnel, the breaks being approximately vertical and proceeding, as might be expected, on the ribs from the “surface” downward and at the center line from the tunnel upward, the failure being from bending moment, not shear.

This is interesting because it shows that breakage at the ribs is not necessarily evidence of shear. It may be only a demonstration of a holoid structure. It is the form of failure whenever coal is blasted down and not an infrequent form of roof demolition. The test on the soap tunnel further showed that as soon as rupture takes place in the roof of the tunnel, there must inevitably come a thrust on the ribs. The tension draws them together till rupture occurs and then the two roof units, in endeavoring to revolve, crowd each other and push on the opposing walls. (See Fig. 3.)

When a holoid structure, by reason of the weakness of the floor or because of a lack of adhesion between the ribs and the floor, ceases to engage the floor in its movements, then its shape as a structural element roughly resembles the Greek letter π and for want of a better name we might term the new element a pyoid structure (See Fig. 4.) The doctors use the word for a totally different purpose with little excuse. “Pyonoid” is the word which they should use to express the attributive “pus-like.”

By reversing the soap tunnel after the roof has been caved the weakness of the pyoid structure is made clear. As soon as pressure is brought on the new roof (formerly the floor of the tunnel), the two ribs are seen to recede markedly and if the floor were intact this would result in a well-developed stress in that element of the holoid structure. (See Figs. 5 and 6, showing progressive demolition.)

Probably it is well here to express a belief in the importance of the holoid. The general notion is that all the beds shear horizontally along the lines of stratification and that it is a mistake to consider the mine or even the roof as a monolith. It is true that most of the accidents in mines are due to the lack of unity in the roof. What we call draw-slate accidents are almost wholly due to this fact. Nevertheless, it is interesting to note how firmly roof and coal are usually “ burned ” to one another. Even when undermined and sheared on both sides the coal often fails to fall, being supported by the vertical shearing strength of the one side still attached and by the adhesion to the roof. The writer is hardly prepared to state when the holoid structure ceases to exist, and of course the time and conditions will vary with the materials under consideration.

It is obvious that with the holoid structure, the ribs being drawn

Fig. 1 shows in broken lines a cross-section of a tunnel and in full lines the same tunnel distorted by pressure. This is a typical holoid structure.
Fig. 2 shows a soap model to which reference is made in the article.
Fig. 3 exaggerates the upper or roof bar in the soap model and shows how the collapse of the roof flrces back the sides by the arch action.
Fig. 4 shows the soap pyoid without load.
Fig. 5 shows the same when the bending of the roof bar takes place, thrusting out the ribs, or legs.
Fig. 6 shows a further action when the revolution of the segments in collapse tends to force back the upper parts of the ribs or legs.
Fig. 7 illustrates a semi-holoid structure.
Fig. 8 illustrates a semi-pyoid structure.

together by the movement of the roof, they must tend more or less to be split vertically and in longwall they will then fall down on the advancing undercut. In certain sub-bituminous mines the writer has noted a tendency toward what he thought was a vertical shear parallel to the headings. This developed rapidly after the work was opened, especially at great depth.

Whether this was only a vertical shear seems doubtful. It may have been due to the holoid character of the structure, the ribs receiving no relief by a horizontal shear between ribs and roof. Instead the roof pulled off the edge of the pillar as the former bent under the load. It was interesting to note that these lines of fracture did not coincide with the normal cleavage of the coal.

This rending along vertical planes eventually throws back the real rib lines far into, the pillar. Where the draw slate and roof proper leave one another, we have probably a plate structure superposing one which is holoid or pyoid in character, and in the longwall the, breaks back of the face which bring coal and roof down together, or which tend so to do if the latter is not duly propped, are failures of the semi-holoid-or semi-pyoid and not of the plate structure above. (See Figs. 9 and 10.)

