S. Yu. Dobrozhinsky.

Materials and Studies

on the Theory and Calculation

of Mine Head Structures.

pp. 1–64 and Table 1.

Tomsk.

Typo-lithography of the Siberian Association of Printing, at the corner of Dvoryanskaya St. and Yamskaya Lane, own building No. 6

1910.

PREFACE.

Despite the enormous development of modern mining, the literature concerning hoisting structures is quite scant. In almost every textbook of mining engineering or hoisting machines, several pages are devoted to mine head buildings; schematic drawings of them are given to explain the general idea of the design, and that is where matters stop. As for the forces acting on these structures, in the textbooks mentioned they usually go no further than constructing the parallelogram of the forces from the tension of the rope by the load and by the machine.

In monographs on certain mining districts one may find more detailed drawings of structures distinguished by their solidity, and explanatory notes to them, concerning chiefly dimensions and design. From the specialized literature the following works may be cited: in German there is Die Seilscheiben-Gerüste der Bergwerke, by A. Eichenauer. Leipzig, 1877 (translated into Russian by mining engineer B. Feivushevich). This is perhaps the only work in which mine head structures are considered in some system, but at the present time this work is partly obsolete, does not correspond to modern views on structural statics, and errors are encountered in its methods of calculation. The recently published French brochure entitled Les chassis à moletts, disposition et calcul, par L. Lemaire. Paris, 1906, gives us examples of the shop calculation of mine head buildings used in Belgium and northern France. Of the same character is Thomson’s Coal-Hoisting Devices for Great Depths, a translation from Glückauf. Rather detailed descriptions and drawings are found in The Mechanical Engineering of Collieries, T. Campbell Futers; several examples were taken by me from The Engineering and Mining Journal, 1903.

That is all—or at least all that I managed to obtain and use.

S. Yu. Luborzhinsky.

Mine head structures, however, are built and do exist, and therefore are being calculated; but how these calculations are carried out in each individual case is known only to the plant that builds them, or rather to its technical office; in view of this, of course, there can be no question of any generally accepted method.

The aim of the present work is to fill the indicated gap; I wish to give interested persons the possibility, if not to design, then at least to check the designs presented to them without wasting unnecessary time searching for methods of calculation. I say “not to design,” since the most rational design itself, if prepared outside the plant, may in practice prove difficult to execute and costly, for only the plant’s technicians are in a position to take account of the available resources of their plant, and therefore it is most advantageous to entrust them with the designing itself. At the same time, the present work may serve as an aid for students in studying the question of headframes and their design.

In addition to the works indicated above, I also made use of the following: Stability of Deformations and Statics of Structures by Professor F. S. Yasinsky, Reference Book by Honored Professor Iv. Time, and Podręcznik Statyki Budowli by Maks. Thullie. During the printing of the present work, there appeared the work Das Fördergerüst by Th. Mörle, which, of course, could not be used.

Introduction.

The gradual development of mine-head structures may be characterized by the following stages: 1) the hand windlass, 2) the horse gin, in which a small pulley frame already appears as a necessary appliance, 3) light wooden headframes in the presence of a steam hoisting machine, then 4) stone or brick mine-head towers and, at the same time, more substantial wooden structures, and finally, 5) iron headframes consisting of various lattice trusses.

Until such time as the working of minerals is carried out at an insignificant depth and the output of the mine is not large, the mine-head structures are made of light construction, since in this case the machines are of small power, the ropes are thin, and the load being hoisted is small.

Where industry developed gradually, as occurred in Western Europe, the output of mines also increased gradually, simultaneously with the increase in the depth of shafts. The mine-head buildings adapted themselves to the new conditions, and their type was also worked out gradually.

In old mines, where it was no longer possible to increase the transverse dimensions of the hoisting compartments, in order to increase production it was necessary to use multi-deck cages, so as in this way to increase the number of mine cars in a single hoist. When new shafts were sunk, they were given dimensions corresponding to the number of cars hoisted simultaneously; however, only in rare cases are more than two cars lifted at once, standing side by side or one behind the other, since with a greater number of them the cross-section of the shaft would become excessively large, which would have an unfavorable effect on its strength and cost.

In general, cages are now built with two, three, and four decks. On each deck two cars are placed, side by side or one behind the other, as a result of which it becomes necessary to build high and substantial mine-head buildings and to use very strong ropes.

Under the above-mentioned conditions, wood ceased to be a suitable building material; in view of this, solid brick towers began to be used, against the walls of which the pulley...

beams. In time, however, it became apparent that structures of this type did not possess the degree of stability that was required of them; and since, at the same time, advances in metallurgy had made iron a generally available building material, it became necessary to return to the old, already partly abandoned type of headframes—but not wooden ones: rather, iron ones, of gigantic dimensions, which nevertheless in many respects resemble their prototype, the wooden headframe.

In Russia, with the diversity of its individual localities in terms of cultural and industrial development, we may at present encounter literally all types of headworks: wooden, masonry, and iron. Masonry shaft towers, however, are found only in Poland. This type corresponds to the stage of development of the coal industry and to the contemporaneous high cost of iron. In the south, wooden headframes still predominate, being replaced by iron ones. In general, it must be assumed that modern industrial life will eliminate the transitional link—the masonry tower—altogether, and that wooden headframes will be replaced directly by iron ones, as is, indeed, observed in America and, as we have already noted, in the Donets Basin.

Investigation of the Effect of the Tension of the Ropes from the Load and the Operation of the Machine

Whatever the structural system of the mine-head building may be, it must always counteract the active forces—the load being raised and the hoisting force of the machine.

Thus, let us have pulleys \(A\), Drawing 1, over which ropes are passed, winding onto and unwinding from the drum of the machine \(B\). For generality, let the drum be taken as conical, and let the rope be of uniform resistance. We shall first consider the case of a symmetrical arrangement of the pulleys, that is, the case when the hoisting compartments are located side by side, and then generalize the conclusions to the remaining cases.

Drawing 1.

Let \(\alpha_1\) be the angle formed by the upper rope with its vertical direction after leaving the pulley, and \(\alpha_2\) the same angle of the lower—

of the rope, \(s_1\) and \(s_2\) are the tensions of both ropes. The resultant applied to the sheave of the upper rope will be given by the expression:

\[ R_1=\sqrt{s_1^2+s_1^2-2s_1^2\cos(180-\alpha_1)} =s_1\sqrt{2(1+\cos\alpha_1)} =2s_1\cos\frac{\alpha_1}{2} \tag{1} \]

The horizontal overturning force, or the force caused by this force, will be

\[ U_1=R_1\sin\frac{\alpha_1}{2} =2s_1\sin\frac{\alpha_1}{2}\cos\frac{\alpha_1}{2} =s_1\sin\alpha_1; \tag{2} \]

the vertical pressure is equal to

\[ P_1=R_1\cos\frac{\alpha_1}{2} =2s_1\cos^2\frac{\alpha_1}{2} =s_1(1+\cos\alpha_1). \tag{3} \]

Exactly the same expressions can be written for the lower rope; it is only necessary to replace the subscripts 1 by 2.

The total overturning force will be

\[ U_1=s_1\sin\alpha_1+s_2\sin\alpha_2; \tag{4} \]

the total vertical pressure—

\[ P_1=s_1(1+\cos\alpha_1)+s_2(1+\cos\alpha_2). \tag{5} \]

In these expressions neither \(\alpha\) nor \(s\) are constant quantities; thus, \(\alpha\) depends on which coil of the rope is being wound onto or unwound from the drum; its value fluctuates between two extreme values for each rope, corresponding to the Max. and Min. winding radii; the tensions \(s\) depend on the position of the cages in the shaft and on whether the rope is carrying an empty or a loaded cage.

To determine the dependence of \(\alpha\) on the indicated circumstances, let \(L\) denote the horizontal distance from the axis of the drum shaft to the projection of the axis of the sheaves; \(H\), the vertical distance from the axis of the sheaves to the horizon of the machine shaft; \(r\), the radius of the sheave, and \(\rho\), the radius of the drum at the given instant; the remaining notations are clear from the drawing. For the upper rope we may write:

\[ \operatorname{tg}\alpha_1=(L+r+L_1):(H+H'); \]

and since

\[ L_1\cos\alpha_1=\rho_1,\quad H'\operatorname{tg}\frac{\alpha_1}{2}=r; \]

then, after substitution, we have

\[ \operatorname{tg}\alpha_1= \left(L+r+\frac{\rho_1}{\cos\alpha_1}\right): \left(H+\frac{r}{\operatorname{tg}\frac{\alpha_1}{2}}\right) \]

whence

\[ H\operatorname{tg}\alpha_1+ \frac{r\operatorname{tg}\alpha_1}{\operatorname{tg}\frac{\alpha_1}{2}} = \frac{(L+r)\cos\alpha_1+\rho_1}{\cos\alpha_1}; \]

after transformation and reduction we obtain

\[ H\sin\alpha_1-L\cos\alpha_1=\rho_1-r. \]

Let us denote by \(\beta\) the angle formed by the plane drawn through the axes of the pulleys and the drum with the vertical straight line; then

\[ L=H\operatorname{tg}\beta. \]

If this expression is introduced into the preceding one, it will take the following form:

\[ \sin\alpha_1-\operatorname{tg}\beta\cos\alpha_1=\frac{\rho_1-r_1}{H}, \]

or else

\[ \sin(\alpha_1-\beta)=\frac{\rho_1-r}{H}\cos\beta; \]

whence

\[ \alpha_1=\beta+\arcsin\left(\frac{\rho_1-r}{H}\cos\beta\right)=\beta+\gamma, \tag{6} \]

if

\[ \sin\gamma=\frac{\rho_1-r}{H}\cos\beta. \]

In view of the fact, however, that

\[ \sin\beta=\frac{L}{\sqrt{H^2+L^2}},\quad \cos\beta=\frac{H}{\sqrt{H^2+L^2}}, \]

S. Yu. Doborzhinsky.

we can write our expression, eliminating the auxiliary angles, as follows:

\[ \alpha_1=\arcsin \frac{L}{\sqrt{H^2+L^2}}+\arcsin \frac{+\rho_1-r}{\sqrt{H^2+L^2}}. \tag{7} \]

For the lower rope, the corresponding expression will be obtained if, instead of \(\rho_1\), we substitute \(-\rho_2\):

\[ \alpha_1=\arcsin \frac{L}{\sqrt{H^2+L^2}}+\arcsin \frac{-\rho_2-r}{\sqrt{H^2+L^2}}. \tag{8} \]

Only one variable quantity, \(\rho_1\) or \(\rho_2\), enters into these expressions; for cylindrical drums, if, as often happens,

\[ \rho = r \]

our expressions assume a simpler form, namely:

\[ \alpha_1=\arcsin \frac{L}{\sqrt{H^2+L^2}}, \]

\[ \alpha_2=\arcsin \frac{L}{\sqrt{H^2+L^2}}-\arcsin \frac{2r}{\sqrt{H^2+L^2}}. \tag{9} \]

For the case in which the sheaves are located at different horizons and at different distances from the axis of the drum, and also when the drums are arranged one behind the other, in general the same derivation is applicable; it is only necessary to replace \(L\) and \(H\), respectively, by \(L_1, H_1\) and \(L_2, H_2\).

Of course, what has been said also applies to the Koepe hoist.

From the formulas given above we see that the angle of inclination of the resultant to the vertical is always

\[ 0 > \frac{\alpha}{2} > 45^\circ, \]

or \(\alpha\) becomes zero only when \(L-r=0\), and a right angle only in the case when \(L-r=\infty\); neither the one nor the other is practically attainable.

The limiting values of \(\alpha_1\) and \(\alpha_2\):

\[ \begin{aligned} &\text{for the upper rope}\quad &&\alpha_1'=\operatorname{Max}\alpha_1 \quad \text{when } \rho_1=\frac{D}{2},\\ &&&\alpha_1''=\operatorname{Min}\alpha_1 \quad \text{when } \rho_1=r,\\ &\text{for the lower rope}\quad &&\alpha_2'=\operatorname{Min}\alpha_2 \quad \text{when } \rho_2=\frac{D}{2},\\ &&&\alpha_2''=\operatorname{Max}\alpha_2 \quad \text{when } \rho_2=r. \end{aligned} \tag{10} \]

In order to be able to compute \(\alpha_1\) and \(\alpha_2\) for any position of the cage, it is necessary to have the relation between \(\rho_1\) and the depth at which this cage is located at the given moment; let this depth be \(h\). If, moreover, we denote the angle of taper of the drum by \(\eta\), then

\[ \tg \eta=\frac{\frac{D}{2}-r}{l}; \]

where \(l\) is the length of the drum. If the width of the grooves on the drum is \(e\), then mine mechanics gives us the following relation between these quantities:

\[ eh\sin\eta=\pi\left(\frac{D^2}{4}-\rho_1^2\right), \]

whence

\[ \rho_1=\sqrt{\frac{D^2}{4}-\frac{eh\sin\eta}{\pi}}. \tag{11} \]

For computing \(\rho_2\), one may use the property

\[ \rho_1+\rho_2=\frac{D}{2}+r. \]

If we were to substitute expression (11) for \(\rho_1\) in (7), we would obtain a formula by which \(\alpha\) could be computed directly. I shall confine myself to indicating only the possibility of this kind.

calculations, since in themselves they are of no interest.

If we were to move both resultants parallel to themselves to a point lying midway between them, then we would have to consider the parallelogram of forces formed by \(R_1\) and \(R_2\), as well as the moments of the forces in their former position with respect to the new point of application. Their common resultant

\[ R=\sqrt{R_1^2+R_2^2-2R_1R_2\cos\frac{\alpha_1-\alpha_2}{2}} \]

is destroyed by the bearing resistances of the headframe, leaving moments which may be expressed as follows: if by \(\lambda\) we denote the mutual distance between the sheaves, then the moment of the horizontal components will be

\[ M_n=\frac{\lambda}{2}\left(R_1\sin\frac{\alpha_1}{2}-R_2\sin\frac{\alpha_2}{2}\right). \]

The moment of the vertical components is

\[ M_p=\frac{\lambda}{2}\left(R_1\cos\frac{\alpha_1}{2}-R_2\cos\frac{\alpha_2}{2}\right). \]

These moments tend to rotate the entire structure: the first about a vertical axis, the second about a horizontal axis parallel to the axis of the structure, in one direction or the other, depending on which of the forces is greater at the given moment. Since the forces under consideration may, as we shall see below, assume very different values, their moments may likewise be expressed by rather large quantities. In view of this, headframes are often designed in such a way that each half is capable of taking up completely, without the assistance of the other part of the structure, its own resultant.

