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7
THE CONSTRUCTION
OF
LARGE TUNNEL SHAFTS
J. H WATSON BUCK
ARTES
LIBRARY
1837 U
VERITAS
SCIENTIA
OF THE
UNIVERSITY OF MICHIGAN
E-PLURIBUS UNUM
QUAERIS PENINSULAM AMOR NAM
CIRCUMSPICE
TF
230
B92
THE CONSTRUCTION
OF
30207
LARGE TUNNEL SHAFTS
A Practical and Theoretical Essay
Josephy wood
RESIDENT
BY
H. WATSON BUCK, M. INST.C.E.
ENGINEER,
LONDON AND NORTH-WESTERN
RAILWAY
'Prius quàm incipias consulto, et ubi consulueris
mature facto opus est'-SALLUST
Capio Lumen
LONDON
CROSBY LOCKWOOD AND CO.
7 STATIONERS'-HALL COURT, LUDGATE HILL
1880
[All rights reserved]
грив
PREFACE.
THIS little work has been written in compliance with the
request of several professional friends of the author, who
were aware that he has had considerable practical ex-
perience in the construction of Tunnels, and has always
taken an interest in the accurate execution of works of
the character forming the subject of the present essay.
The author has now much pleasure in publishing the
results of his investigation of this important subject, which
does not appear to have been alluded to in any treatise
upon stone-cutting hitherto published. The practical man
will at once perceive the great advantage of being able to
work every stone in the base of a large shaft, by means of
templates, above ground; thus insuring accurate work-
manship, and effecting a great saving of time, which is
iv
PREFACE.
always of importance where excavation has to be tem-
porarily supported by timber, and especially so in tunnel
work. The analytical processes given serve to show
that the rules are founded on correct mathematical
principles.
RUGBY, 1879.
J. H. W. B.
CHAPTER
PREFACE.
CONTENTS.
PAGE
iii
1. DESCRIPTIVE GEOMETRY APPLICABLE TO FIRST PRINCIPLES
II. THE LEADING DIMENSIONS OBTAINED BY Calculation
1
5
III. THE DEVELOPMENT OF THE INTRADOS AND EXTRADOS, ETC.,
PLAN, CROSS SECTION, AND LONGITUDINAL SECTION
10
IV.
THE METHOD OF WORKING THE SIDE BEDS AND SOFFITS OF
THE VOUSSOIRS
15
V. THE METHOD OF WORKING THE BACKS OF THE VOUSSOIRS 20
VI.
METHODS OF WORKING THE FACES OF THE VOUSSOIRS . 23
VII.
THE METHOD OF WORKING THE TOP BEDS OF THE VOUS-
SOIRS BY MEANS OF A FULL-SIZED PLAN, WITH RULES FOR
DRAWING THE SAME
29
VIII.
FORM AND DIMENSIONS OF SKEWBACKS ON VERTICAL SIDE
WALLS, AND THE METHOD OF WORKING THE SAME
34
vi
CONTENTS.
CHAPTER
PAGE
IX. THE METHOD OF SETTING OUT THE SPIRALS AND VOUSSOIRS
UPON THE Laggings
38
X. OBSERVATIONS WITH REFERENCE TO FORM OF ARCH, ETC. .
40
XI. RULES FOR ORDERING THE STONE
43
XII. ADDENDA .
48
""
Errata
Page 2, last line, for sine read cosecant
12, line 15, for (Fig. 4a) read (Fig. 5)
44, to first formula add + e + 0·16
Plate 2, call line between Figs. 4 and 4a, AB
4, Fig. 17, call left side of stone, LM
THE CONSTRUCTION
OF
LARGE TUNNEL SHAFTS.
CHAPTER I.
DESCRIPTIVE GEOMETRY APPLICABLE TO FIRST
PRINCIPLES.
SUPPOSING it be required to erect a semicircular arch with
à circular opening forming the base of a shaft or tower,
the centre of which is a point in the crown of the arch.
Divide the circumference of a quadrant of the circular
opening on plan (Fig. 1) into any number of parts (twelve
in the figure), and from each point, thus given (1, 2, 3, 4,
&c.), draw vertical lines to intersect a line representing the
cross section of half of the soffit of the arch, drawn ver-
tically underneath; making the radius A,H, on the same
straight line as the radius AH. These intersections are
marked a, b, c, d, &c. Produce HC to C' (Fig. 2). Let
H'a', H'b', H'c', H'd', &c., be made respectively equal to
B
2
THE CONSTRUCTION OF
the arcs A,α, A,b, A₂c, And, &c. (Fig. 1.) Then the junc-
tions of the lines drawn horizontally from A, 1, 2, 3, 4, &c.,
and vertically from H', a', b', c', d', &c., will give points in
the curve representing the development of one quarter of
the soffit of the circular opening (1', 2′, 3′, 4', &c.).