The breaking of even the holoid roof is not necessarily a sudden, unheralded event, such as one might anticipate from a cursory consideration of the problem. It is clear that the action of the moments cannot destroy the roof without revolving in a degree the elements into which the roof is broken, and any revolution inevitably binds these elements against one another so that they are less able to fall. Either a recession of the ribs or a further demolition of the revolving elements must take place or the roof will not fall. One form of demolition which frequently occurs in shallow workings is vertical shear along the cracks already made by the bending-moment stresses. But horizontal shears may make it possible for the roof, masses to revolve and yet fit the space they occupied by a counter revolution of the strata past each other. Or again, the whole mass may be broken up by the rubbing of the opposing faces of the elements as they try to fall.

It is this last action which is in evidence in coal brought down by a shot when it is broken considerably in falling, and that vertical shear is not an important cause of the fall of coal is shown by the fact that there is a distinct tendency for the coal to roll away from the side ribs. It is necessary now to consider the plate structure in which the roof is considered as a vast plate, a monolith in itself but resting without adhesion on its supports. Whenever there is a, mined area the roof is depressed, and being elastic it tends to rise on the surrounding supports, resting its weight on the edges of the surrounding ribs of the excavation. This area of quasi-elevation is followed by another area of depression

Fig. 9 shows a semi-holoid such as often occurs in longwall, supporting a cumoid which, being separated from the semi-holoid, has freedom of movement except for the restraint of frictional contact.

Fig. 10 shows a semi-pyoid likewise surcharged by an independent cumoid.

Fig. 11 is a plan of a circular excavation. The sagging of the roof into this area produces an area of quasi-elevation; that is, an area which would be elevated if the mineral itself and the floor surrounding the excavation were incompressible and indestructible. Outside this area is one of depression, the stiffness of the measures lightening the burden on the area of quasi-elevation and loading with increased heaviness the edges of the excavation and the area of depression. It exhibits the fact that the action of the roof is cumoidal, or wavelike, and not confined to the area of excavation. The depressed area is in its turn followed by another area of quasi-elevation, and this again by an area of depression. The size of these waves of course decreases as the area of excavation is left behind.
Fig. 12 shows a cumoid bending over an opening and crushing two pillars.
Fig. 13 shows a cumoid bending over a pillar, the center of which is a point of quasi-elevation. The roof tends to break over such a support.
Fig. 14 shows a weak cumoid, such as a loose draw slate which has broken away from the stiffer cumoid above and probably has insufficient moment of inertia for self support.
Fig. 15 is a Luten bridge, showing the lower tension member, or paving, which ties the abutments together and prevents their recession, thus adding to the strength of the bridge.
Fig. 16 shows a broken cumoid failing to fall owing to an arching action.

surrounding the central depression and the area of quasi-elevation. Thus the roof plate is bent into a series of waves around the central area of disturbance just as the surface of the water is rippled round the point where a stone has fallen and disturbed its equilibrium. (See Fig. 11.)

Of course, the elevations are only relative, not actual, and naturally, like all undulations, as they recede from the point of disturbance they die down. But it is essential to remember that while the holoid structure is to a large extent a closed force chain, this is not nearly so true of the plate, the stresses in which are less localized and circumscribed.

This structure we may dub as cumoid (from kuma, a wave). The remarkable feature about such a structure is that it develops points of great stress far away from the disturbing cause, and it may break over the pillar instead of in the opening. The appearance of the Forth Bridge, Scotland, is known to almost every one. The light structural work over the midspan contrasts most forcibly with the heavy structure over the piers. It is the case with all cantilevers and continuous-arch structures, and the roof in the mine is like a continuous arch, only it is continuous not only in one, but in every direction.

It is the peculiarity of the cumoid structure that the stresses it involves may be greater farther from the point of disturbance than at some nearer point. But, like the holoid structure and the pyoid, it puts the greater burden on the pillar’s edge. The center of the pillar between two large open spaces may be relieved from much of the normal pressure because of the bending of the cumoid roof over the pillar. (See Fig. 13.)

There is some evidence that in actual operations in some sections of the country the roof soon breaks by horizontal shear into two or more separate cumoid structures, of which of course one is free of external load, while the others, though below other cumoids, may or may not be loaded. If the stiffness of the upper cumoid, or cumoids, exceeds that of the lower, the loading may be relieved from the open spaces and the upper cumoids may restrain the lateral, and therefore the vertical, movement of the lower cumoids, thus adding to their resistive strength.