For the complete determination of \(R_1\) and \(R_2\), it is also necessary to be able to calculate the tensions \(s_1\) and \(s_2\). Thus, let \(G\) be the useful load, \(Q\) the dead load, \(W\) the weight of the rope, and \(T\) the friction; then, for the loaded cage being raised,

\[ s_1=G+Q+W_1+T; \]

for the empty cage being lowered,

\[ s_2=Q+W_2-T. \]

In these equalities \(W\) and \(T\) are variable quantities; \(T\) depends on many factors, but chiefly on the speed of motion of the cage, which in turn is connected with the depth at which it is located, and with the depth of the mine itself. However, in view of the fact that \(T\) amounts to only about \(5\%\) of all the other forces determining the tension, in order not to complicate the question we shall introduce, instead of its exact value, a correction in the form of a numerical coefficient; \(W\) also depends exactly on the depth. For cylindrical ropes it may be given by the expression

\[ W = wh, \]

where \(w\) is the weight per running unit of rope, and \(h\) is the magnitude of its overhanging part in the mine. For conical ropes

\[ W = (Q + G)\left(e^{\frac{\delta h}{k}} - 1\right), \]

in which \(k\) is the coefficient of resistance from tension, and \(\delta\) is the specific weight.

If we make use of these expressions, then for cylindrical ropes we obtain:

\[ \begin{aligned} s' &= 1.05(G + Q + wh),\\ s'' &= 0.95(G + wh), \end{aligned} \tag{12} \]

and for conical ropes

\[ \begin{aligned} s' &= 1.05\,G + Q e^{\frac{\delta h}{k}},\\ s'' &= 0.95\,Q e^{\frac{\delta h}{k}}. \end{aligned} \tag{12a} \]

Thus, the stresses do not depend on how the rope is passed over the sheave, whether from the lower or from the upper part of the drum. The maximum rope stress will occur, of course, at the beginning of hoisting, when \(h\) is equal to the full depth of the mine; the minimum tension is in that rope to which the empty cage is suspended, when it is still at the surface.

Such tensions occur at the beginning of every hoisting operation; at its end the Max. may be on the side of the empty cage, if only

since the total weight of the rope exceeds the useful load. With respect to this rope, Max. will always be at the lower position of the cage, and Min.—at the upper.

From all that has been said we are entitled to conclude that the resultants \(R_1\) and \(R_2\) periodically change their magnitude not only in direction, but also in magnitude, depending on the fact that the tensions \(s_1\) and \(s_2\) and their angles of inclination change. Even in the case of employing cylindrical drums or the Koepe hoist, the total resultant \(R\) changes somewhat in its direction and in its magnitude, because the components \(R_1\) and \(R_2\) are variable in magnitude. Ultimately we may arrive at the conclusion that the tendency of some builders to make the inclined leg take up the entire force \(R\) is impracticable; one can strive only to choose the most suitable direction for this leg.

Let us now consider the horizontal and vertical forces caused by the tensions of the ropes. Let \(\alpha_1'\) and \(\alpha_2'\) denote the angles at the upper position of the cages, and \(\alpha_1''\) and \(\alpha_2''\) at the lower; it is clear that:

\[ \alpha_1'' > \alpha_1' \quad \alpha_2' < \alpha_2'' . \tag{13} \]

This is easily verified from formulas (7) and (8) and simply from the drawing. Thus, for the beginning of hoisting, with the forward stroke of the machine,

\[ U = \operatorname{Max} s_1 \sin \alpha_1'' + \operatorname{Min} s_2 \sin \alpha_2', \]

\[ P = \operatorname{Max} s_1 (1 + \cos \alpha_1'') + \operatorname{Min} s_2 (1 + \cos \alpha_2'), \]

toward the end of the same hoisting,

\[ U = s_1 \sin \alpha_1' + s_2 \sin \alpha_2'', \]

\[ P = s_1 (1 + \cos \alpha_1') + s_2 (1 + \cos \alpha_2''); \]

for the next stroke (occupied) of the machine, at its beginning,

\[ U = \operatorname{Min} s_1 \sin \alpha_1' + \operatorname{Max} s_2 \sin \alpha_2'', \]

\[ P = \operatorname{Min} s_1 (1 + \cos \alpha_1') + \operatorname{Max} s_2 (1 + \cos \alpha_2'') \]

and at the end

\[ U = s_1 \sin \alpha_1'' + s_2 \sin \alpha_2', \]

\[ P = s_1 (1 + \cos \alpha_1'') + s_2 (1 + \cos \alpha_2'), \]

Comparing these expressions, it is easy to see that the greatest horizontal tension occurs at the beginning of hoisting in the forward run, i.e. when the loaded cage of the upper rope is at the bottom of the shaft and the empty cage of the lower rope is at the surface. The greatest vertical pressure arises at the beginning of hoisting in the rear run of the machine, when the loaded cage of the lower rope is below.

What has been said applies to normal hoisting; if, however, during hoisting materials are being lowered into the shaft, then in every run of the machine the indicated maximum forces \(P\) and \(U\) appear at its beginning and at its end, so that for this case, if friction is neglected, one may write for conical ropes

\[ \operatorname{Max} U = (G+Q)\sin \alpha_1' + (G+Q)e^{\frac{\partial h}{k}}\sin \alpha_2'', \]

\[ \operatorname{Max} P = (G+Q)e^{\frac{\partial h}{k}}(1+\cos \alpha_1'') + (G+Q)(1+\cos \alpha_2'') \tag{14} \]

These two forces can never appear simultaneously, but, since the lowering of material is a common occurrence in mines, the headframes must, when necessary, satisfy the requirements of both these forces. For the case of cylindrical ropes,

\[ \operatorname{Max} U = (G+Q)\sin \alpha_1' + (G+Q+wh)\sin \alpha_2'', \]

\[ \operatorname{Max} P = (G+Q\,wh)(1+\cos \alpha_1'') + (G+Q)(1+\cos \alpha_2'). \tag{15} \]

However, besides resisting the above-mentioned forces, the headframe building must sometimes also withstand certain extraordinary stresses.

We know that a rope may break either from the action of emergency stresses, when at the beginning of hoisting the machine is set in motion with excessive speed—then the break may occur both on the side of the loaded cage and on the side of the empty cage; or the break may take place during the hoisting itself, when the cage is jammed somewhere in the guides; or, finally, from the impact of the cage against the sheaves, if it has been raised at great speed to their level. In all these cases there develops a force—tension—capable of breaking the rope. Since ropes are designed with a safety factor of 5–6 with respect to the load \(G+Q\) for ropes of equal resistance and \(G+Q+wh\) for ropes of equal cross-section,

the breaking forces must exceed these forces by five to six times; thus

\[ \begin{gathered} \sigma_k = n(G + Q),\\ \\ \sigma_c = h(G + Q + wh), \end{gathered} \tag{16} \]

where \(n = 5—6\) and more. It is clear that the most dangerous place for ropes of equal resistance with respect to these extraordinary tensions is that part of the rope directly adjoining the cage, since in the other, higher-lying parts the stress is distributed over a section of greater area.

Conversely, cylindrical ropes are subject to the danger of rupture near the sheaves, since in the parts located lower down the stretching force does not include the full weight of the rope. Our reasoning, of course, applies only to new, fully good-quality ropes, for a rope that has been in service may break at any other place, although, apparently, the parts at the cages suffer most of all; in that case, however, the breaking forces will be less than \(\sigma_k\) or \(\sigma_c\).

From the action of emergency stresses both ropes cannot break simultaneously, for these stresses first appear in the rope moving upward, and then on the side of the descending cage.

From jamming in the guides, only the rope moving upward can break; for the descending cage this can occur only if the cage for some reason became stuck, and then suddenly broke loose and fell freely downward; it is clear that the simultaneous coincidence of these two indicated contingencies is scarcely probable.

Let us suppose that the rupture of the rope occurs during the lowering of material; such an assumption simplifies the question; it is appropriate, for by adopting it we erect a more stable structure. Thus, we have to consider two cases: 1) the rope breaks at the cage located at the bottom of the shaft, 2) at the cage at the surface.

The load will have two magnitudes: the full \(\dfrac{\sigma}{n}\) and the incomplete \(\dfrac{\sigma_0}{n}\), when the cage is at the mouth of the shaft. In view of the assumption made, exactly the same breaking tension \(\sigma\) is one and the same in all cases.

Let there be the forward stroke of the machine, the beginning of hoisting:

break below:

\[ \begin{aligned} U_a&=\sigma\sin\alpha_1''+\frac{\sigma_0}{n}\sin\alpha_2',\\ P_a&=\sigma(1+\cos\alpha_1'')+\frac{\sigma_0}{n}(1+\cos\alpha_2'); \end{aligned} \qquad\left\}\ \text{Ia}\quad \begin{gathered} \text{break of the upper rope,}\\ \text{incomplete load on the lower one,} \end{gathered} \]

break above:

\[ \begin{aligned} U_b&=\frac{\sigma}{n}\sin\alpha_1''+\sigma\sin\alpha_2',\\ P_b&=\frac{\sigma}{n}(1+\cos\alpha_1'')+\sigma(1+\cos\alpha_2'); \end{aligned} \qquad\left\}\ \text{Ib}\quad \begin{gathered} \text{break of the lower rope,}\\ \text{full load of the upper one;} \end{gathered} \]

at the end of hoisting, if the break is below:

\[ \begin{aligned} U_c&=\frac{\sigma_0}{n}\sin\alpha_1'+\sigma\sin\alpha_2'',\\ P_c&=\frac{\sigma_0}{n}(1+\cos\alpha_1')+\sigma(1+\cos\alpha_2''); \end{aligned} \qquad\left\}\ \text{IIc}\quad \begin{gathered} \text{break of the lower rope,}\\ \text{incomplete load on the upper}\\ \text{rope;} \end{gathered} \]

break above:

\[ \begin{aligned} U_d&=\sigma\sin\alpha_1'+\frac{\sigma}{n}\sin\alpha_2'',\\ P_d&=\sigma(1+\cos\alpha_1')+\frac{\sigma}{n}(1+\cos\alpha_2''); \end{aligned} \qquad\left\}\ \text{IId}\quad \begin{gathered} \text{break of the upper rope,}\\ \text{full load of the lower}\\ \text{rope.} \end{gathered} \]

Let us now consider the reverse run of the machine, the beginning of hoisting:

break below

\[ \begin{aligned} U&=\frac{\sigma_0}{n}\sin\alpha_1'+\sigma\sin\alpha_2'',\\ P&=\frac{\sigma_0}{n}(1+\cos\alpha_1')+\sigma(1+\cos\alpha_2''); \end{aligned} \qquad\left\}\ \text{IIc}\quad \begin{gathered} \text{break of the lower rope,}\\ \text{incomplete load on the}\\ \text{upper one;} \end{gathered} \]

break above

\[ \begin{aligned} U&=\sigma\sin\alpha_1'+\frac{\sigma}{n}\sin\alpha_2'',\\ P&=\sigma(1+\cos\alpha_1')+\frac{\sigma}{n}(1+\cos\alpha_2''); \end{aligned} \qquad\left\}\ \text{IId}\quad \begin{gathered} \text{break of the upper rope,}\\ \text{full load of the lower one.} \end{gathered} \]

At the end of the hoisting, if the break occurs below:

\[ \begin{aligned} U &= \sigma \sin \alpha_1'' + \frac{\sigma_0}{n}\sin \alpha_2',\\ P &= \sigma(1+\cos \alpha_1'')+\frac{\sigma}{a}(1+\cos \alpha_2'), \end{aligned} \qquad \left\{ \begin{array}{l} \text{Ia}\quad \text{break of the upper rope;}\\ \text{incomplete load of the lower one;} \end{array} \right. \]

if the break occurs above:

\[ \begin{aligned} U &= \frac{\sigma}{n}\sin \alpha_1''+\sigma \sin \alpha_2',\\ P &= \frac{\sigma}{n}(1+\cos \alpha_1'')+\sigma(1+\cos \alpha_2'), \end{aligned} \qquad \left\{ \begin{array}{l} \text{Ib}\quad \text{break of the lower rope;}\\ \text{full load of the upper one.} \end{array} \right. \]

Thus we have only the cases Ia,b and Ic,d; moreover it is clear that the greatest vertical pressures are obtained when the lower rope breaks, and the greatest horizontal forces—when the upper one breaks.

It is easy to prove that

\[ P_b > P_c . \]

Comparing their expressions, we obtain the inequality

\[ \frac{\sigma}{n}(1+\cos \alpha_1'')+\sigma(1+\cos \alpha_2') > \frac{\sigma_0}{n}(1+\cos \alpha_1')+\sigma(1+\cos \alpha_2''), \]

which necessarily holds in view of the existence of the inequalities

\[ \cos \alpha_1'' > \cos \alpha_1', \quad \cos \alpha_2' > \cos \alpha_2'', \quad \sigma > \sigma_0 . \]

Thus we may say that

The greatest vertical pressure arises when the lower rope breaks, when its cage is approaching the sheaves, and the cage of the upper rope has descended almost to the bottom of the shaft.

The same reasoning shows us that

\[ U_d > U_a \]

In fact, the inequality

\[ \sigma \sin \alpha_1' + \frac{\sigma}{n}\sin \alpha_2'' > \sigma \sin \alpha_1'' + \frac{\sigma_0}{n}\sin \alpha_2' \]

will hold, since the inequalities

\[ \sin \alpha_1' > \sin \alpha_1'',\quad \sin \alpha_2'' > \sin \alpha_2',\quad \sigma > \sigma_0 \]

hold.

Thus, we have the right to say that

The greatest horizontal overturning force occurs when the upper rope breaks, when the cage is located near the sheaves, and the other cage is at the bottom of the shaft.

On the basis of all that has been said, one may give the following expression for the greatest horizontal force and the vertical force simultaneous with it:

\[ \begin{aligned} \operatorname{Max} U &= \sigma\left(\sin \alpha_1' + \frac{\sin \alpha_2''}{n}\right),\\ P &= \sigma\left[(1+\cos \alpha_1') + \frac{(1+\cos \alpha_2'')}{n}\right]. \end{aligned} \tag{17} \]

For the maximum vertical force,

\[ \begin{aligned} U &= \sigma\left(\frac{\sin \alpha_1''}{n}+\sin \alpha_2'\right),\\ \operatorname{Max} P &= \sigma\left[\frac{(1+\cos \alpha_1'')}{n}+(1+\cos \alpha_2')\right]. \end{aligned} \tag{18} \]

These maximum forces cannot appear simultaneously; in view of this, the structures are calculated and built so that they may take either group of forces separately.