Let 1', 2', 3', 4', &c. (Fig. 2) represent points from each
of which it is desired to draw a joint of a course of stones,
such that it shall be at right angles to a tangent to the
curve of development at that point; it is required to mark
upon the development the position of the joint at the
point 8'. Draw D8E on plan (Fig. 1) tangent to the
point 8 (corresponding to 8' on the development). Pro-
duce the vertical lines already drawn (Fig. 1) until they
meet the line D8E, and from the points of intersection
(D, x, x, x, x, &c.) draw horizontal lines until they meet
the vertical lines on Fig. 2, produced; then the latter in-
tersections (F, x', x', x', x', &c.) will give points in a curve
representing the development of a line upon the soffit of
the arch, formed by the intersection with the soffit of a
vertical plane suspended from the tangent D8E (Fig. 1).
The curve thus formed is necessarily tangent at the
point 8' to the curve 1, 2, 3', 4', &c., and is a harmonic
curve, the abscissæ to which are the arcs Aa, Ab, A‚c,
&c., and the ordinates to which are the sines of these arcs
increased in the proportion in which the secant of the
angle DH8 exceeds the sine of the same angle. A tangent
codec ane
LARGE TUNNEL SHAFTS.
3
to this curve at the point 8' will be also tangent to the
curve 1', 2′, 3′, 4', &c., and it may be drawn in the follow-
ing manner.
Upon the line F M as axis, draw the curve of sines FSL
making the abscissæ equal to the arcs A₂α, A,b, A½c, A‚d,
&c. (Fig. 1), and the ordinates equal to their sines ɑ,ɑ, b₂b,
c₂c, d₂d, &c. (Fig. 1). Then the ordinates of this curve
bearing a uniform ratio to the ordinates of the harmonic
curve x', x', x', x', &c., and the two curves having a common
axis FM; by a property of such curves, tangents drawn
to the curves from any point in the axis, will make their
contacts in a straight line perpendicular to the axis. With
the centre M describe a quadrant PTL equal to A,C,R.
From the point S where the curve of sines FSL intersects
the perpendicular h'N, draw ST parallel to the axis F M,
and from the point T draw T V parallel to h'N or perpen-
dicular to the axis F M. Again make M O equal to M V,
and join PO. Through the point S draw W S parallel to
Then the line W S is tangent to the curve of sines
FSL at the point S.*
PO.
* Let it be required to draw a line touching this curve at the point F
(Fig. 2a).
Assume the proximate point ƒ, and draw FG I and ƒgi parallel, and FE,
GH, foe, and gyh perpendicular to the axis.
The elementary triangleƒ F being similar to F ET,ƒ$: $F::FE: ET;
but from the genesis of the curve the minute segment F is equal to the
elementary arc G g of the circle, and consequently foorgy: Gg:: FEET.
Now the elementary triangle gyG is similar to OHG, and gy: Gg:: OH: OG;
wherefore FE : ET:: OH: OG, and the subtangent.E T is thus given. A
B 2
4
THE CONSTRUCTION OF
Join W 8'. Then W 8' is tangent to the harmonic
curve at the point S, and it is therefore also tangent to
the curve of development at the same point. Draw 8'Y
perpendicular to W 8'. Then the line 8'Y gives the direc-
tion of the joint at the point 8'. Now if horizontal lines
be drawn from the intersections of 8'Y with the vertical
lines on Fig. 2, to intersect the vertical lines on Fig. 1,
they will give points in the plan of the spiral of which
8'Y is the development.
simple construction results. Make O K equal to OH, join C K, and parallel to
it draw FT, which will be the tangent required. For the triangle TFE is
obviously similar to KO C, and consequently FE : ET:: OK or OH: OC
or OG.
Cor.- At the point A the cosine OK or OH is equal to the radius O L, and
hence the curve at its origin crosses the axis at the inclination of half a right
angle. (Leslie's Geometry of Curve Lines.)
LARGE TUNNEL SHAFTS.
5
CHAPTER II.
THE LEADING DIMENSIONS OBTAINED BY CALCULATION.
We now propose to show how to obtain the development
of the circular opening, and the directions of the joints, &c.,
by means of calculation..
The axis H'C' is equal to the arc A,C₂, and H'"', H'k', H'',
&c., are equal to the arcs Al, Ask, Ai, &c.; the ordinates
l'11′, k'10', ¿'9', being equal to the sines of the angles
11 HC, 10 HC, 9 HC, &c., or cosines of the angles
A H 11, A H 10, A H 9, &c.
Let the angle AH 8 = 0, and the angle A,H₂h
also let the radius H Cr, and the radius H₂R = R.
Then Rr: sin : sin A
1;
=
A,
r. sin e
sin ▲ =
A
to radius unity
R
Let the arc Anh = a, and 3.14159, &c. = π.
Then 360 A 2πR: a.
a = A
2 π R
360
(1)
(2)
6
THE CONSTRUCTION OF
Let the ordinate 8'h' 0.
0 = r.cos 0
Join M T (Fig. 2).
Then by similar triangles-
MO: MPSN: NW.