In an interesting paper read before the winter meeting of the West Virginia Coal Mining Institute, the late F. C. Keighley called attention to the fact that when the lower roof broke or was preparing to break in the mined spaces of the Gonnellsville region it frequently weakened the lower roof in the rooms and headings nearby, despite the strong support afforded by large pillars. This caused in the narrow places many falls which had to be loaded out.

It would seem, therefore, that the breaking of the lower roof or its initial stressing tears the lower roof from the upper and from the ribs, converting it into a cumoid structure which is too weak to stand the strains to which it is exposed. In cases, however, these primary failures may be due to substitution of a pyoid for a holoid structure. For it must not be forgotten that where the bottom, ribs, and roof are one and indivisible, the floor is an clement of strength and prevents the roof from breaking. Just as the lower flange in a rail helps the ball of the rail to support the weight, so does the floor help sustain the roof so long as the former is unbroken.

Fig. 17 shows the large amount of distortion needed to permit the fall of one leaf of a broken cumoid, or conchoid, 1,000 ft. deep.
Fig. 18 shows how this distortion is secured by repeated horizontal shears at places of weakness.

One of the patented features of the Luten bridge construction is its holoid structure. Luten does not use the term, but nevertheless the construction is closely analogous to the structure I have been describing. Luten builds not one bridge but two, one for vehicles to pass on and one to bind the feet of the piers together so as to form a perfect holoid. Destroy the lower bridge and the upper or traveling bridge is in a precarious condition. (See Fig. 15.)

The holoid structure gives the strongest of roofs; the pyoid is less strong; the cumoid is still weaker. Every horizontal shear makes the roof progressively less stable. Strange to say, vertical cracks permit the replacement in many cases of a less stable by a more stable structure. The fractures in a vertical direction, which would ordinarily be thought more detrimental than those in any other direction, prove not unimportant, it is true, yet in a way strengthening.

When a cumoid of great depth finds itself inadequate to its own support as a cumoid and begins to rend vertically from the surface downward to its supports and upward from the span centers to the surface, then the cumoid beam becomes an arch, and the cumoid plate becomes a dome and until it fails in its new structural relations, it cannot continue to extend those rents which threatened its stability. Let us say that the cumoid has become a conchoid (from conchus, a shell), the lines of stress forming figures resembling the shell of a bivalve.

As the elements of which the roof consists crowd each other into the opening it is impossible for the roof to rupture or fall until the various strata cease to act as a unit and begin to slide past each other. (See Fig. 16.) If we imagine an element of rock stretching from a rib to the half span, a distance, let us say, of 100 ft.; if we further suppose the excavated coal or other mineral to be 10 ft. thick, and regard the rib, floor, and falling rock element to be so adamant that none of the three will crush, heave or bend, then the inclination of the element when one corner reaches to the floor will be 10 per cent.

If the excavation is 1,000 ft. below the surface, the displacement of the block at the surface would have to be 100 ft. (See Fig. 17.) At a greater depth the displacement would be more. It is needless to say such a displacement cannot occur even where the crop, broken roof, or the sags in nearby workings make a certain amount of lateral adjustment possible. But horizontal shear along weak planes, such as beds of clay or coal or stratification planes, will permit such a number of readjustments in the element of rock itself that it will be able to come down. It will, in fact, rather tend to tilt than actually upset forward. (See Fig. 18.)

This, then, is how the end comes unless the rock is too strong to shear horizontally. Subsidence ceases, when, the required deformation has taken place. The choking up of the falls probably has but little to do with the final result, though it may have its effect in some cases. In fact, the theory as given is comforting to the conservationist, as the beds above the one extracted suffer but little, being exposed only to the horizontal shearing process, which is by no means destructive of the integrity of the measures.

However, the rupture at the half span may often result in falls of moderate size at that point, for when the monolithic roof is weakening by successive horizontal shears, we have a series of superposed cantilever cumoids entirely unequal to the task of self support. If the action which forms these cumoids does not let them all down to the floor suddenly, they will be sure to fracture successively from the bottom upward, and such seams of mineral as partake of this independent action will be destroyed.