In the case of conical or cylindrical drums, in addition to the forces that we have just considered, there are also lateral overturning forces, arising from the fact that the ropes deviate now to the left, now to the right from the plane of the sheaves.

Thus, if by \(\gamma_1\) we denote the angle of the extreme left deviation, and by \(\gamma_2\) that of the extreme right deviation (for the left sheave and drum), then

\[ \tg \gamma_1=\frac{l_1}{L-\frac{d}{2}-r},\qquad \tg \gamma_2=\frac{l_2}{L-\frac{d}{2}-r}, \]

under the condition

\[ l=l_1+l_2, \]

where \(l\) is the width of the drum, and \(l_1\) and \(l_2\) are its parts on both sides of the plane of the pulley. When the drums are arranged with their broad sides inward, the same expressions are obtained, only in the reverse order. The axial stress in the plane of the pulley is equal to the stress in the vertical parts of the rope \(s\). At the beginning and at the end of each hoist these stresses take the values \(s_1\) and \(s_2\); in view of this, the lateral tension for the extreme positions of the ropes will be

Fig. 2.

\[ v_1=s_1 \operatorname{tg}\gamma_1+s_2 \operatorname{tg}\gamma_2=v_2, \tag{19} \]

where the directions of \(v_1\) and \(v_2\) are directly opposite to each other.

To determine the magnitude of these forces \(v\), let us simplify the problem by assuming that the drum is cylindrical. In this case

\[ \gamma_1=\gamma_2,\quad r=\frac{d}{2}=\frac{D}{2},\quad l_1=l_2=\frac{l}{2}; \]

and therefore

\[ \operatorname{tg}\gamma=\frac{l}{2}:(L-2r); \]

but since the machine is usually arranged so that

\[ \frac{l}{L}=\frac{1}{20}\ \text{to}\ \frac{1}{30}, \]

then, taking as an average value \(L=25l\), and \(l=\infty\,3\frac{D}{2}\), we shall have

\[ \operatorname{tg}\gamma=\infty\,\frac{1}{47}, \]

whence

\[ v=(s_1+s_2)\,0.02. \]

This quantity is small in comparison with \(s_1\) and \(s_2\), but it may not be small absolutely. Of course, the greatest value of \(v\) is obtained when the rope breaks, namely, if the rope breaks at the cage that is coming to the surface while the other is in the lowest position; then the rope tension will be \(\sigma,\frac{\sigma}{n}\), of course—under the assumption of lowering material, and therefore

\[ \operatorname{Max} v=\sigma\left(\operatorname{tg}\gamma_1+\frac{\operatorname{tg}\gamma_2}{n}\right). \tag{20} \]

The arguments presented also show that the tension of the ropes between the machine and the pulleys is somewhat greater than their tension in the vertical part, independently of friction and of the stiffness,

\[ S=\frac{s}{\cos\gamma}. \]

For cylindrical drums,

\[ S=\frac{s}{\sqrt{1+\left(\frac{l}{2L-2l}\right)^2}}. \]

For spiral drums, as well as for Koepe hoisting, and also Graven’s, lateral forces do not exist, since for them \(\gamma=0\).

Forces in the case of balanced ropes.

Let us now consider the forces taken up by the headframe structures in the case of hoisting by means of balanced ropes. Here two systems must be distinguished: 1) balancing by means of ropes attached to the upper part of the cages, and 2) by means of ropes suspended beneath the cages.

Let us consider the case of upper balancing. Let \(wx\) be the weight of the entire hoisting rope, \(g_1\) the weight of the counterweight rope, and \(g_2\) the weight of the thin steel cords passing around the auxiliary pulleys \(c\) and \(d\) (Fig. 3, p. 20). The tension at the beginning of hoisting, when the rope \(g_1\) is on the right-hand side, will be

\[ S_1=Q+G+wh+g_2-g_1,\qquad S_2=G-g_2. \]

Toward the end of hoisting the rope \(g_1\) passes to the left-hand side, and

\[ S_1=Q+G-g_2, \]

\[ S_2=G+wh+g_2-g_1. \]

In these expressions \(Q\) is the full load, and \(G\) is the weight of the cages with the mine cars. As is known, for complete balancing it is necessary that

\[ wh+2g_2=g_1, \]

that is, that the weight of the counterweight rope be equal to the weight of one main rope plus twice the weight of the cord; this condition is difficult to achieve

in practice; moreover, in order that the empty cage could not be lifted from the bottom of the shaft by the counterweight, the inequality must hold

\[ g_1 < G + g_2 . \]

Fig. 3.

The tensions in the auxiliary ropes will be, at the beginning and end of each lift, depending on the side on which the flat balancing rope is situated,

\[ s_1 = g_2,\quad g_1 = g_2 = s_2. \tag{21} \]

or conversely. Thus, in the normal operation of the hoisting device, when raising the mineral and lowering materials into the shaft, we have to take into account the tensions \(S_1, S_2, s_1\), and \(s_2\), or the resultants \(R_1, R_2, r_1\), and \(r_2\), applied at the centers of the main and auxiliary sheaves, where the resultants

\[ r_1 = 2(g_1 - g_2),\quad r_1 = 2g_2 \]

have a vertical direction. For calculating the headframe, everything said earlier remains in force; namely: the greatest horizontal and vertical forces are obtained when the ropes break at the surface. From the balancing ropes the mine building is burdened with the load

\[ r_1 + r_2 = 2g_1 . \]

For calculations we shall apply formulas (17) and (18).

In the case of lower balancing, the rope tension is a constant quantity for each lift and is equal at any moment to

\[ S_1 = G + Q + wh, \]

\[ S_2 = G + wh. \tag{23} \]

Here complete balancing takes place; the tension of the ropes changes only depending on whether a loaded or an empty cage is suspended from it. Obviously, during the lowering of material,

\[ s_1 = s_1 = \frac{\sigma}{n}G + Q + wh. \]

In the present case it is immaterial where the jamming of the cage, or in general the breaking of the rope, occurs; therefore the greatest horizontal force is obtained in general when the upper rope breaks, and the greatest vertical force when the lower one breaks. Since the drums here are cylindrical, the angles \(\alpha_1\) and \(\alpha_1\) are constant quantities. Formulae (17) and (18) may be used for this case with only the reservations just stated. The Koepe hoisting arrangement presents a certain peculiarity. Since the rope on the sheave replacing the drum is not fastened, but is held solely by friction, then in the event of the cage jamming somewhere in the shaft, the rope tension can reach only a value equal to this friction, after which the rope will begin to slip over the sheave and no break will occur. The ratio of the greatest tension to the least is given by the formula

\[ \frac{s_1}{s_2} = e^{f\eta}, \]

in which the quantities \(s_1\) and \(s_2\) are known, i.e. may be calculated from formula (23). Thus, once the angle of wrap \(\eta\) has been determined and even taken somewhat larger than required by the formula, then in the event of jamming, even during the lowering of material,

\[ \operatorname{Max} s = (G + Q + wh)e^{f\eta}; \tag{24} \]

in other words, its magnitude must remain close to the normal tension. I consider it also necessary to note that, owing to the indicated feature of the design, extreme stresses from an incautious starting of the machine during hoisting with the Koepe system are likewise impossible. Here the following remark must be made: before deciding to adopt a lighter construction of the headframe with the Koepe system, it is necessary to consider whether, in view of a further deepening of the mine beyond the limit at which this system can be used, it will not be necessary to change over to ordinary drums. As regards the placement of sheaves at different levels, this is ...

This circumstance cannot have any substantial influence on the calculation and design of pulley frames. Formulas (17) and (18), in the form in which they are given, are not used in practical calculations; they are of more theoretical interest. For calculations it is more important to have an expression for the horizontal and vertical forces acting on the individual pulleys. We shall denote these forces by \(V_1\), \(V_2\) and \(P_1\), \(P_2\), where the subscript 1 refers to the pulley of the upper rope, and the subscript 2 to the lower one.

Resolving the indicated expressions, we obtain

\[ \operatorname{Max.} U_1=\sigma \sin \alpha'_1,\qquad P_1=\sigma(1+\cos \alpha'_1). \tag{25} \]

At the same time, on the other pulley,

\[ U_2=\frac{\sigma}{n}\sin \alpha''_2,\qquad P_2=\frac{\sigma}{n}(1+\cos \alpha''_1). \tag{25} \]

For rupture of the lower rope, when the greatest vertical pressure arises,

\[ U_2=\sigma \sin \alpha'_2,\qquad \operatorname{Max} P_2=\sigma(1+\cos \alpha'_2), \]

\[ U_1=\frac{\sigma}{n}\sin \alpha''_1,\qquad P_1=\frac{\sigma}{n}(1+\cos \alpha''_1). \tag{26} \]

For the case of an inclined shaft, the calculation of the active forces appearing at the pulleys is somewhat different from the preceding one. If by \(\alpha_1\) and \(\alpha_2\) we denote the inclination of the ropes to the vertical, and by \(\beta\) the inclination of the shaft to the vertical, then, retaining the remaining designations, for the magnitudes of the resultants we find the following expressions:

Fig. 4.

\[ R_1=2s_1\cos\frac{\alpha_1+\beta}{2},\qquad R_2=2s_2\cos\frac{\alpha_2+\beta}{2}. \]

The horizontal overturning force for the upper rope will be

\[ V_1 = s_1(\sin \alpha_1 - \sin \beta) = 2s_1 \cos \frac{\alpha_1+\beta}{2}\,\sin \frac{\alpha_1-\beta}{2}, \]

and the vertical pressure—

\[ P_1 = s_1(\cos \alpha_1 + \cos \beta) = 2s_1 \cos \frac{\alpha_1+\beta}{2}\,\cos \frac{\alpha_1-\beta}{2}; \]

the inclination of the resultant to the vertical will be expressed as follows:

\[ \delta_1 = \frac{\alpha_1-\beta}{2},\qquad \delta_2 = \frac{\alpha_2-\beta}{2}; \]

it is clear that, depending on whether the differences \(\alpha_1-\beta\) and \(\alpha_2-\beta\) are positive or negative, the resultants may be inclined to the right or to the left. In the case where \(\alpha_1-\beta=0\) or \(\alpha_2-\beta=0\), the corresponding resultant becomes vertical. At the same time, one of the horizontal components becomes zero. In general, the horizontal forces are the smaller, the less the angles \(\alpha\) and \(\beta\) differ from one another.

Since all the remaining preceding conclusions can, in general, very easily be extended also to the case of headframes at inclined shafts, I shall confine myself only to the addition just made, without entering into details which would merely repeat these conclusions in a more general form.

Other external forces borne by mine-head buildings.

Besides the active forces \(R_1\) and \(R_2\), which are the direct result of lifting the load, mine-head buildings have to resist wind pressure and their own weight. This latter force is directed vertically and is transmitted from the upper parts of the structure to the lower ones; in calculating the separate parts, this circumstance must be borne in mind. We give here a table of the ratios of the weights of iron headframes, their heights, and the useful load lifted at one time.

Mine name	Height, m	Useful load	Weight of headframe
Gneisenau I	38.0	4400 kg	119730 kg
” II	38.0	4400 ”	107003 ”
Lothringen	26.0	3300 ”	78100 ”
Monopol	26.5	4400 ”	75000 ”
Minister Stein	32.0	6600 ”	93000 ”
Pluto	30.4	1100 ”	65000 ”
Massen	30.4	2200 ”	90000 ”
Kaiser Friedrich	44.0	4400 ”	115000 ”
Kaiserstal	38.0	4400 ”	108000 ”
Von der Heydt	41.4	4400 ”	90000 ”
Salz Glückauf	30.0	2200 ”	100000 ”

This table is borrowed from Thomson’s work.

As for wind pressure, as is known, it is directed at a certain angle to the horizon; this angle is usually taken equal to \(10^\circ\).

If the plane opposing the motion of the wind is set at an angle \(\delta\) to the vertical, that is, at an angle \(90^\circ-\delta\) to the horizon, and if the pressure on a square unit of the plane \(ab\), normal to the direction of the wind, is denoted by \(q\), then the pressure per sq. unit of our plane, according to the ratio of the sides in the triangle \(abc\), will be (Fig. 5)

\[ q_1 = q \cos(\delta - 10^\circ); \]

\(q\) is taken for calculations as 180 kg per sq. meter.

Fig. 5.

If the wall of a building subjected to the force of the wind is smooth, as, for example, in masonry headhouse buildings or in headframes covered with plank sheathing or sheet iron, so that the friction of the wind against this obstacle may be neglected, then, resolving \(q_1\) into pressures normal and parallel to the wall, for the normal pressure we obtain

\[ q_2 = q \cos^2(\delta - 10^\circ). \]

We resolve this normal pressure into horizontal and vertical components; the vertical component is

\[ p = q \cos^2(\delta - 10^\circ)\sin\delta, \]

horizontal component

\[ v=q\cos^2(\delta-10^\circ)\sin\delta. \tag{27} \]

The forces \(vF\) and \(pF\), where \(F\) is the area subjected to wind pressure, may rightly be assumed to be applied at the center of pressure of the surface part of the structure under consideration, and then distributed among the adjacent joints on the assumption that this part constitutes a rigid body.

It is easy to see that the horizontal pressure of the wind directed from the machine toward the sheaves during hoisting will be compensated by the horizontal component \(V\) of the hoisting force. It can be dangerous only in the case when the machine is inactive, when the cages are set on the detents. Conversely, during hoisting the dangerous direction is that from the headframe toward the machine, since in this case both horizontal forces are added together. Some danger is also presented by the direction perpendicular to the longitudinal axis of the mine-head building, which is added to the forces arising from the deviation of the ropes; see formulas (19) and (20). In order to make the headframe more stable in this direction, the distance between the legs at the bottom is made somewhat greater than at the sheaves.

It is much more difficult to deal with wind pressure on open headframes (without sheathing) than in the case we have just considered. In low wooden headframes this force is most often neglected, in view of the enormous margin of strength with which such headframes are built. For iron structures of substantial dimensions, however, it is already difficult to neglect this force. If the part of the headframe subjected to wind action is inclined to the vertical at an angle \(\delta\), then, owing to the different shape of the iron members of which this part consists, some planes will be inclined at an angle \(\delta\), while others at an angle \((90^\circ-\delta)\); furthermore, it is known that the resistance of lattice obstacles is in reality greater than the resistance of the area occupied by the members of the lattice themselves.