But MO = MV = cos P M T or A₂H₂h
and SN
=
V T = sin P M T or A₂H₂h
Also M P = radius H₂R = R
..cos A: R:: sin A: N W ;
1
= cos A,
= sin A.
or subtangent NW R.
sin A
COS A
=
R. tan A;
(3)
and N 8' will be seen by inspecting Fig. 1 to be equal to
DH − 8' h′ = r. (sec 0
-
-
cos 0).
Also
tan Y8'h' tan N W 8'
N 8'
N W
r. (sec ✪ — cos 0)
R. tan A
Let the angle Y8′h' = ß
Then
tan B =
r.(sec ✪ — cos 0)
R.tan A
to radius unity
(4)
Let the distance Y h' D
D = 0.tan B
(5)
The angle ẞ is the angle at which the joint intersects
-тур
The clements
L
LARGE TUNNEL SHAFTS.
7
the ordinate at the point 8', and corresponds to the angle
of the intrados in an oblique arch.
Let the angle of the extrados = &.
The tangents of the angles ẞ and being respectively
arcs of the intrados and extrados, tan & may be found by
proportion thus
Let e = the thickness of the voussoirs.
Then R Re: tan B: tan ;
:
and
tan =
R+ e
R
tan ẞ
to radius unity.* . (6)
Let the difference between the angles & and B = 8, the
distance from each other at which the parallel and twist-
ing rules for working the side beds are to be applied on
the soffit = l, and the quantity by which the breadth of one
end of the twisting rule must exceed the other end = t
(the other end being made the same breadth as the parallel
rule)-
t = l.tan &*
(7)
Let the distance between the other or unequal ends of
the rules coinciding with the extrados
=
12
1₂ = 1. !
sec o
sec B
*
(8)
(The parallel and twisting rules, with their application,
are described in Chapter IV.)
* Buck on Oblique Bridges.
8
THE CONSTRUCTION OF
These are all general expressions, and apply even
when the radius of the interior of the shaft and that of the
arch are equal, i.e. when r = R, in which case the curve
of development is a true curve of sines.
It will be seen by reference to the corollary to the
problem forming the foot-note to Chapter I, that in the
latter case, if it should be desired to draw a joint at the
extremity of the axis, it will form with a line perpen-
dicular to the axis an angle of 45°, which will therefore be
its angle of intrados, and the top of its springer or lowest
voussoir will be a right angle on the soffit.
The tangent for 45° being 1 to radius unity, the ex-
pression for the corresponding angle of the extrados be-
comes
tan &
R+ e
R
We will now give an example showing the application
of the above formulæ.
Let the span of the semicircular arch be 25 feet, and
the diameter of the circular opening 20 feet, and let
the quadrant of the circular opening on plan be divided
so as to give nine voussoirs in addition to half the width
of a keystone at both A and C (Fig. 1).
The most convenient system will be found to be to
collect the formula, with their application to this particular
case, in a tabular form, thus-
LARGE TUNNEL SHAFTS.
6
r=10 ft.
(1)
R=12.5 ft.
e=2 ft.
(2)
(3)
(4)
(5)
(6)
Sin A
a
O
Tan B
D
Tan &
R+e=14.5 ft.
(7)
t
7=2 ft.
(8)
6
12
Sin 0
A
Tan A
Cos e
Sec 0
β
φ
Tan &
Sec❘ Sec B
r.sin e
R
2πR
r. (sec ◊ – cos 0)
Δ
r. cos e
O. tan B
360
R. tan A
R+e
Ꭱ
7.
seco
tan B
φ - β
7. tan 8
sec B
Ft.
Ft.
Ft.
430 0784-0627 336 0629
•7854-9969 9.9692 1.0030
⚫0784
•2357
•3941
.5564
4 29 7821
13 16 2.2923|
21 313-6413]
29 64-6898
⚫7298
36 85.5497
13 30 2334 1868 10 46 1901 2.3496-9723 9.7237 1.0284|
22 30 3826-3061 17 50 3217 3.8899-9238 9.2388] 1.0823]
31 30 5225-4180 24 43 4603 5.3925-8526 8.5264 1.1728
40 30 64945195 31 18 6080 6.8691-7604 7.6040 1.3150
49 30 7604-6083 37 28 7664 8.1741 6494 6.4945 1-5397
58 30 8526-682143 1 9330 9.3849 5225 5.2250 1.9138)
67 30 9238-7391 47 39 1.0970 10-3958 3826 3.8268 2.6131
76 30 9723-777951 41.2378 11-1376 2334 2.3345 4.2836
85 30 9969-7975 52 54 1.3222 11.5411-0784 0.7846 12.7454
90 01·0000-800053 8
11.5908-0000
⚫9293
1.1929
1.6264
2.6176
7.6639
Ft.
Ft.
