=29.= A =head-frame= of wood, iron, or steel is built over a shaft or
slope mouth to carry the sheaves over which the hoisting ropes are
conducted from the mine to the drum of the hoisting engine; it also
usually carries the upper portion of the cage guides or, in the case of
a slope, the tracks for cars.
[Illustration]
[Illustration: FIG. 27]
A head-frame must be strong enough to bear the strain brought on it due
to the total load hoisted and the pull of the engine in hoisting this
load; it must also be rigid in construction to withstand the severe
vibration and shock to which it is subjected on account of the rapid
hoisting and the jar due to the landing of the cages.
[Illustration: FIG. 28]
The amount and direction of stresses that a head-frame must resist
are usually determined by applying the parallelogram of forces as
follows: Fig. 28 is a simple head-frame at a slope; _a_ is the drum
of the hoisting engine with the rope coming from its upper side and
running over the head-sheave _b_ down to the slope cage _c_. Assuming
that the angles _e_, _f_ made by the two portions of the rope with the
horizontal are equal, and that the pull on each part of the rope is
20,000 pounds, to determine the amount and direction of the resultant
of the two rope pulls, proceed as follows: Extend the rope lines to the
point of intersection _g_ and from there lay off the two lines _g h_
and _g k_, to some definite scale, representing the pull of the rope.
If a scale of 2,000 pounds to ⅒ inch is taken (⅒ inch = 2,000 pounds),
_g h_ and _g k_ will each be 1 inch long. Complete the parallelogram
by drawing _h l_ parallel to _g k_ and _k l_ parallel to _g h_. The
diagonal _g l_ represents the direction and amount of the force acting
on the head-frame due to the pull of the two portions of the rope.
The diagonal, by measurement, is 1½ inches or ¹⁵/₁₀ inches long, and
since each tenth inch equals 2,000 pounds, the stress on the head-frame
in the line of the diagonal _g l_ is 2,000 × 15 = 30,000 pounds. The
figure also shows that the direction of this force is vertical, hence
there is no tendency for the frame to be pulled over to either side
and, theoretically, side bracing is not needed.
[Illustration: FIG. 29]
=30.= Consider now the case of a vertical shaft, Fig. 29, in which, as
before, _a_ is the drum, _b_ the head-sheave, _c_ the cage, and _d_ the
head-frame, and assume the same pull of 20,000 pounds on each part of
the rope. As before, extend the lines of the rope, which are the lines
of force along which the pulls due to the engine and the load act,
until they intersect at _g_. From this point lay off on these lines
distances representing the stresses in the rope to any scale. Using the
same scale as before, ⅒ inch = 2,000 pounds, the lines _g h_ and _g k_
representing the two forces will be each 1 inch long. Completing the
parallelogram by drawing _h l_ parallel to _g k_, and _k l_ parallel to
_g h_, and drawing the diagonal _g l_ through _g_, the resultant, _g
l_ = ¹⁹/₁₀ inches, represents a stress of 38,000 pounds. The direction
of the resultant is also determined, being in the line of the diagonal
_g l_. If the head-frame shown in Fig. 28 were used for this case,
it would be overturned by this resultant force, unless the leg on the
opposite side of the shaft from the engine were securely anchored, so
an inclined brace _m_ is added to resist this overturning action. The
resultant of all forces acting on the head-frame should generally fall
within the structure if the greatest stability is to be secured, but
when this cannot be done it is necessary to resist the overturning pull
by anchoring the head-frame to its foundations much more securely than
is the case where the resultant falls within the structure.
The direction of the resultant force may be obtained by drawing a line
through the intersection of the lines of action of the forces at _g_
and the center of the head-sheave _b_, as may be seen in Figs. 28 and
29.
[Illustration: FIG. 30]
=31.= In Figs. 28 and 29, the pull of one hoisting rope running from
the top of the drum was considered, but in most cases it is necessary
to consider the pull from two hoisting ropes, one running from the top
and one from the bottom of the drum _f_, as shown in Fig. 30. _a b_ and
_a′ b′_ represent the directions of action of the two forces acting
on the hoisting ropes, while the two vertical forces _a c_ and _a′ c_
acting down the shaft are approximately equal to the two forces acting
toward the drum. There are, therefore, two resultants _a d_ and _a′
d′_, the directions of which are determined by lines from _a_ and _a′_
through the center of the sheave _e_. The amounts of these resultant
forces can be determined by the parallelogram of forces as shown in
Figs. 28 and 29. A resultant that is a mean between _a d_ and _a′ d′_,
both in position and amount, is sometimes taken, or the greater value
as determined from _a d_ or _a′ d′_ and the greatest inclination as
given by _a′ d′_ may be used, as being the worst theoretical conditions
to which the frame may be subjected. A head-frame usually has a
vertical post approximately parallel to the vertical pull of the rope
in the shaft, and an inclined member _g h_ approximately parallel to
the resultant determined by the parallelogram of forces. If _g h_, Fig.
30, is parallel to the resultant, the vertical leg _h i_ is under no
strain and merely supports the end of _g h_. If the resultant falls
between _g h_ and _h i_, both of these legs will be under compression.
If the resultant falls outside of _g h_, the leg _g h_ will be under
compression and _h i_ will be under tension. The head frame will be
most stable when the resultant falls between _g h_ and _h i_, but this
cannot always be accomplished in building the frame on account of the
conditions at the head of the shaft; nor is it always advisable to do
so from structural considerations.
=32.= Since wood is much better adapted to withstand compressive than
tensile stresses and since steel is adapted to withstand either tensile
or compressive stresses, it is much more important that the members of
timber frame conform as closely as possible to the theoretical line
worked out in Figs. 28, 29, and 30 than in the case of a steel frame.
Take, for instance, the case shown in Fig. 31, where for some local
reason it is impossible to put an inclined strut in or near the line
of the resultant stress to withstand the pull that tends to overturn
the head-frame. In a steel structure, _a_ can very easily be made a
tension member by anchoring its lower end to a heavy foundation. This
resists the tendency to overturn and makes a very stable structure.
In practice, braces can generally be located parallel to the line of
resultant strain, Fig. 29, or outside this line, as shown in Fig. 30,
so that the strain due to the pull of the rope will come mainly on the
inclined brace and not on the upright. To distribute the stress on the
foot of the different parts of the frame, an inclined brace is usually
set farther from the shaft than the parallelogram of forces locates it,
and so placed that about two-thirds of the strain due to the pull of
the rope comes on the brace and one-third on the upright parts of the
frame. In order to give the frame a more stable base and because the
base must be larger than the top of the frame to bring the foundations
back from the shaft mouth, usually the members _h i_ are also slightly
inclined.
[Illustration: FIG. 31]
Wherever permanency of head-frames is required, if steel is obtainable
at a price at all comparable with wood, steel structures are being
used, as timber frames rot.