It is difficult to determine exactly in this case the magnitude of the pressure and its point of application. If by \(k\) we denote the coefficient of increase of pressure for lattice obstacles, then the former formulas (pressure per unit area) for parts inclined at an angle \(\delta\) take the form

\[ p'=kq\cos^2(\delta-10^\circ)\sin\delta, \]

\[ v'=kq\cos^2(\delta-10^\circ)\cos\delta, \]

for the parts inclined at an angle \((90^\circ-\delta)\), we shall have

\[ p''=kq\sin^2(\delta-10^\circ)\cos\delta, \]

\[ v'=kq\sin^2(\delta-10^\circ)\sin\delta. \]

Thus we see that in this case part of the pressure is directed upward. If by \(f_1\) we denote the total area of the parts inclined at an angle \(\delta\), and by \(f_2\)—at an angle \((90^\circ-\delta)\), then the total pressure on the part of the headframe subjected to the action of the wind will be

\[ \begin{aligned} P'&=kq\left[f_1\cos^2(\delta-10^\circ)\sin\delta -f_2\sin^2(\delta-10^\circ)\cos\delta\right],\\ V'&=kq\left[f_1\cos^2(\delta-10^\circ)\cos\delta +f_2\sin^2(\delta-10^\circ)\sin\delta\right]. \end{aligned} \tag{28} \]

In this case we have neglected the friction of the wind on the surface, that is, the force directed along them, just as in the case of sheathed or solid walls. It is clear that, before the design is drawn up, it is impossible to make use of exact data for \(f_1\) and \(f_2\); therefore one usually somewhat increases the forces \(V_1\), \(V_2\), \(P_1\), and \(P_2\) according to experience, or simply calculates the pressure as though the lattice barriers were replaced by solid ones. If such a calculation for some reason gives rise to doubts, then the completed design sketch may be checked by formulas (28).

On Mine Head Structures in General.

The construction of mine head structures depends to a considerable degree on the material from which it is built. In addition, the depth from which hoisting is carried out, and the quantity of mineral raised at one time, play the principal role here. Let us briefly consider the advantages and disadvantages of wooden, masonry, and iron structures.

Wooden buildings are advantageously used in those places where cheap structural timber of the required length is available at hand. Of course, cheapness in this case is only a relative concept; in a remote steppe locality one sometimes has to use expensive timber simply because iron is still more expensive. A second indispensable condition justifying the use of a wooden frame is the small output of the mine at a small depth, since for large mines, in particular—

...for coal mines, when large loads must be lifted from considerable depths, wooden structures do not provide adequate strength and stability. However, where output at ore deposits is insignificant, wooden headframes may also be used for fairly considerable depths. An important condition for the use of wooden headframe structures is the presence of at least two hoisting shafts, so that, when a headframe that has become unfit is replaced by another, there is no prolonged stoppage in production, which is always associated with losses.

Wooden headframes, moreover, are almost always used as temporary structures when sinking new shafts. In this case their dimensions must be such that they do not interfere with the erection of the permanent headframe. If this condition is observed, the transition from working with the temporary frame to working with the permanent one is accomplished in only a few days.

The advantages of wooden headframes are their relative cheapness, simplicity of construction, and the possibility of employing ordinary carpenters for their erection.

Their disadvantages include, first of all, their short service life. In the temperate climate of England these headframes last from 15 to 18 years; in western Europe, sheave frames made of soft wood serve for about 6 years, and those made of oak for about 10 years. As regards Russia, no precise data can yet be given, but apparently the duration of service of wooden headframes here is somewhat longer than in the west of the continent, which may be explained partly by the dryness of the air and partly by the length of the winter season, during which decay does not occur in parts exposed to the open air and is slowed in enclosed spaces.

Round timber, not deprived of the outer part of the wood, resists atmospheric influences better than hewn beams. It is advisable to coat the timber with tar, pitch, or, still better, carbolineum. Wooden frames deteriorate rapidly from the moist, warm mine air; therefore their service life in upcast shafts is shortened.

In the event of a mine fire, the timbering that has spread to the shaft means that a wooden headframe is subject to inevitable destruction; in addition, it may be destroyed by a fire on the surface arising independently of the mine, which, of course, entails a stoppage in production even if other hoisting appliances are available.

All this, taken together, compels us to regard wooden headframes as something temporary and short-lived.

Stone or brick structures with a system of beams on which the pulley bearings are placed, or so-called headframes over the shaft, owe their appearance to that period in the development of industrial culture when timber was already expensive, while iron was not yet cheap, with a relatively high productivity of the mines. The pulley beams rest on the walls of the building; sometimes they simply lie on top, and sometimes they are embedded in them. The defect of these arrangements consists in the fact that the shocks during hoisting are transmitted to the entire building, as a result of which it must be very solid. Over time the beams become loosened in their sockets, the walls develop cracks, and so forth. These pressures occur especially in those cases when, with an increase in output and deepening of the shafts, use is made of towers built earlier. The movement of the beams in such cases, despite all manner of props and fastenings, is sometimes so considerable that at times it is dangerous to be on the pulley platform during hoisting. To increase the stability of such structures they are provided with buttresses, so that the horizontal pressure is transmitted not only perpendicularly to the walls, but also to the buttresses. Diagonal anchor ties are also used, by means of which the pressure from the beams is transmitted to the side walls, thereby replacing buttresses, while the walls on which the beams are fastened experience chiefly vertical pressure. Nevertheless, despite all the indicated shortcomings, even now one may sometimes choose the tower system, especially when the working is expected to be carried on for a long time at shallow levels and brick or building stone is cheap.

Stone shaft-head buildings provide sufficient protection against fire; inside them there is a space in which the workers are protected from winter frosts and cold. In addition, by choosing suitable architectural forms for them, we are able to give the mine a solid, monumental appearance on the surface. These advantages sometimes lead to resorting to a mixed system, in which the pulleys are placed on iron headframes erected inside the stone building, entirely independently of it. Such a solution to the question is, perhaps, the most rational.

There are also transitional types, in which the tower walls are used partly to support the pulley beams, which are at the same time supported by vertical trusses.

Iron mine headframes are used for deep workings with high output. They are covered on the outside sometimes with wood, zinc or iron sheet, and more rarely are left completely open. The sheaves must necessarily be protected from rain and snow by a roof. At the present time it is still difficult to indicate the service life of iron mine headframes. Existing structures have been operating in perfectly good order for several decades already. These headframes provide complete protection against fire, and, if built with some reserve of strength, make it possible to increase the productivity of mines within broad limits without substantial alterations. It is also easy, by using bolted joints instead of some of the riveted ones, to adapt iron headframes for rearrangements, so that in some cases they may become competitors of wooden headframes even in small, multi-shaft workings. The necessary condition for using such headframes is the availability of free capital.

The dimensions and form of mine head structures depend chiefly on the dimensions of the shaft and on its depth. The height of the headframe building is determined, first, by the level of the trestle along which the mine cars are rolled to the concentrating mill, or, in general, by the level at which the cars are unloaded, and, secondly, by the height of the position of the sheaves above the level of the catches. This dimension must necessarily satisfy the condition that, in the event of too rapid a hoist, the engineer should have time to reverse and stop the cage, without allowing it to strike the sheaves. With the former small machines, with one steam cylinder and a flywheel, the sheaves were placed above the unloading platform at a height of 9–12 meters, which equaled \(1 \tfrac{1}{2}\) to 2 times the circumference of the drum rim when the diameter of the latter was \(1 \tfrac{1}{2}\) to 2 meters. With the large-diameter drums used at the present time, especially spiral drums, the diameter of which often reaches 10 meters, there is no possibility of adhering to this rule, for the height of the building would become very great. With four-deck cages, a trestle height of 8 meters and 10 meters, the headframe with a spiral drum would be too 60 meters high. With modern reversing coupled engines, which are comparatively easy to control, there is no need to apply the former rule: from the practical point of view it is quite sufficient to place the sheaves at a distance of at least half the circumference of the greatest cross-section of the drum above the upper part of the cage set on the catches.

The transverse dimensions of mine headframes depend 1) on the transverse dimensions and arrangement of the hoisting compartments and 2) on the

productivity of the mine. The first condition is self-evident; as for the second, it is clear that the headframe must correspond in its dimensions to the unloading platform, on which there must be enough room to place reserve cars and for the free movement of loaded and empty ones. The surface unloading station is to a certain extent a symmetrical reflection of the underground yard, but, because space here is not so constrained, its dimensions are somewhat larger.

In concluding the general remarks on headframes, I consider it necessary to add that the guides for the cages should preferably not be included, at the surface, as one whole with the headframe—that is, they should not be fastened directly to its vertical legs—in order to avoid harmful impacts during hoisting in the event that they are in a defective condition.

The classification of mine head structures is most easily made according to the material from which they are built, and accordingly we shall have two systems sharply differing from one another: 1) tower-type mine-head buildings, of stone or brick; 2) mine headframes, of iron or wood. The mixed headframe-tower system, about which we have already said a few words, does not deserve to be set apart into a special group. In accordance with the indicated subdivision, we shall first take up the description and calculation of the tower system, as the simpler one, and then the headframe system, in describing which we shall indicate its more detailed subdivision.

Tower System

In this system the sheave bearings are placed on two, three, or four beams connected with one another, the ends of which are embedded in the walls of the tower. From the manner in which the sheaves are fastened, this system is often called the beam system, although it cannot be regarded as exclusively characteristic of masonry mine-head buildings, since exactly the same sheave beams are also found in pyramidal headframes.

If the hoisting compartments are located side by side, then four beams are arranged alongside one another, and one bearing is placed on each of them; or, if there are only three beams side by side, two bearings are placed on the middle one. If, however, the hoisting compartments are arranged one behind the other, then it is possible to make do with only two beams placed side by side, with two bearings on each.

It is sometimes asserted that the sheave beams should be elastic, and that the advantages of such an arrangement are as follows: 1) reduction of shocks to the cage and hence safety of hoisting, 2) preservation of the hoisting rope because it does not experience impacts, 3) gradual transmission of the pressure of the load being raised to the sheave frame.

The first of these advantages can have only some significance when an empty cage is being lifted from the keps during descent; as for lifting a loaded cage from the bottom of the shaft, however great the ability of the beams to spring may be, it must disappear owing to the great distance between the cage and the sheaves, since the rope, being stretched, elongates and therefore itself acts as a spring. Flat ropes possess this property to a somewhat lesser degree than round ones, but even in them it is so great that the elasticity of the beams brings no substantial benefit. Since, when an empty cage is lifted from the keps, the tension is not great, the advantage in question is merely fictitious. The same may be said of the second; as for the third consideration, if we examine it more closely, we shall see that it too gives nothing positive but, on the contrary, is harmful. In fact, in order that the beams may spring, a gap is left between them (drawing 6, p. 33) in the space between \(a\) and \(b\); with

under the tension of the rope the upper beam \(A\) bends, since it is not able to withstand the whole load, and presses on beam \(B\); then both together press on the third, \(C\), and so on. As a result, the impact will be transmitted from one beam to another not gradually, but by jolts; the bearings with the pulleys will be in motion until all the beams are pressed tightly against one another. In this process the beams soon become damaged, the pulley axles are easily broken, and even the ropes suffer more than with ordinary arrangements. Another defect of elastic beams consists in the fact that, in order to attain a strength equal to that of ordinary beams, they must be made much thicker than ordinary ones. If the beams are not pressed tightly one against another and a gap remains between them, then the safe resistance of such a composite beam will be, obviously, less than the safe resistance of a solid beam of the same dimensions. Moreover, from the constant bending and straightening of the individual beams the strength of the material is reduced, so that, for proper safety, twice as much material must be expended on such elastic beams as on ordinary ones, which, of course, is associated with unnecessary and, as we saw above, entirely useless expenditure.

If we wished to arrange elastic beams that bend immediately, it would be necessary to make them relatively thin, and consequently little stable in comparison with the other parts of the structure, so that this method also is not applicable.

To weaken the jolts during hoisting, a spring was sometimes placed between the bearings and the beam; springs are also placed in the bearings themselves or under the ends of the beams in sockets. In practice these methods too have not justified themselves and at the present time they have been abandoned. With constant bending and straightening the springs soon weaken and settle, and usually not uniformly; most often there occurs the phenomenon that on one side the spring has already weakened, while on the other it is still acting. The pulleys then tilt, whereby breakage of the axles, slipping of the rope off the pulley, and even its rupture may easily occur. The indicated defects are aggravated by the circumstance that the pulleys are located high and are inspected once a day, at most twice, so that tilting may not be noticed in time. If all the foregoing is compared, one may come to the conclusion that elastic pulley frames generally present no substantial advantages.

Let us now proceed to a description of the construction of sheave frames and of the masonry towers themselves; in doing so, where necessary, we shall confine ourselves to several typical arrangements.

Wooden sheave frames for hoisting compartments situated side by side are shown in Fig. 7. They consist of four built-up beams on which the bearings are placed. The ends of these beams rest on timbers \(d\), built firmly into the walls, to which they are also connected by bolts. The timbers making up each beam are connected to one another by iron keys \(b\) and drawn together by bolts \(a\). All four beams are usually tied by crosspieces \(c\) into one whole, these crosspieces resisting the lateral displacement of the beams. Round iron keys should be preferred to wooden ones, because the latter loosen when they dry out. As soon as loosening of the iron keys is noticed, they are removed, the nuts of the bolts are tightened, and the keys are driven in again, thereby attaining the original strength. Wooden keys have the disadvantage that they dry out just like the beams, and are made too thin for the holes. New keys, however, are difficult to fit exactly to the old holes, whereas round iron ones always fit well. If it is desirable to use wooden keys, they should always be made of thoroughly dry wood.