0910 5 120 43 0125 0250 1.00411-0030 2-0021
2734 15 182 20355 0710 1.0367|1-02742-0181
•4572 24 343 30532-1065 1.0995 1.0749 2-0458)
6455 32 51 3 450655·1310|1.1903|1-1444 2.0801
8466 40 154 70719-1439 1.3102|1-2381 2-1164
42 546-0356 1.0780 47 94 15.0743 1486 1.4704 1.3651 2.1542
50 26.2332 1.3838 54 94 7.0719-1439 1.70741-5568 2.1935
58 25 6.2242 1.8867 62 53 400640 1281 2.1359 1.9093 2.2373]
69 66-1107 3.0364 71 462 40.0465 0931 3.1960 2.8031 2.2802
82 346-0131 8.8902 83 35 1 1.0177-0355 8.9479 7.7296 2·3155
10
THE CONSTRUCTION OF ·
CHAPTER III.
THE DEVELOPMENT OF THE INTRADOS AND EXTRADOS, ETC.,
PLAN, CROSS SECTION, AND LONGITUDINAL SECTION.
FIG. 3 is the development of the intrados, showing the
joints of the course of stones drawn in accordance with
formulæ 2 and 5. The length of the stones on the soffit
(2 feet) being marked at each joint, and the points thus
given being joined, the curve of development of the backs
of the stones on the soffit is obtained. Intermediate points
should be calculated for, in order to obtain the curve
accurately. The outer curve is shown in Fig. 3 by a
dotted line; the backs of the stones, however, can be cut
in steps, so as to present faces at right angles to the thrust
of the arch upon which the shaft stands. These are shown
in Fig. 3, the apices of the angles at the joints coinciding
with the dotted line.
The spiral lines upon the plan of the intrados (Fig. 4)
are drawn in accordance with the principle already given
for transferring to the plan from the development the
LARGE TUNNEL SHAFTS.
11
direction of any joint, and represent the curves which
would be assumed by the blade of a flexible straight-edge,
applied successively to each point D given by formula 5
(marked upon the line AB drawn upon the laggings) and
the front of its corresponding joint, and by applying the
straight-edge in this manner, the position and form of the
soffit of each stone can be accurately marked upon the
laggings, and the stones be set thereto, when dressed, as
hereafter explained.
The backs of the stones in Fig. 3 are transferred to the
plan of the intrados (Fig. 4) by drawing vertical lines to
the axis H'C' (Fig. 3) and transferring their points of
intersection to the soffit line (Fig. 5), of which H'C' is the.
development. Lines from these points are again drawn
vertically to Fig. 4, and the backs of the stones in Fig. 4
are given by the intersections of these lines with horizontal
lines drawn from the corresponding points in Fig. 3. This
will be readily understood from the description given of
the mode of development, in Chapter I.
We will now explain the mode of obtaining, by calcu-
lation, the development of the extrados, with the develop-
ments of the spirals representing the direction of the
joints shown thereon as in Fig. 3a.
Upon the axis H'C' mark the lengths of the arcs of the
extrados for the same angles (4) as those for which the
12
THE CONSTRUCTION OF
arcs of the intrados (a) have been already given by
formula 2.
These are found by proportion, thus—
Let A = arc of extrados
R+ e
A = a.
Ꭱ
(9)
From these points set off the ordinates already given by
formula 3, after which, using the following formula,
D₂ = 0. tan
2
(10)
instead of formula 5, the developments of the spirals
giving the directions of the joints upon the extrados are
obtained. These are transferred to the plan (Fig. 4a) in
the same manner as the spirals upon the intrados, except
that the points of intersection of the vertical lines drawn
from the backs of the stones to the axis H'C' are first
transferred to the line of the extrados (Fig. 4a), instead of
to the line of the soffit or intrados.
5
The lengths of the stones on the extrados (2 in pre-
ceding table) being marked at each joint, and the points
thus given being joined, the curve of development of the
backs of the stones is obtained. The dimensions and
formulæ referring to the extrados, with their results, are
collected in the following table.
LARGE TUNNEL SHAFTS.
13
R = 12.5 ft.
e = 2 ft.
(9)
A
Re 14.5 ft.
(10)
D2
Tan &
12
a
R+c
O. tan o
a.
R
Ft.
Ft.
Ft.
•7854
•9110
9.9692
⚫0910
•9073
2.0021
2.3496
2.7656
9.7237
•2734
2.6591
2.0181
3.8899
4.5123
9.2388
•4572
4.2249
2.0458
5.3925
6.2553
8.5264
•6455
5.5040
2.0801
6.8691
7.9681
7.6040
•8466
6.4376
2.1164
8.1741
9.4819
6.4945
1.0780
7.0014
2.1542
9.3849 10.8865
5.2250
1.3838
7.2305
2.1935
10.3958 12.0591
3.8268
1.8867
7.2201
2.2373
11.1376 12.9196
2.3345
3.0364
7.0884
2.2802
11.5411 13.3877 0.7846 8.8902 6.9752
2.3155
11-5908 13.4453
Intermediate points should also be calculated for,
corresponding to the intermediate points in the develop-
ment of the intrados.