Fig. 8 shows the same kind of iron sheave frame. Four beams \(a\) are joined by crosspieces \(b\) into one whole and rest with their ends on iron and cast-iron plates \(d\), built into the walls of the head tower, and in addition are connected with them by bolts \(f\). Fig. 9 likewise shows an iron sheave frame for two hoisting compartments situated side by side. Bearings rest on four beams \(a\); the beams \(a\) are connected to one another and to the two beams \(b\) by bolts. The beams \(b\), built into the walls of the head tower, to which it is connected by bolts \(f\), serve as supports for the beams \(a\). Fig. 10 shows a wooden sheave frame with struts for hoisting compartments situated side by side. The beams \(a\), on which the bearings rest, are supported by beams \(c\) with the aid of struts \(b\); the beams \(c\) rest on timbers \(d\), built into the wall, and are connected to the latter by bolts \(f\). The beams \(a\) are drawn tightly to the beams \(c\) by strong tension bolts, so as to eliminate deflection in the latter. The separate beams are connected by bolts \(i\) and keys \(k\) into one whole; all four beams are connected by bolts with crosspieces \(h\), which prevent lateral displacement. In these

In some cases a combination of the suspension and inclined systems is also used (Drawing 11). The beams \(a\), on which the bearings rest, are supported in many places by struts resting on the beams \(b\); the latter are connected with the beams \(a\) by tie bolts \(c\). The arrangement and connection of the parts here are the same as in the inclined system just described. An iron frame of the same type is shown in Fig. 12. Here the beams are latticed and are likewise embedded in the walls and connected by bolts \(f\). All four beams are connected to one another by crosswise riveted angle irons \(c\).

In all the above-described systems of beam pulley frames, the horizontal tension during hoisting is transmitted directly to the walls in which the beams are fixed, or else to walls perpendicular to them. These are often provided with buttresses on the side of the machine, as we have already mentioned. A somewhat different arrangement is shown in Drawing 13. This is an iron pulley frame with braces; the beams \(a\) rest on cast-iron plates \(b\) and \(c\), not connected by bolts with the wall \(h\). The cast-iron plate \(c\) is connected by means of diagonal ties \(d\) with the shoes \(g\), which embrace the corners of the walls. In this way the lateral force is transmitted in the direction of the arrows to the walls \(f\), while the walls \(h\) are subjected chiefly to vertical pressure. Buttresses here are not needed in most cases. The four beams \(a\) are connected to one another by crosswise riveted angle irons \(k\).

In all pulley frames with side-by-side compartments for the cages, four or three beams are arranged in one horizontal plane; the axes of the pulleys are perpendicular to the length of the beams and parallel to the axis of the drum \(c\). However, with conical drums this last condition is not strictly observed. In pulley frames with hoisting compartments arranged one behind the other, as shown in Drawing 14, one pair of beams lies so much higher than the other that the rope running from the upper pulley \(a\) to the drum \(b\) does not touch the lower pulley \(c\). The pulleys diverge from the center line of the drum \(b\); the beams \(dd\) and \(ff\) are arranged parallel to the corresponding pulleys. In all other respects, in terms of construction, this arrangement does not differ from the preceding one.

Calculation of Tower-Type Mine Head Structures

The calculation of these structures reduces to the calculation of 1) the beam frame and 2) the stone or brick building itself. Despite the generally very similar arrangement of the beams in all frames, they are sharply divided into two types: a) frames whose pulley beams are fixed immovably into the walls of the building; b) frames in which the front or rear ends of the pulley beams can slide freely in the direction of action of the horizontal forces. The first construction is statically indeterminate with respect to the action of horizontal forces; in the second, all stresses in the parts of the structure are easily determined by means of simple composition and resolution of forces. In fact, if the ends \(E\) and \(F\) are connected immovably with the walls, then we are not able, without considering deformation, to determine how the horizontal force is transmitted to both supports. If the right ends \(A_1\) can slide, while the left \(B_1\) are fixed (Fig. 68), then it is clear that the horizontal force is transmitted to \(B\), and from this point, by means of diagonal ties, can be transferred to the side walls. If the end \(B\) were suspended, then the entire force would be taken up by the support point \(A\). Since, however, even the strongest fixing with fastening by bolts, of which we spoke in describing the frames, soon becomes loosened because the magnitude of the external forces acting on them changes from minute to minute, we may confidently assert that the horizontal force in all cases, with the exception of structures with diagonal ties, is transmitted to the rear wall of the building, and that the parts \(BO\), \(BO_1\), and \(BO_2\) of the beams are subjected to compression, while the parts \(AO\), \(A_1O\), and \(A_2O\) are indifferent with respect to horizontal forces. We shall have to return again to the question under consideration when investigating pyramidal headframes, in which analogous phenomena occur.

A simple calculation proves that the friction force produced by the tension of the rope may be insufficient to hold the beam in place if the hoisting machine is placed too far from the pulleys; so that during hoisting it will move somewhat in the direction of the horizontal force, and as a result of this displa—

motion all the horizontal force is taken by the rear wall (adjacent to the machine). In reality, if \(f\) is the coefficient of friction of iron on cast iron, then, if the fastening of the beams is loose, the beams will lie motionless as long as the inequality

\[ f\,2s\cos^2\frac{\alpha}{2}-2s\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}>0; \]

holds; after reductions and transformations this expression becomes the following:

\[ f>\operatorname{tg}\frac{\alpha}{2}. \]

The coefficient of friction of iron on cast iron is at most \(0.16\); on the other hand, since \(\alpha\) quite often reaches \(60^\circ\), the condition indicated by us is most often not satisfied.

In our calculation we did not take into account the pressure resulting from the weight of the beams themselves together with the sheaves; in view of this, the condition under which the beams will remain motionless is somewhat more favorable than the theoretical one, but nevertheless in reality the phenomena of which we spoke above do occur.

The sheave beams are subjected to complex bending; the weight of the sheave and of the beams themselves acts in the vertical direction, while the resultant of the tensions of the ropes is inclined to this direction. The calculation would be rather complicated; in view, however, of the fact that it is carried out with a very large factor of safety, in practice the beams are calculated only for vertical bending; in addition, it is sometimes necessary to check the parts subjected to compression, that is, the parts from the sheaves to the rear wall, for longitudinal bending or for compression, depending on the length of these parts. For the calculation of the vertical forces, Max. \(P\) is used, and for the horizontal forces—Max. \(U\) according to formulas (25)—(26). If by \(G\) we denote the weight of one sheave, by \(E\) the weight of the beam, and the distance of the axis of the sheaves from the walls by \(l_1\) and \(l_2\), where

\[ l_1+l_2=l, \]

where \(l\) is the length of the beams, then we obtain

\[ \begin{aligned} p_1'&=\frac{1}{2}\left[E+\frac{l_1}{l}(G+P_1)\right],\\ p_2'&=\frac{1}{2}\left[E+\frac{l_1}{l}(G+P_2)\right], \end{aligned} \tag{29} \]

\[ \begin{aligned} p_1''&=\frac{1}{2}\left[E+\frac{l_2}{l}(G+P_1)\right],\\ p_2''&=\frac{1}{2}\left[E+\frac{l_2}{l}(G+P_2)\right]. \end{aligned} \tag{29} \]

In these expressions \(p_1'\) and \(p_2'\) denote the pressure on the right-hand supports, and \(p_1''\) and \(p_2''\)—on the left-hand ones (see Drawing 16).

In the case where there are three parallel beams, the middle one of them is subjected to a load equal to the weight of the full pulley and to the sum of the vertical forces \(P_1+P_2\), so that the pressure on the right and left walls is determined by the expression

\[ \begin{aligned} p_{1,2}'&=\frac{1}{2}\left[E+\frac{l_1}{l}(2G+P_1+P_2)\right],\\ p_{1,2}''&=\frac{1}{2}\left[E+\frac{l_2}{l}(2G+P_1+P_2)\right]. \end{aligned} \tag{29a} \]

Drawing 16.

Since we have introduced the condition that the beams sooner or later loosen in their sockets, for the calculation one should use the formulas derived for the case of a beam freely resting on supports. Thus, if

\[ \frac{G+P_1}{2E}<\frac{l_2-l_1}{2l_2}, \]

then, according to the theory of strength of materials, we have

\[ W_k=\left[\frac{1}{2}(G+P_1)\frac{l_1}{l}+\frac{\varepsilon}{2}\right]^2\frac{l}{2E}; \]

in the case, however, where

\[ \frac{G+P_1}{2E}>\frac{l_2-l_1}{2l_2}, \]

then

\[ W_k=\left(G+P_1+\frac{E}{2}\right)\frac{l_1l_2}{l}. \tag{30} \]

In view of the fact that \((G+P_1)\) or \((G+P_2)\) are very large in comparison with \(2E\), so that even in the case of the smallest \(l_1\) there will be place for

the second inequality, then in general one must use formula (30). For other beams the same formulas are applicable, of course, with the corresponding modifications. This remark also applies to the case where the hoisting compartments are arranged one behind the other.

The calculation of a beam structure with diagonal ties, of course, differs in no way from that indicated above; the ties themselves may be calculated on the basis of the following considerations.

Drawing 17.

If the tearing force appears on sheave \(A\), to which \(\operatorname{Max.} U = U_1\) corresponds, then the horizontal force on the other sheave \(B\), equal to \(U_2\), is in magnitude much smaller than the first. The resultant of these forces does not intersect at one point with the continuations of the directions of the ties. Let us denote the distance between the ends of the ties fixed to the cast-iron plate by \(e\); moreover, let the length of the building, measured in the direction of the force, be \(b\), the width \(a\), and the angle of mutual inclination of the ties \(\alpha\); if we transfer the point of application of the resultant \(U_1 + U_2\) to the middle of the distance \(\frac{e}{2}\), then the tension of each tie may be calculated on the basis of the formula

\[ s = \frac{U_1 + U_2}{2 \cos \frac{\alpha}{2}}, \tag{31} \]

which is obtained with the aid of the parallelogram of equal forces stretching the ties. In this case there remains a moment of forces not equal to zero

\[ M = \frac{e}{2}(U_1 - U_2), \]

tending to rotate the plate and skew the ties. Since \(U_1\) is approximately 5 times greater than \(U_2\), we are dealing with fairly considerable forces; usually one assumes

\[ U_1 = U_2 = \operatorname{Max.} U \]

and calculates the ties by the formula

\[ s = \frac{\operatorname{Max.} U}{\cos \frac{\alpha}{2}}, \tag{32} \]

whereby, of course, an enormous margin of strength is obtained. The value of the angle \(\alpha\), depending on the transverse dimensions of the building and the distance between the ends of the tie-rods \(e\), can be determined from the formula

\[ \operatorname{tg}\frac{\alpha}{2}=\frac{a-e}{2b}. \]

Let us now consider the calculation of several typical compound trusses under the conditions of their use as sheave beams.

An inclined beam (Fig. 18) may be calculated as a continuous beam supported at four points, that is, in the wall seats and at the points of contact with the braces. In handbooks there are also empirical and semi-empirical formulas for this case.

If at the points \(A\), \(B\), \(C\), and \(D\) we imagine hinged joints, then the stresses in the parts of the truss may be calculated in the following way. We resolve the resultant \(R\) into horizontal and vertical components; then the part \(AB\) may be calculated as a beam subjected to a concentrated load (Fig. 18)

\[ P=R\sin\varepsilon. \]

At the angles \(A\) and \(B\) there act vertical forces

\[ \frac{l_2}{l}R\sin\varepsilon,\quad \frac{l_1}{l}R\sin\varepsilon; \]

in view of this, the compressive stresses in the braces will be

\[ s_1=\frac{l_2}{l}R\frac{\sin\varepsilon}{\sin\alpha},\quad s_2=\frac{l_1}{l}R\frac{\sin\varepsilon}{\sin\beta}. \]

Fig. 18.

In the part of the upper chord \(AO\) there is a horizontal force

\[ s_1'=\frac{l_2}{l}R\frac{\sin\varepsilon}{\operatorname{tg}\alpha}, \]

and in the part \(OB\)—

\[ s_2'=R\cos\varepsilon-\frac{l_2}{l}R\frac{\sin\varepsilon}{\operatorname{tg}\beta}. \]

The right support takes up the horizontal pressure

\[ v_1=\frac{l_2}{l}R\frac{\sin \varepsilon}{\operatorname{tg}\alpha} -\frac{l_1}{l}R\frac{\sin \varepsilon}{\operatorname{tg}\beta} +R\cos \varepsilon \]

The lower panel is stretched by the greater of the forces \(s'_1\) or \(s'_2\); the right or left support, respectively, takes up the difference of these forces, depending on their relative magnitude,

\[ v_2=-\frac{l_2}{l}R\frac{\sin \varepsilon}{\operatorname{tg}\alpha} +\frac{l_1}{l}R\frac{\sin \varepsilon}{\operatorname{tg}\beta}. \]

The vertical pressure at point \(D\) is equal to the pressure at \(A\), and the pressure at \(C\) is equal to the pressure at point \(B\). By choosing the relative dimensions of the truss in the appropriate manner, it is possible, for a given inclination \(R\), to transfer the horizontal force to \(C\), if we assume that \(v_1=0\), then \(v_2=R\cos\varepsilon\); in this case the conditional equation must be satisfied

\[ \frac{l_2}{\operatorname{tg}\alpha} -\frac{l_1}{\operatorname{tg}\beta} +\frac{l_2+l_1}{\operatorname{tg}\varepsilon}=0; \]

usually one takes \(\alpha=\beta\), then

\[ \frac{\operatorname{tg}\varepsilon}{\operatorname{tg}\alpha} =\frac{l_1+l_2}{l_1-l_2}. \]

This condition should be taken into consideration for preliminary considerations in design.

Reverse tie truss. Let us retain the previous notation; let, moreover, \(\alpha_1\) and \(\alpha_2\) be the angles of inclination of the ties to the main beams; let the tensions in the ties be \(s_1\) and \(s_2\); from triangle \(abc\), in view of the fact that \(\angle bac=\alpha_1+\alpha_2\), we have

Drawing 19.

Drawing 20.

\[ R_0^2=s_1^2+s_2^2-2s_1s_2\cos(\alpha_1+\alpha_2); \]

Moreover, the stresses in the upper part of the panel are equal to one another. Their magnitude will be

\[ s' - s_1 \cos \alpha_1 = s'' - s_2 \cos \alpha_2; \]

whence we determine

\[ s_1 = \frac{R_0}{ \sqrt{ 1 + \frac{\cos^2 \alpha_1}{\cos^2 \alpha_2} - 2 \cos(\alpha_1 + \alpha_2)\frac{\cos \alpha_1}{\cos \alpha_2} }}, \]

\[ s_2 = \frac{R_0}{ \sqrt{ 1 + \frac{\cos^2 \alpha_2}{\cos^2 \alpha_1} - 2 \cos(\alpha_1 + \alpha_2)\frac{\cos \alpha_2}{\cos \alpha_1} }}, \tag{32} \]

where \(R_0\) is the force directed along the post; it may be directed along the inclined resultant force, and then \(R_0 \leq R\); more precisely, of course, one should compute the force directed along the post by resolving \(R\) into a horizontal component and one directed along the direction of the force \(R_0\). The advantage of the inclined post consists in the fact that part of the horizontal forces is taken up by the ties. In practice, structures with a vertical post are more often used; then, evidently,

\[ R_0 = P + G, \]

where \(G\) is the self-weight of the truss. If the panel \(DC\) is a continuous beam, then the above calculation is not exact: part of the pressure is taken up directly by the upper panel, so that the force falling to the share of the ties is smaller than according to formula (32). Practitioners often do not take deformation into account and simply distribute the load a priori, assuming, for example, that the upper panel takes one half of it, and the ties the other. In this case the upper panel is calculated as a beam on two supports, subjected to a concentrated load and to its own weight, and is checked for compression from the force \(s' = s''\), which is the result of transferring part of the load to the ties and the post, plus the horizontal pressure \(U\). Of course, in reality the division of the load among the parts of the truss will be different, but in view of the enormous margin of strength with which the calculation of mine head structures is made, there are no harmful consequences from such a simplification. A more exact calculation of a continuous panel as a beam on three supports is rarely applied.

In lattice trusses the pulleys are placed on the upper panel. In most cases trusses of this kind are used with diagonal crossing members and, in view of

therefore statically indeterminate. The calculation may be carried out by a simplified and approximate method, by dividing the given structure into simpler trusses. For example, the truss shown in Fig. 12 may be divided into the two parts shown below in Figs. 21 and 22; the part \(ABCD\),

Fig. 21.

Fig. 22.

in which the upper chord is connected with the lower by means of sheet iron, represents a rigid body upon which external forces act, namely—one half of the vertical pressure from the hoist \(\left(\frac{1}{2}P\right)\), one half of the weight of the sheave \(\left(\frac{1}{2}G\right)\), and the weight of the part \(ABCD\); thus, if the total active force is \(\frac{1}{2}(G+P+g)\), then, at the adjacent joints, with the method of calculation indicated by us, it is necessary to take the load

\[ \frac{1}{8}(G+P+g); \]

the calculation itself may be carried out by one of the well-known methods of Cremona, Ritter, Zimmermann, etc. The members are calculated in accordance with the stresses obtained, and the upper and lower chords—according to the sum of both calculations. As regards the proper weight of the truss itself, we are justified in dividing it into parts and considering it as applied at the separate joints.

In tower structures, designs of the kind shown in Fig. 9 are used rather often. In this case the pressure \(R\) is transmitted partly to the sheave beams, and partly to the lower cross beams. An exact calculation of the structure may be based on the fact that the deflection of both systems of beams must be the same in those parts where they are in contact. If the deflection arrows at these points are determined from the general equations and then equated to one another respectively, we obtain equations giving the mutual relation of the dimensions of the beams being calculated. The indicated method of calculation is too complicated and therefore is not used in practice. Instead, it is preferred simply to assume that load which

is taken up by the upper and lower beams; for example, the total load is divided into two parts and, in accordance with this, the transverse dimensions of the beams are calculated. We indicated an analogous calculation above for inverted trussed frames. In the construction we are considering, part of the horizontal forces \(U\) is transmitted to the side walls of the tower, which speaks in favor of this construction; moreover, the transverse beams play the role of anchor ties. It is, however, rather difficult to determine precisely what part of the horizontal forces is transmitted to the side walls.

The calculation of the walls of the tower and of its foundations reduces to determining their thickness, sufficient to resist the vertical compressive forces and the horizontal overturning forces. The pressure is determined from the maximum vertical force, occurring when the lower rope breaks and determined by formulas (18). To the pressure transmitted to the wall by means of cast-iron plates there is added the pressure of the weight of the part of the walls located above the sheave beams, whose thickness may be comparatively small. The sum of these pressures, distributed over a unit of area, determines for us the thickness of the walls at the points where the sheave beams are fastened; and although in other parts of the front and rear walls it would be possible to take a smaller thickness, nevertheless, taking into account the existence of the horizontal overturning force and the shocks from the periodic hoisting and the variable magnitude of the acting forces in general, they are most often given the same dimensions. Toward the bottom the thickness is increased by offsets, as in all buildings. The base of the building and the foundations must be of proper dimensions. In view of the shocks mentioned earlier, and also in view of the weakening of the soil by the mine, it is not superfluous to reduce the coefficient of soil resistance by half against the usual one. The side walls, if the transverse beams are not embedded in them, may have normal dimensions determined by the height of the building, etc.

Let \(H\) be the height of the fastening of the sheave beams, \(H_1\) the height to the center of wind pressure; then the greatest moment tending to overturn the building (Fig. 22) is

\[ M_a=\operatorname{Max} U H+v H_1, \tag{33} \]

where \(\operatorname{Max}. U\) is determined according to formulas (18), and \(v\) is the horizontal component of the wind pressure. The axis of rotation is at \(O\). If by \(Q_1, Q_2, Q_3\) we denote the weight of the parts of the wall of equal

of thicknesses, by \(L_1,\ L_2,\ L_3\)—the horizontal distances of their centers of gravity from the axis of rotation, by \(L\)—the distance of the centers of the plates on which the ends of the beams rest from \(O\), that is, from the outer edge of the wall facing the machine, and if \(P\) is the vertical pressure that occurs at Max. \(U\), and \(p\) is that part of the pressure which is transmitted to the wall under consideration, then the moment resisting overturning will be

\[ M_r=\sum Q_i L_i+pL . \tag{34} \]

It is clear that the building is stable if

\[ M_r > M_a . \tag{35} \]

If this condition is not satisfied, then, since increasing the thickness of the walls involves large expense, an effort is made to bind the entire building with anchor ties so that it constitutes a single whole; this is best achieved by means of diagonal ties of the lattice construction previously considered by us, transmitting the horizontal pressure to the side walls. The front wall is not taken into account in the calculation, since the horizontal pressure is transmitted to it only to an insignificant extent, as the sockets are most often loosened, and its structural connection with the other walls is rather weak.

Fig. 23.

For greater stability the rear wall is often provided with buttresses, which are arranged in the direction of the pulley beams, and sometimes also at the corners as continuations of the transverse walls. Buttresses increase, on the one hand, the weight of the wall, but their chief benefit consists in the fact that, thanks to them, the axis of rotation is shifted backward, and thus the lever arm of the resisting moment is increased by \(a\)—the width of the buttress at the bottom. Of course, the thickness of the walls at the points where the plates are fixed in them must be checked for resistance to the horizontal force Max. \(U\). It is also useful to check the ratio of the overturning moment to the resisting moment for the upper parts of the wall.

Headframe System.

Mine-head structures of this type are usually subdivided into two-, three-, and four-legged headframes. If this subdivision is retained, then a group of six-legged headframes must also be added to it.

The simplest construction of a two-legged headframe is shown in Fig. 24. In this system both the inclined legs \(a\) and the vertical legs \(b\) rest on the horizontal sills \(c\). The crossbeams \(d\) serve for a more reliable connection of these and the other legs with one another and, moreover, their parts projecting forward support the platform beneath the sheaves. The vertical legs are connected with one another by crosspieces \(e\). The sheaves are fastened by means of beams \(f\) and crosspieces \(e\) and \(g\) to the vertical legs.

Fig. 25 shows a wooden headframe, also of the English type. The hoisting compartments are placed side by side; both legs \(a\) rest in cast-iron shoes \(b\), bolted to the stone foundation; the legs \(a\) are brought closer together at the top, are tied by the crosspiece \(h\), and are set approximately in the direction of the resultant tensile force. The entire weight of the headframe is transmitted chiefly to two posts \(g\), vertical in profile and brought together at the top; they are connected with each other by the transverse member \(l\) and the brace \(m\), and with the leg by inclined crossbeams \(i\) and cast-iron shoes \(r\). The bearings are placed on four beams \(o\), situated in the plane of the legs \(a\); these beams are supported by struts \(n\).

Fig. 26 shows a construction very similar to that described earlier (see Fig. 24). Here the inclined and vertical legs rest, exactly as in Fig. 24, on sills. The inclined legs, in addition, are further supported by braces \(a\), a crossbeam \(b\), and posts \(c\).

In the construction of Fig. 27, the inclined rear legs rest on a high stone foundation, while the front leg \(b\) rests on concrete foundations. The guides for the cages \(c\) stand on horizontal sills \(e\), under the ends of which similar sills are again placed, resting on concrete bases. At the top, the guides adjoin the legs \(b\), which here are brought closer together and are connected with them by means of crossbeams \(h\) and transverse members \(k\). The upper ends of the front and rear ...

the legs rest against the longitudinal beams $l$, which are also supported by props $m$ with the transverse beam $n$ and by struts $r$, supporting them from the other end. On these, as well as on the middle beams $i$, lie the sheave bearings. Thus, in the structure we are describing, the guide frame assumes a certain, though insignificant, role in supporting the sheave beams. The sheaves are arranged so that the resultant of the forces passing through them intersects the points of connection of the front legs with the rear ones approximately at $A$. The inclined legs are tied together by means of a series of cross-braces and transverse members, and are further tightened with anchor bolts. The front legs are connected with the rear ones by means of diagonal braces $s$ and also by anchor bolts.

If, instead of two inclined legs, three are used, with the middle one supporting the inner bearings or the beams on which they rest, then a three-legged headframe is obtained, which in general does not differ from the structures described above.

Let us pass on to the description of iron headframes of the same type. In Fig. 28 a two-legged iron headframe with adjacent compartments is shown. The lower part of the legs $a$ is made in the form of bearing plates and rests in cast-iron shoes $b$, connected to the stone foundation by bolts. Both legs $a$ are connected to each other by a series of diagonal and transverse rods. On the side of the hoisting engine they are held by three strong ties $c$, on which chiefly the entire weight of the whole structure rests. These ties on the machine side are projected vertically, are made of strip iron of suitable thickness, and are connected to one another. For the stability of the structure, on the front side there are also two strong ties $k$, placed along the sides and connected to the foundation. The legs $a$ are somewhat brought closer together at the top; the outer bearings stand directly on the upper ends of the legs, while the two inner ones are supported by a beam connected by special fastenings to the legs.

Fig. 29 depicts an English headframe, all parts of which are lattice trusses, or are riveted from sheet iron. The inclined legs $a$ rest against a high-raised foundation, which at the same time serves as the foundation for the hoisting engine. The front legs $b$, projecting from the side, are connected vertically with the rear (inclined) legs by means of longitudinal beams of box section $c$, riveted from sheet iron, on which the sheave bearings rest. These same beams are supported in

at their ends by braces \(e\) and \(d\). The middle bearings are fastened to the column \(f\), which rests on the transverse beam \(g\).

The truss of the inclined legs consists of two triangular trusses; each of them plays the role of the legs of wooden headframes. In Fig. 29a the front and rear legs are connected to each other by braces \(h\).

The headframe shown in Fig. 30 differs from that described above only in structural details; the basic idea is one and the same.

Fig. 31 represents a headframe of the same English type, but of a somewhat different construction, which has some similarity to the wooden headframe shown in Fig. 26. The legs of the inclined truss \(a\) and of the vertical truss \(b\) are connected by means of an angle \(c\); triangular brackets \(d\) and \(e\) are riveted to them, to which the side bearings are fastened. The middle bearings are supported by a column \(f\), which rests on the transverse beam \(g\) and is supported by struts \(h\). On the side of the inclined truss the column \(f\) is stayed by brackets \(k\) and is fastened to the trusses by means of the struts \(h\) and the transverse beam \(g\). The inclined truss is supported by vertical posts \(l\) and \(m\) and is connected with the legs \(b\) by means of braces \(r\) and \(n\). The guides \(s\) are connected with the headframe, but play no role in supporting the pulleys and in resisting the active forces.

Fig. 33 represents a construction intended for hoisting a large quantity of mineral from a great depth. Two hoisting machines can operate simultaneously. There are four pulleys, which, alternating, are arranged on two levels. The pulley bearings are installed on brackets \(a\); the brackets of the outer bearings are riveted to the legs of the vertical truss, and the brackets of the middle ones to three columns. Thus each column supports two pulleys. The columns are installed on a horizontal cross-truss, which, in turn, is supported by struts bearing against the legs of the vertical truss. The inclined supporting truss has the same arrangement, so that the columns have second points of support in their upper parts. Both the inclined legs and the vertical ones converge upward and bear against solid stone foundations. All parts of the structure are lattice trusses and consist either of iron angles arranged along the verticals of a quadrilateral and connected by means of diagonal and transverse rods, or of two trough-shaped members connected in the same way. In general, box trusses are obtained.

The guides are connected to the headframe, but relatively weakly, so that in the event they are broken by a cage that has for some reason become stuck in them, the headframe itself cannot in any way suffer damage.

In Fig. 34 is shown the arrangement of a four-legged headframe, in which two legs are ordinarily placed in the direction of the resultant and carry almost the entire load being raised. The other two legs \(f\), to which the entire structure with the sheaves is transmitted chiefly, are so placed that they intersect at point \(b\) of the parallelogram of forces. In accordance with this, legs \(a\) are made more substantial than legs \(f\). Each pair of legs is connected at the top by a transverse beam \(g\) with four cast-iron shoes \(h\), on which the bearing blocks are placed. All four legs rest with their lower ends in cast-iron shoes \(i\), fastened by bolts to the stone foundation; at the top the legs are brought together and connected among themselves by cross-pieces and braces into one whole. This arrangement may be regarded as a transitional form from a two-legged to a four-legged headframe.

In Fig. 35 is depicted the arrangement of an iron four-legged headframe with cage compartments arranged side by side. Four strong corner posts \(a\) are connected with one another and form, as it were, a truncated pyramid; the posts—legs—rest with their lower ends on cast-iron shoes \(b\), fastened by bolts to the stone foundation; at the top these posts are connected by a beam quadrilateral \(e\), on which the sheave beams \(d\) are placed. The corner posts \(a\) are, moreover, connected with one another by cross-braces \(f\) and ties \(e\).

When the compartments are arranged one behind the other, the truncated pyramid receives somewhat different proportions. In this case the corner posts are connected by two beam quadrilaterals at two levels; on each of these quadrilaterals there are two sheave beams; the distance between these beam quadrilaterals is somewhat greater than the diameter of the sheave.

In Fig. 36 is shown a wooden pyramidal headframe in which there is no beam quadrilateral; beams \(c\) are fitted on the front posts \(a\) and the rear posts \(b\), and on these the sheave beams \(d\) are arranged. The geometrical invariability of the whole truss is achieved by means of longitudinal and transverse horizontal ties \(g, h, g_1, h_1\) and diagonal struts \(k, l, m, n\). Since tensile forces may occur in the ties, iron anchor ties are passed along them. The legs bear against stone foundations.