The development thus given is that of a circular ring,
the extrados of which forms part of the same curved
surface as the extrados of the arch into which it is inserted,
and the front and back of which are radiated in the same
manner as the arch, and illustrates the principle upon
which the twisting rules are applied, but in all practical
cases the front of the circular ring will form part of the
same curved surface as the interior of the shaft standing
14
THE CONSTRUCTION OF
upon it. The backs of the stones can be cut in steps
similar to those of the intrados.
Each side bed of the triangular extension of the stones
necessary to cause the front of the circular ring to form
part of the same curved surface as the interior of the shaft,
will form part of the same spiral surface as the bed of
which it is the extension, and can be worked without
difficulty with the same twisting rules at the same time.
The top of this triangular extension is shown in plan
(Fig. 4a), and forms the bed upon which the front of the
shaft rests. The method of drawing it is shown in Figs.
8 and 11, and the mode of working it in Chapter VII. It
is hereafter called the "top bed."
Figs. 5 and 6 show the Cross Section and Longitudinal
Section of the arch hitherto referred to. Figs. 7, 8, 9, 10,
and 11, show general rules for obtaining the developments,
plan, cross section, and longitudinal section, geometrically,
the positions of the joints on the development of the
intrados being fixed by formula 5; and it is by a repetition
of the methods there given for projecting the curves and
directions of the joints that Figs. 4, 4a, 5, and 6, are drawn.
It will be seen that the joints on the face and soffit
are not straight lines, but curves. Unless, however, the
drawings are to be made to a large scale, it will not be
necessary to show the curvature.
LARGE TUNNEL SHAFTS.
15
CHAPTER IV.
THE METHOD OF WORKING THE SIDE BEDS AND
SOFFITS OF THE VOUSSOIRS.
PREPARE parallel and twisting rules for one side bed ac-
cording to formula 8; and also a template for each side
bed of the stone it is intended to work, made to the com-
plement of its angle B (corresponding to the angle of the
intrados in an oblique arch) as hereafter described; and
having worked one side bed by means of the parallel and
twisting rules applicable thereto, and afterwards having
obtained the soffit from the bed by means of the template
made to the complement of its angle of intrados, apply to
the soffit a flexible zinc plate made to the form of the
soffit of the stone, as given upon the development of the
intrados drawn full-size, and, having with it marked upon
the soffit the finished form of the stone, obtain the second
bed from the soffit by means of the template made to the
complement of its angle of intrados.
We cannot here do better than give an extract from
16
THE CONSTRUCTION OF
Buck on 'Oblique Bridges,' Chapter III., describing in
detail the mode of using the parallel and twisting rules,
and templates; only making such alterations as are ren-
dered necessary by the differences in the circumstances of
the case.
The beds of the voussoirs consist of what are usually
called winding beds. The mode of obtaining such winding
beds is familiar to workmen, and is done by placing two
rules, one of which has its edges parallel, and the other
diverging, at a determinate distance, and then each is
sunk into a draft in the stone until their upper edges are
in one plane, when the under edges will be in the intended
winding surface or bed; this being done, the superfluous
parts of the stone on the other parts of the bed are to be
dressed off until a straight-edge applied from one draft to
the other, and in contact therewith, being always kept
parallel to the soffit, shall in every part of the bed coincide
with the surface thereof.
2
These rules are shown by Figs. 12 and 13. Fig. 12 is
the parallel rule, and Fig. 13 the twisting rule. Their
lengths AB (Fig. 12) and A,B, (Fig. 13) must be equal
to the depth of the voussoir (it is usually made three
inches), and the other end of the twisting rule, B₂G (Fig.
13) must be increased by the quantity FG = 7. tan 8.
The application of these rules to the bed of a stone is
shown by Fig. 14, where CDEF is the winding bed of
LARGE TUNNEL SHAFTS.
17
the stone; AB is the parallel rule, and A,B, is the twist-
ing rule; the distance between the ends A and A, on the
soffit is 1, and the distance between the ends B and B₂ is l2.
In order that the workmen may not be liable to make
any mistake in applying these rules with the proper degree
of divergence, as well as to obviate the necessity of measur-
ing the distances AA, and B B2, we connect them, when
used, by light iron rods, one of which is fastened at one end
to one of the rules, and the other rod by one of its ends to
the other rule, each rod having a hook of this form
which is made to drop into an eye in the side
of the other rule, so that (these rods being made of the
proper length) when they are dropped into their respective
eyes, the rules must necessarily have the proper degree of
divergence or radiation. These rods are represented in
Fig. 14.
One bed having been thus worked, the soffit must
next be obtained from it, and to effect this, let the follow-
ing template be made. Prepare two moulds, as shown in
Fig. 15, where AC is the radius of the cylinder, and DB
is its thickness, or the depth of the voussoirs. The stock
of the mould AB must be made to fit the curve of the
soffit, and may be of any convenient length; BD should
be equal to the depth of the voussoir with both edges
radiated to the centre. These two moulds should there
be framed together, as represented in perspective by Fig.16,
C
18
THE CONSTRUCTION OF
which must be done in such a manner that the angle A CB
shall be equal to the complement of the angle of the in-
trados. This being done, the edges of the two blades BD
and CE (Fig. 16) will exactly coincide with the spiral bed
of the stone which has been already worked by the twisting
rules, and the stone being placed with its soffit uppermost,
let this template be inverted, and applied with its blades BD
and CE to the worked bed, the strip BC (Figs. 16 and 17)
being at the same time in contact with the soffit DF (Fig.