In Fig. 37 is shown a wooden pyramidal headframe provided with supporting legs \(a\). In the construction under consideration, the posts \(b\)

connected at the top by transverse beams \(c\) and longitudinal ties \(e\), so that here there is a beam quadrilateral. The ties \(e, e_1, e_2\) are double; they embrace the posts from two sides and are fastened to the posts with bolts. The diagonal struts, likewise bolted to the posts, bear against them. The supporting legs \(a\) are connected with the legs \(b\) by means of crossbars \(f\). All the legs stand on stone foundations.

A similar six-legged headframe, but of a somewhat different construction, is shown in Fig. 38. Here there is a beam quadrilateral, namely: transverse beams \(c\) are laid on the posts \(b\), while the longitudinal \(e\) are fastened to the latter with a tenon; in addition, the posts are also tightened by longitudinal anchors. On the transverse beams lie the sheave beams. Both the transverse and the longitudinal beams are supported in the middle by struts \(d\) and \(g\), which bear against the lower horizontal ties. In addition, there are also diagonal struts. Under the unloading platform, for which the ties \(e_2\) and \(i_2\) serve as supporting beams, there is also a system of braces and struts bearing against the posts \(b\) and the ties \(e_3\) and \(i_3\), which in turn are supported by braces. The inclined legs \(a\) are connected with the legs \(b\) by means of an exactly similar truss. All the legs bear on longitudinal sills \(h\), under which there are transverse sills \(k\).

Fig. 39 shows a construction of the same type, only the legs \(b\) are placed here vertically. The beam quadrilateral is composed somewhat differently, namely—the longitudinal beams are set on the posts, to which the transverse ones are bolted, and on these latter the sheave beams already rest. The supporting legs diverge downward.

Sometimes the supporting legs are moved somewhat apart in order to allow the sheaves to be placed so that the resultant force passes as close as possible to the supporting legs. A construction of this kind is shown in Fig. 40. There are three supporting legs; their lower ends bear on cast-iron shoes \(e\), bolted to the stone foundation. The outer legs \(ff\) are brought together at the top, while the third leg is projected in front vertically; all three legs are joined by crosspieces \(h\) and cross-braces into one whole. The entire weight of the headframe is transmitted chiefly to the vertical legs; the middle legs are connected by strong cast-iron shoes \(l\) with the inclined legs and with the horizontal beams \(m\), on which the bearings already rest.

The iron headframe shown in Fig. 41 also belongs to the type under consideration. The supporting legs widen downward; this

the widening begins at approximately \( \frac{1}{3} \) of the full height of the headframe. Both the vertical and the inclined legs are connected to one another by a system of transverse and diagonal bracing, so that each pair of adjacent legs, taken separately, together with the bracing forms a plane lattice truss.

In Fig. 42 a headframe of a somewhat different type is shown, for hoisting large quantities of mineral from great depths. The supporting legs enter the space between the vertical legs and directly support the sheave beams at the points where the bearings are placed upon them. The inclined legs are lattice trusses, narrowed along the edges and connected by means of transverse and diagonal bracing into one whole. The vertical legs are similar trusses and are connected with one another by means of the same kind of crosspieces and cross-braces. The guides (not shown in the drawing) are independent of the vertical legs, although attached to them, but in such a way that their breakage cannot affect the legs themselves. The panels of the legs consist of trough-shaped iron; the transverse and diagonal bars are of angle iron; the crosspieces are of precisely the same construction. The abutments are of stone; the lower parts of the legs are connected to them by means of cast-iron plates and similar shoes, and are fastened with bolts. The arched truss connecting the inclined legs in their lower parts is used in order to leave free access to the shaft from the side of the hoisting machine. Hoisting is carried out by means of butterflies.

With this I conclude the examples of designs of mine headframes; of course, the typical schemes presented by no means exhaust all their varieties, since in this respect the field of activity for designers is not limited. Let us consider a few more distinctive types.

In Fig. 43 a construction is shown in general view whose purpose is to eliminate lateral overturning forces when long drums are used. Behind the hoisting machine, on a strong stone foundation, are arranged the bearings of the sheaves, over which the ropes are passed from below. The lateral forces are taken up by these sheaves and foundations. The rope to the guide sheaves and from them is led over rollers. The legs of the mine headframe are projected vertically onto a plane perpendicular to the axis of the entire structure. An interesting design was proposed by the firm Siemens & Halske for electric hoisting (Fig. 44). The electric motor with the drums and all the apparatus is placed on solid beams fastened toיזיז

to the rear legs of the headframe. The entire mechanism is housed in a light building of iron and glass. A hoist of the Koepe type. The headframe is of the so-called two-legged type; the pair of legs to which the hoisting device is attached is projected vertically, while the second pair of legs diverges downward. The guides are built independently of the legs.

Mixed Headframe-and-Tower System

If the sheave beams are fastened in massive stone walls of the mine head tower and, in addition, are supported by posts or legs, then a mixed tower-and-headframe system is obtained, which was mentioned earlier. This system presents a large number of varied forms of mine-head buildings; the most common arrangements are those in which the sheave beams are supported by four legs connected to one another. At the bottom these legs are fastened to buttresses or walls by means of special beams, or else to cast-iron towers. With such an arrangement, a pyramidal headframe is joined to a tower. Since in this case the horizontal forces are taken up only partly by the headframe, it is often necessary here to provide buttresses to take up part of the horizontal forces. These buttresses are of much smaller thickness than in a simple tower system. An important drawback of this arrangement must be considered to be the fact that, with it, the space inside the mine head tower is cramped.

Quite often structures are encountered in which the lower part consists of a tower or strong pillars, connected by arches, serving as the base for a pyramidal headframe. This construction is used more often in cases where the height of the building is greater than the length of the available timber, and the use of iron is uneconomical.

Theory of Mine Headframes.

Mine headframes, as we have been able to see from the data given above concerning their construction, and even without this simply a priori, proceeding from their purpose for lifting loads in one definite, most often vertical, direction, must constitute a spatial, geometrically unchanging truss, subjected to the active forces \(R_1\) and \(R_2\), the wind pressure, and carrying its own weight. Of course, their principal task is stability with respect to the forces \(R_1\) and \(R_2\), by which above all the form and construction of the headframes are determined; the other forces are secondary and have relatively little importance in the choice and development of the type and construction of the structure.

Headframes are distinguished by a certain symmetry, and consequently headframes with lifting compartments arranged side by side may be divided by a plane drawn midway between these compartments and parallel to their axes into two entirely symmetrical parts, each of which resists its own force \(R\), applied on the axis of the sheave, independently or with the assistance of the neighboring part. In the case of compartments arranged one behind another, the plane of symmetry passes through the axes of these compartments; in this case the headframe is somewhat more compact: both forces \(R_1\) and \(R_2\) lie almost in one plane, sometimes even strictly in one plane, for example in the Koepe hoisting system.

If we begin to consider one such half of a headframe with parallel compartments, or an entire headframe with compartments one behind another, simplifying them to the ultimate limit, we shall see that at the basis of all headframes without exception lie simple geometrical figures, namely—a triangle and a trapezoid. These figures, obtained by the intersection of the headframe by planes parallel to the plane of symmetry, are of course ideal, since in order to obtain them it is necessary to ignore all constructional details.

Thus, the essential part of a headframe is a prism cut off along the sides, or a truncated pyramid (Fig. 45). There are also headframes representing a combination of a prism with a pyramid.

A schematic section of the headframe in the direction indicated above gives us those basic figures of plane trusses which, when combined two at a time or three at a time, give us the final spatial truss—the headframe. For headframes with compartments arranged side by side, these schemes are shown in Fig. 46, \(d, e\).

Fig. 45.

These truss schemes will have to be considered in greater detail before passing on to the headframes themselves. Thus it is necessary to examine the conditions of action of forces applied at the vertices, or simultaneously at a vertex and somewhat lower, on one of the sides of a triangle; in trapezia these forces are applied at one upper side, or at the upper side and to the tie located somewhat lower. I consider it necessary to make the reservation that everything now said applies strictly to the hoisting apparatus itself, and by no means to the guides of the cages at the surface, which are continuations of the hoisting compartments of the shaft, although in the structural sense they have to be connected with the headframe.

Fig. 46.

Prismatic Headframes.

The basic figure for three-legged headframes must be considered to be the triangle, which, as is known, represents a form that is statically determinate and geometrically unchangeable if all three joints are hinged. It must, however, be observed that in the construction of headframes hinged connections are avoided. In our derivations, however, we shall in most cases regard the connections as hinged.

Depending on the direction of the force \(T\) and on the type of triangular form, different stresses may predominate in its parts. If

let us denote by \(\varepsilon\) the angle of inclination to the vertical line of the force \(T\), as is customary in the chapter on forces; \(T\) and \(\varepsilon\) have somewhat different meanings here, than the angle \(\frac{\alpha}{2}\) and the force \(R\). Let us denote by \(\eta\) and \(\xi\) the angles made by the directions of the sides \(AB\) and \(BC\) with the vertical, and the stress in \(AB\) by \(\sigma_1\), and in \(BC\) by \(\sigma_2\). Then the equations of equilibrium of the forces at node \(B\) will be (see Fig. 47)

Fig. 47.

\[ T \cos \varepsilon + \sigma_1 \cos \eta + \sigma_2 \cos \xi = 0, \]

\[ T \sin \varepsilon + \sigma_1 \sin \eta + \sigma_2 \sin \xi = 0; \]

\[ \sigma_1 = -T \frac{\sin \xi \cos \varepsilon - \cos \xi \sin \varepsilon} {\sin \xi \cos \eta - \cos \xi \sin \eta} = -T \frac{\sin(\xi-\varepsilon)} {\sin(\xi-\eta)}, \]

\[ \sigma_2 = +T \frac{\sin \eta \cos \varepsilon - \cos \eta \sin \varepsilon} {\sin \xi \cos \eta - \cos \xi \sin \eta} = +T \frac{\sin(\eta-\varepsilon)} {\sin(\xi-\eta)}. \tag{36} \]

These are general expressions; in them, angles to the right of the vertical should be taken as positive, and to the left as negative. In the particular case when the resultant force \(T\) passes between the front and rear legs, but on the right side of the vertical, we must have

\[ \xi > 0,\qquad \eta < 0,\qquad \varepsilon > 0, \]

and therefore formulas (36) take the form

\[ \sigma_1 = -T \frac{\sin(\xi-\varepsilon)} {\sin(\eta+\xi)}, \]

\[ \sigma_2 = -T \frac{\sin(\eta+\varepsilon)} {\sin(\eta+\xi)}. \tag{37} \]

Both stresses are compressive, and in magnitude \(\sigma_2 > \sigma_1\); if \(\xi=\varepsilon\), then \(\sigma_1=0\), which, however, cannot be achieved in practice.

* In the drawings, instead of \(T\), the Greek capital rho has been used.

If the resultant \(T\) passes on the outside of the rear legs, so that \(\eta < 0\), \(\xi < 0\), \(\varepsilon > 0\), then

\[ \begin{aligned} \sigma_1 &= +T \frac{\sin(\xi+\varepsilon)}{\sin(\eta-\xi)},\\[4pt] \sigma_2 &= -T \frac{\sin(\eta+\varepsilon)}{\sin(\eta-\xi)}; \end{aligned} \tag{38} \]

the stress \(\sigma_2\) is compressive, and \(\sigma_1\) is tensile (see Fig. 48c).

With the aid of a construction it is very easy to obtain both stresses; it is only necessary to resolve the force opposite to \(T\) along the directions of the legs (Fig. 48a).

Fig. 48.

From the triangle \(BEF\) (Fig. 48a) we have the equalities

\[ \frac{\sigma_1}{\sin(\xi-\varepsilon)} = \frac{\sigma_2}{\sin(\eta+\varepsilon)} = \frac{-T}{\sin(180-\eta-\xi)}, \]

which are identical with formula (37); from the triangle \(B_2E_2F_2\) (Fig. 48c)—

\[ \frac{-\sigma_1}{\sin(\xi+\varepsilon)} = \frac{\sigma_2}{\sin(\eta+\varepsilon)} = \frac{-T}{\sin(\eta-\xi)}; \]

this equality is identical with formula (38).

Thus, both legs will be compressed if the resultant lies between them; if, however, it passes beyond the headframe, then the leg to which the resultant is nearer is subjected to compression, while the other leg is in tension. Since, from the constructive point of view, it is not always possible to build long trusses working under tension of variable magnitude, passing into compression during the inactivity of the structure, or at least this is not always readily attainable, builders usually try to arrange

...the legs so that they are only compressed. This observation applies first of all to wooden headframes. Let us consider the formulas we have derived. Let the angles \(\xi\) and \(\varepsilon\) be constant; then it is clear that with an increase of \(\eta\) the stress \(\sigma_1\) in both of the indicated cases decreases: we may write that, for \(\varepsilon = \mathrm{Const.}\) and \(\xi = \mathrm{Const.}\),

\[ \operatorname{Min}\sigma_1 \text{ will occur when } \eta + \xi = 90^\circ, \]

or

\[ \text{”}\qquad \text{”}\qquad \text{”}\quad \eta - \xi = 90^\circ. \]

What has been said may be represented graphically in the following manner (Fig. 49):

Fig. 49.

In the graphical investigations presented, we decompose directly the force \(T\) along the directions of the legs, thus obtaining not stresses, but forces directed along these legs and producing reactions equal to them but opposite in direction.

Fig. 49 (I) gives us the possibility of drawing the following conclusion: if it is desired to relieve the front leg at the expense of the rear one, then the angle \(ABC\) should be increased by deflecting the front leg. The limit of this deflection is \(90^\circ\), for then \(\min \sigma_1\) is obtained. Further deflection leads to an increase of \(\sigma_2\), but with a simultaneous increase also of \(\sigma_1\). Equal stress in both legs will occur in the case when the angle formed by the legs is divided by the force \(T\) in half, that is, when

\[ \varepsilon = \frac{\xi - \eta}{2}. \]

In the second case, when the resultant \(P\) passes on the outside, the most unfavorable arrangement will occur when the angle \(ABC\) is small; by moving \(AB\) away, we achieve a decrease of both \(\sigma_1\) and \(\sigma_2\) until

\[ \xi - \eta = 90^\circ; \]

further deviation is associated with a decrease of \(\sigma_2\), but at the same time \(\sigma_1\) also increases.