17); then let a line be drawn on the stone by the stock
AC (Fig. 17), and this line so drawn will be at right
angles to the axis of the cylinder. Let another line be
drawn by the side AB, and this line will be parallel to
the axis of the cylinder. Remove the template, and let
chisel drafts be sunk in the soffit on the line CA, to fit
the curve of the stock of the template, and also on the line
AB (which will be perfectly straight), to fit the side
AB, so that when these drafts are sunk to the proper
depth, the template on being applied with its blades to
the previously worked bed, and the diagonal strip to the
arris of the soffit DF, the stock CA, and the side A B,
shall all be exactly in contact in every part at the same
time. Two segmental pieces, each of about the length of
CA, and alike curved to the radius of the intrados of the
cylinder, as shown in Fig. 18, should be in readiness one
to apply to the draft AC, and the other upon a line GH
(Fig. 17), drawn at a convenient distance from, and
LARGE TUNNEL SHAFTS.
19
parallel to, CA. These segmental pieces must be of
exactly the same dimensions, and each must have a centre
line marked on it, as at C (Fig. 18). They must now
be applied thus: one upon the draft CA, and with its
centre C coinciding with a line IK parallel to AB; and
the other segment is to be applied upon the line GH,
drawn parallel to CA (the more remote the better), and
with its centre C also coinciding with the line IK. This
second segment should then be sunk in a chisel draft,
until its upper edge is out of winding with that of the
other segment in the draft CA. This being done, the
superfluous stone on the soffit should be dressed off,
until a straight-edge applied all over the soffit, parallel
to AB, shall coincide with bottoms of the chisel drafts
CA and GH, as well as with every other part of the
soffit between them and beyond them, and the soffit will
be finished.
To obtain the bed from the soffit, apply the template
with its diagonal strip CB in contact with the arris LM,
and its stock CA, and side AB in close contact with the
soffit. Let drafts be sunk in the second bed of the stone,
until the bottoms of them exactly fit the blades of the
template; these drafts in the bed will then answer the
same purpose for this bed as those did which were ob-
tained by the twisting rules for the first bed, and are to
be made a similar use of.
c 2
20
THE CONSTRUCTION OF
CHAPTER V.
THE METHOD OF WORKING THE BACKS OF THE VOUS-
SOIRS.
IF the backs of the stones are to be worked in steps, the
side of the step at right angles to the axis of the cylinder,
and parallel to the axis of the curve of development, which
is in a vertical position when the stone is fixed in its place
in the arch, can be obtained from the soffit by applying a
square, with one blade on the soffit parallel to the line
marked thereon representing the other step. If the soffit
has been truly worked, the blade of the square thus ap-
plied upon it will always be in contact therewith, wherever
placed.
The radiation of the side which is parallel to the axis
of the cylinder, and at right angles to the axis of the curve
of development, can also be obtained from the soffit by
means of one of the moulds shown in Fig. 15, always
applying it parallel to the side just worked, the curved
portion being in contact with the soffit, and the inner side
LARGE TUNNEL SHAFTS.
21
of the radiating blade being in contact with the line
marked thereon, representing the bed to be worked; or it
may be obtained by means of the template represented in
Fig. 16, but without the radiating blade EC and the strip
BC, the two curved moulds being always parallel to the
side just worked, and the inner edge of the blade DB being
always in contact with the line upon the soffit represent-
ing the bed to be worked.
If the backs of the stones are to be curved, transfer
from the development to the soffit the finished form of the
stone, as before, together with a line drawn across it
parallel to the axis of the curve of development. Then
by means of the template described above, applied so that
the inner edge of the blade DB may always be in contact
with the curved line representing the back of the soffit of
the stone, and also so that the two curved moulds may
always be parallel to the line drawn across the stone, the
varying radiation of the back of the stone may be obtained.
The blade DB in the position shown is only applicable to
half the number of stones. For the other half it will re-
quire to be transferred to the other side of the curved
mould upon which it stands.
(The backs of the voussoirs can also be worked
by means of the obtuse templates, described in Chapter
VI.)
In each case, chisel drafts are to be sunk into the back
22
THE CONSTRUCTION OF
of the stone, to enable the blade to be in contact therewith
in several positions; after which, the superfluous stone is
to be dressed off, so that the blade may be in contact in
any position in accordance with the above rules.
LARGE TUNNEL SHAFTS.
23
CHAPTER VI.
METHODS OF WORKING THE FACES OF THE VOUSSOIRS.