If we were to agree to consider

\[ \eta=\mathrm{Const.},\quad \varepsilon=\mathrm{Const.}, \]

then, taking \(\xi\) as variable, on the basis of the same considerations we would find that, by changing the inclination \(BC\), we change the stress in both legs. Strictly speaking, this problem is no different from the preceding one. In general, the conclusions we have drawn show that it is possible to adapt, to a certain degree, the design of a headframe to the material at hand.

Since the forces are transmitted to the supports in the direction of the legs, the lower tie \(AC\) may be eliminated; in this case the ends of the legs must be secured immovably and unchangeably in two directions (if, as we do, one considers a plane truss). This is achieved by using cast-iron shoes or plates fastened to strong stone foundations by means of anchors. In small wooden headframes, in order to avoid the use of shoes and foundations, lower ties are often employed in the form of wooden sleepers into which the ends of the legs are let in (Figs. 26 and 38, described earlier).

These sleepers, receiving the pressure of the legs \(\sigma_1\) and \(\sigma_2\), are themselves subjected to tensions \(\sigma'_3\) and \(\sigma''_3\), directed along them and tending to split off the parts lying on the outer side of the legs; moreover,

\[ \begin{aligned} \sigma'_3&=\sigma_1\sin\eta =\mathrm{T}\frac{\sin(\xi-\varepsilon)}{\sin(\eta+\xi)}\sin\eta,\\ \sigma''_3&=\sigma_2\sin\xi =\mathrm{T}\frac{\sin(\eta+\varepsilon)}{\sin(\eta+\xi)}\sin\xi. \end{aligned} \tag{39} \]

Fig. 50.

The vertical pressures at the points \(A\) and \(C\) (Fig. 50) will be

\[ \begin{aligned} p_1&=\sigma_1\cos\eta =\mathrm{T}\frac{\sin(\xi-\varepsilon)}{\sin(\eta+\xi)}\cos\eta,\\ p_1&=\sigma_2\cos\xi =\mathrm{T}\frac{\sin(\eta+\varepsilon)}{\sin(\eta+\xi)}\cos\xi. \end{aligned} \tag{40} \]

The force \(T\) imparts to the whole structure a tendency toward horizontal displacement,

\[ \mathrm{T}\cos\varepsilon=\sigma''_3-\sigma'_3=\Omega . \]

and the vertical pressure

\[ T \sin \varepsilon = p_1 + p_2 = \Pi . \]

The horizontal force can move the sill if the sill is laid directly on the ground without any fastenings, in the case when it exceeds the frictional force. The expression for the analogous case was derived by us on p. 36. If this condition of stability is not satisfied, then the sill must be secured artificially.

If we were to lay the sill not horizontally, but at some angle \(\delta\) to the horizon, then the stresses in this sill are determined somewhat differently, namely:

\[ \begin{aligned} \sigma_3' &= \sigma_1 \sin(\eta + \delta),\\ \sigma_3'' &= \sigma_2 \sin(\xi - \delta),\\ p_1 &= \sigma_1 \cos(\eta + \delta),\\ p_2 &= \sigma_2 \cos(\xi - \delta). \end{aligned} \tag{41} \]

The total shearing force along the plane of the sill that interests us, and the force normal to it, are determined by the expressions

\[ \begin{aligned} s &= T \sin(\delta - \varepsilon),\\ p &= T \cos(\delta - \varepsilon); \end{aligned} \tag{42} \]

when \(\delta = \varepsilon\), \(s\) becomes equal to zero; however, such a value of \(\delta\) is practically inconceivable. In reality, the stability of the headframe can be increased somewhat by giving the ground a slight slope away from the hoisting machine, by cutting away a certain part of the surface, or by laying a solid material so that it will not shift. In the case under consideration, stability will be given by the inequality

\[ f\left[T\cos(\delta-\varepsilon)+\frac{G}{2}\cos\delta\right] > T\sin(\delta-\varepsilon)-\frac{G}{2}\sin\delta, \]

where \(G\) is the whole headframe; moreover, it is assumed that half of it is included in the resultant \(T\), and the other half is taken up directly by the supports.

Let us now consider the case of a prismatic headframe for compartments arranged one behind another. The principal figure of the frame of such

the headframe will be a triangle with one force applied at the apex and the other—directly to the rear leg (Fig. 51), since the sheaves are arranged one at the apex and the other lower down on the leg. Let the resultant \(T_1\) of the upper sheave be inclined to the vertical at an angle \(\varepsilon_1\), and the resultant of the lower one \(T_2\)—at an angle \(\varepsilon_2\). The heights of the sheaves above the horizon are \(H\) and \(H_1\); the remaining notation is as before. We shall write the equations of moments with respect to the supports \(A\) and \(C\), without regard to the direction of the forces, and in accordance with section \(ab\)

Fig. 51.

\[ \sigma_1 L\cos\eta + T_1 L_1' + T_2 L_1'' = 0, \]

\[ \sigma_2 L\cos\xi + T_1 L_1 + T_2 L_2 = 0. \]

In these equations, \(L\) denotes the distance between the supports; \(L_1, L_2, L_1', L_2'\)—the perpendiculars dropped from the joints in the directions of the forces \(T_1, T_2, \sigma_1, \sigma_2\). According to the drawing they are equal to

\[ L_1 = AD\cos\varepsilon_1,\quad L_2 = AE\cos\varepsilon_2,\quad L_1' = CD\cos\varepsilon_1,\quad L_2' = CE\cos\varepsilon_2. \]

Furthermore,

\[ AD = H(\operatorname{tg}\eta + \operatorname{tg}\varepsilon_1);\quad CE = H_1(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_2);\quad AC = L = H(\operatorname{tg}\xi + \operatorname{tg}\eta); \]

\[ CD = H(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_1);\quad AE = H(\operatorname{tg}\xi + \operatorname{tg}\eta) - H_1(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_2). \]

After substitution, our equations will take the form

\[ -\sigma_1 H(\operatorname{tg}\eta + \operatorname{tg}\xi)\cos\eta = T_1 H(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_1) + T_2 H_1(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_2)\cos\varepsilon_2, \]

\[ -\sigma_1 H(\operatorname{tg}\eta + \operatorname{tg}\xi)\cos\xi = T_1 H(\operatorname{tg}\eta + \operatorname{tg}\varepsilon_1) + T_2 H(\operatorname{tg}\xi + \operatorname{tg}\eta)\cos\varepsilon_2 - T_2 H_1(\operatorname{tg}\xi - \operatorname{tg}\varepsilon_2)\cos\varepsilon_2. \]

If in these expressions, in place of the sums and differences of the tangents, we insert their expressions as functions of the sine and cosine of the sums of the angles, then after reduction we obtain

\[ -\sigma_1 H\frac{\sin(\eta+\xi)}{\cos\xi} = T_1 H\frac{\sin(\xi-\varepsilon_1)}{\cos\xi} + T_2 H_1\frac{\sin(\xi-\varepsilon_2)}{\cos\xi}, \]

\[ -\sigma_2 H \frac{\sin(\eta+\xi)}{\cos\eta} = T_1 H \frac{\sin(\xi-\varepsilon_1)}{\cos\eta} + T_2 H \frac{\sin(\eta+\xi)}{\cos\xi\cos\eta}\cos\varepsilon_2 - T_2 H_1 \frac{\sin(\xi-\varepsilon_2)}{\cos\xi} \]

and hence, finally,

\[ \begin{aligned} \sigma_1 &= - T_1 \frac{\sin(\xi-\varepsilon_1)}{\sin(\eta+\xi)} - T_2 \frac{H_1}{H}\frac{\sin(\xi-\varepsilon_2)}{\sin(\eta+\xi)},\\[6pt] \sigma_2 &= - T_1 \frac{\sin(\eta+\varepsilon_1)}{\sin(\eta+\xi)} + T_2 \frac{H_1}{H}\frac{\sin(\xi-\varepsilon_2)\cos\eta}{\sin(\eta+\xi)\cos\xi} - T_2 \frac{\cos\varepsilon_2}{\cos\xi}. \end{aligned} \tag{43} \]

If we put \(T_2=0\), then these expressions pass into equations (37), derived for the case when the force does not emerge on the same side of the angle formed by the legs; they pass into these same formulas if we suppose the force \(T_2\) to have been transferred to the vertex, that is, if we put

\[ T_1=0,\quad H_1=H. \]

This may perhaps serve as a check on the correctness of the derivation. If both forces emerged on the same side of the headframe, then, by analogy with expressions (38), we may write

\[ \begin{aligned} \sigma_1 &= - T_1 \frac{\sin(\xi+\varepsilon_1)}{\sin(\eta-\xi)} + T_2 \frac{H_1}{H}\frac{\sin(\xi+\varepsilon_2)}{\sin(\eta-\xi)},\\[6pt] \sigma_2 &= - T_1 \frac{\sin(\eta+\varepsilon_1)}{\sin(\eta-\xi)} - T_2 \frac{H_1}{H}\frac{\sin(\xi+\varepsilon_2)\cos\eta}{\sin(\eta-\xi)\cos\xi} - T_2 \frac{\cos\varepsilon_2}{\cos\xi}. \end{aligned} \tag{44} \]

Combining, respectively, equations (43) and (44) with (37) and (38), we can easily, without new derivations, compile expressions determining the stresses in the cases when one of the forces is placed inside the headframe and the other emerges outside it.

Moreover, without analyzing our derivations more closely, on the basis only of their complete analogy with formulas (37) and (38), we can say that by increasing or decreasing the angle formed by the legs, we can change the stress, and there will be

\[ \operatorname{Min}\ \sigma_1 \text{ when } \eta+\xi=90^\circ \text{ or when } \eta-\xi=90^\circ. \]

The following graphical method of calculation may be proposed. Continue \(T_1\) and \(T_2\) (Fig. 52, p. 62) in opposite directions, and on these extensions set off equal magnitudes \(LB\) and \(DK\). Resolve the force \(LB\) along the extensions of the directions \(AB\) and \(CB\).

The force \(DK\)—along the direction of the leg \(BC\) and the horizontal. In turn, we resolve the horizontal force \(DG\) into a force applied at the vertex \(B\) and at the support \(C\); for this purpose, on the extension of \(BC\) we lay off \(BF\), equal to \(DC\), and connect \(G\) with \(F\). The intersection of the horizontal straight line drawn through \(B\) with \(GF\) gives us the required length \(BE\), since

Drawing 52.

\[ \frac{DG}{BE}=\frac{DF}{BF}=\frac{BC}{DB}=\frac{H}{H_1}, \]

whence

\[ BE = DG\frac{H_1}{H}; \]

finally resolving \(BE\) along the directions \(AB\) and \(BC\), we shall have

\[ \sigma_1 = BN + Ba, \]

\[ \sigma_2 = Dc + BM - Bb. \]

Let us determine these forces as functions of \(T_1,\ T_2,\ H,\ H_1\) and of the angles; from the triangle \(BML\) we have

\[ \frac{ML}{\sin(\xi-\varepsilon_1)} = \frac{MB}{\sin(\eta+\varepsilon_1)} = \frac{-T_1}{\sin(180-\eta-\xi)}; \]

and since \(ML=BN\), then

\[ BN=-T_1\frac{\sin(\xi-\varepsilon_1)}{\sin(\eta+\xi)}, \]

\[ BM=-T_1\frac{\sin(\eta+\varepsilon_1)}{\sin(\eta+\xi)}. \]

From the triangle \(DKG\) we have

\[ \frac{DG}{\sin(\eta-\varepsilon_2)} = \frac{GK}{\sin(90+\varepsilon_2)} = \frac{-T_2}{\sin(90-\xi)}, \]

and since \(GK = Dc\), then

\[ Dc = -\mathrm{T}_2 \frac{\cos \varepsilon_2}{\cos \xi}, \]

\[ DG = -\mathrm{T}_2 \frac{\sin(\xi-\varepsilon_2)}{\cos \xi}, \]

so that

\[ BE = -\mathrm{T}_2 \frac{H_1}{H}\frac{\sin(\xi-\varepsilon_2)}{\cos \xi}. \]

Finally, from the triangle \(BEa\) it follows that

\[ \frac{Ba}{\sin(90-\xi)} = \frac{Ea}{\sin(90-\eta)} = \frac{BE}{\sin(\eta+\xi)}; \]

again, in view of the fact that \(Ea = Bb\), we have

\[ Bb = -\mathrm{T}_2 \frac{H_1}{H} \frac{\cos \eta}{\cos \xi} \frac{\sin(\xi-\varepsilon_2)}{\sin(\eta+\xi)}, \]

\[ Ba = -\mathrm{T}_2 \frac{H_1}{H} \frac{\sin(\xi-\varepsilon_2)}{\sin(\eta+\xi)}. \]

Now let us make the substitution in order to obtain the formulas derived earlier.

To determine the support reactions, let us write the equation of moments of the forces at the vertex \(B\)

\[ Bd\,\mathrm{T}_2 + \sigma_3 H = 0; \]

but

\[ Bd = (H-H_1)\frac{\sin(\xi-\varepsilon_2)}{\cos \xi}, \]

and consequently we shall have the horizontal force

\[ \sigma_3 = -\mathrm{T}_1\left(1-\frac{H_1}{H}\right) \frac{\sin(\xi-\varepsilon_2)}{\cos \xi}, \]

apart, of course, from the forces \(\sigma_1\) and \(\sigma_2\).

With the aid of the construction it is easy to find both the force \(\sigma_3\) and its resultant with \(\sigma_2\). The method shown in drawing 53 (p. 64) is clear: \(Cg = DB\)—this is that part of the force \(DG\) which, when resolved

\(T_2\) was assumed to be taken directly by support \(C\). If the lower tie is absent—which, however, may be the case in all substantial structures—then the foundations must be calculated so that they take the forces: 1) in support \(A\), of magnitude \(\sigma_1\), and 2) a force directed along the leg into support \(C\),

Drawing 53.

\[ \rho=\sqrt{\sigma_3^2+\sigma_2^2-2\sigma_3\sigma_2\sin\gamma} \tag{45} \]

at an angle to the horizon \(\gamma\), determined from the equation

\[ \sin\gamma=\frac{s}{\sigma_2}\cos\eta . \tag{46} \]

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 14.

Fig. 15.

Fig. 12.

Fig. 6.

Fig. 25.

Fig. 40.

Fig. 25.

Fig. 38.

Drawing 31.

Fig. 44.

Fig. 41.

Fig. 13.