2
TO WORK the face of the stone, which is part of the
same curved surface as the interior of the shaft, let a
template of a portion of the arch be prepared, as shown in
perspective in Fig. 19, its length and breadth being about
twice the length of the stone. Mark upon it a line ABC
parallel to the axis of the cylinder. Then from this set
out a line B B₂ at the angle of the intrados of one of the
side beds, by means of a flexible zinc plate made from the
development of the intrados. Place the stone soffit down-
wards upon the template, so that the bed may be in contact
with this line. Take another template made as shown in
perspective in Fig. 20, representing a rectangular vertical
plane intersecting the intrados on the line ABC (Fig. 19),
and apply it to the former template, as shown in Figs. 19
and 21, so that the blades may be in contact with the line
ABC, and the curved moulds with the template of the arch;
also so that the point D may coincide with the point B,
24
THE CONSTRUCTION OF
already marked upon the soffit of the stone, and the blade
DE be in contact with the stone along the line B L, the
superfluous stone being dressed off until this is effected.
Mark upon the stone the line BL along the blade of the
template, and on it set off-
BG= R. (cos Y
-
cos Z),
Y being the angle (▲) of the upper bed, and Z that of
the lower.
Then from the point F, already marked upon the soffit,
let a chisel draft be sunk to the point G, until a convex
mould, struck with radius, is in contact with the stone
along the whole of the line F G. Draw upon the template
of the arch another line MO, parallel to B C, and apply
to it and to the stone, as before, a similar template to
that shown in Fig. 20, representing a rectangular vertical
plane intersecting the intrados on the line M O, the blade
DE, in this case, being in contact with the back of the
chisel draft FG at the point P, and also in contact with
the stone along the line M N, the superfluous stone. being
again dressed off to effect this. Mark the line M N on the
stone along the blade of the template. Then let another chisel
draft QR be sunk, parallel to FG, or so that PQ = GR,
and apply to it the same convex mould which was applied
to FG, until the mould is in contact with the stone at the
back of the two chisel drafts M N and B L, and also along
LARGE TUNNEL SHAFTS.
25
the whole of the line H R. Now dress off the superfluous
stone, until a straight edge applied in any position
parallel to BL or MN is in contact with the stone from
top to bottom, and the face will be completed.
The templates represented in Fig. 20 may be struck out
in the following manner (Fig. 22).
Make
a b = R. (sin Z- sin Y),
Y being again the angle (▲) of the upper side bed, and Z
that of the lower.
From the point a draw af perpendicular to ab.
Make
af = R. (cos Y - cos Z).
Then, from the point b, draw bl any required length,
also perpendicular to a b. Apply a mould struck with
radius R to the points ƒ and b, and with it draw the curve
efb any required length. Then fbl will be the form of the
template to be applied at the point B (Figs. 19 and 21),
and emn will be the form of the template applicable to any
other point M (Figs. 19 and 21), mb being the distance
from the point B at which it is intended to apply the
second template, measured on the template of the arch
above referred to, at right angles to the axis of the cylin-
der, and corresponding to OC (Fig. 19) and M B (Fig. 21).
When the radius of the shaft is large in proportion to
26
THE CONSTRUCTION OF
that of the arch, this method of working the face may
often not be applicable to some of the lower voussoirs, on
account of the acuteness of the angles of the templates
required (Fig. 20), and the small space available for
applying them between the stone and the template of the
arch. In these cases, obtuse templates must be used
instead of acute ones. If the curve efmb (Fig. 22) be
produced indefinitely to o, lbo will be the form of the obtuse
template to be applied at the point B (Figs 19 and 21),
and nmo will be the form of the obtuse template applicable
to any other point M (Figs. 19 and 21), mb being the
distance at which it is intended to apply the second tem-
plate, measured on the soffit of the stone, at right angles to
the axis of the cylinder, from a line drawn on the soffit,
through the point B, parallel to the axis of the cylinder;
this line having been transferred from the development
of the intrados to the soffit by means of a flexible zinc
plate.
The best method of forming these templates is shown
in Fig. 20a. A template thus constructed can be used to
work the face of every voussoir in one half of the ring, AB
being a movable stock which can be set to any required
angle (a similar template, but with the stock A B attached
to the other end of the curved mould B D, being required
for the other half). When a template of this description
is used, no template of the arch is necessary. The stone
LARGE TUNNEL SHAFTS.
27
must be turned soffit uppermost, and a line parallel to the
axis of the cylinder transferred thereto from the develop-
ment of the intrados by means of a flexible zinc plate,
after which the template must be applied to it, so that
the sides B C, DE may be parallel, and the curved moulds
BD, CE at right angles, to this line, and in contact with
the stone; also so that the inner edge of the stock A B,
at the point B, may be in contact with any required point
in the curve of the face, already marked upon the soffit.
In working the faces of some of the lower voussoirs, it
will be necessary to make the chisel draft M N (Fig. 21)
on the lower side of the chisel draft B L, instead of on the
upper side. Let the stock A B (Fig. 20a) be set to the
angle bo (Fig. 22). Then let a strip of wood of the
same thickness as the stock A B be fastened to it, as shown
by dotted lines in Fig. 20a, its breadth being
R. (sin Z-sin Z),.
Z being the angle (A) of the lower side bed of the
voussoir being worked, and Z, that of any convenient
point in the front of the soffit of one of the voussoirs
situated below it. Apply the template (with this addition)
to the stone, so that its sides may again be respectively
parallel and at right angles to the line before mentioned;
also so that a line joining the inner edge of the stock A B
at the point B (Fig. 20a) and the corresponding point B
28
THE CONSTRUCTION OF
in the face of the stone (Figs. 19 and 21) may be a con-
tinuation of this line, and its length equal to
0 - 029
O being the ordinate (formula 3) applicable to the lower
side bed of the voussoir being worked, and O, that
applicable to the point in one of the voussoirs below it,
before mentioned. A chisel draft will have to be sunk, as
before, until the inner edge of the added strip, under these
conditions, is in contact with the stone; a line must then
be drawn on the stone, along the inner edge of the added
strip, and the convex mould struck with radius r, before
mentioned, applied in two places at right angles to the line
drawn along the edge of the strip, and to BL (Fig. 21),
so that the mould may be in contact with the backs of
the chisel drafts as before. The face of the stone can then
be finished off in the manner before described.
(The obtuse templates can also be used for working the
backs of the voussoirs, by setting the stock AB so that
its inner edge may radiate to the centre of the circle of
which the top of the curved mould BD forms part.
part. See
Chapter V.)
LARGE TUNNEL SHAFTS.
29
CHAPTER VII.
THE METHOD OF WORKING THE TOP BEDS OF THE VOUS-
SOIRS BY MEANS OF A FULL-SIZED PLAN, WITII RULES
FOR DRAWING THE SAME.
It is now necessary to describe the manner of determining
the quantity to be taken off the top of the stone, so that any
line drawn upon it, radiating to the centre of the shaft, when
the stone is fixed in its place, may be truly horizontal.
This is of great importance.
The best method for practical purposes, is to draw out,
full size, upon a platform, a plan of the front and back of
the top bed of the whole ring, marking upon the line of
the front the position of each joint on the soffit (which
may be easily done by means of compasses, as they are
equidistant upon the plan), and drawing a number of lines
radiating to the centre, across the top bed, as shown in
Fig. 23.
The positions of the joints determine the places at
30
THE CONSTRUCTION OF
which the template for working the top beds is to be
applied, as will be shown hereafter.
The back of the top bed can be drawn by setting out
the ordinates given by formula 3, at the distances
(R+ e) sin A
for each case, measured from the centre of the shaft, thus-
R = 12.5
e = 2 R+ e = 14·5
Sin A
(R+e) sin ▲
0
Ft.
Ft.
•0627
•9102
9.9692
•1868
2.7094
9.7237
•3061
4.4390
9.2388
•4180
6.0610
8.5264
•5195
7.5337
7.6040
•6083
8.8207
6.4945
•6821
9.8405
5.2250
⚫7391
10.7169
3.8268
•7779
11.2795
2.3345
•7975
11.5643
0.7846
·8000
11.6000
0.0000
Intermediate points should be calculated for, as before.
The front of the top bed will of course be a circle, with
its centre coinciding with the centre of the shaft.
Frame together two convex moulds struck with radius
", and attach to the upper one a feather-edged blade
LARGE TUNNEL SHAFTS.
31
radiating to the centre of the circle, and at right angles to
the side of the template, as shown in perspective in Fig.
24. Apply the template, thus formed, to the stone, so
that the moulds may be parallel to FG or HR, and the
side XX parallel to MN or BL (Fig. 21), and at a
distance FP or SP, equal to the distance of one of the
radiating lines upon the plan, from one of the joints of
the stone across which it is drawn, measured on the circle,
from which it can be transferred to the stone by a convex
mould; also so that the template may be in contact with
the face, and the blade in contact with the top of the stone
for a distance equal to that part of the radiating line
contained between the two curves on the full-sized plan
(Fig. 23), and at a determinate height hereafter given, the
top of the stone being dressed off until this is effected.
The points of intersection of the top bed with the side
bed may be found by measuring on the side bed, along the
line of the extrados (accurately delineated by means of
the template made to the complement of the angle of
intrados for that bed), commencing at the back, the length
l2, also applicable to that bed; the extremity of which
distance gives the back of the top bed at its intersection
with the side bed; and in the case of the lower side bed,
the top bed can be obtained by applying the template so
that the blade may be in contact with the extremity of the
distance 2, and the side X X parallel to MN or BL
32
THE CONSTRUCTION OF
(Fig. 21), also the template in contact with the face, as
before, the superfluous stone being again dressed off to
effect this. A sufficient number of lines in the top bed
having thus been found, dress off the remainder of the bed
between them, and the stone will then be completed.
It will sometimes be found convenient to place together
two adjacent stones, and apply the template for the case
of a radiating line crossing the side bed.
One template will work the whole of the voussoirs, but
the blade should be movable, so that it may be attached
to the opposite side of the mould when required.
Let
H = the height at which the radiating blade of the
template should be applied above any point M
(Fig. 21).
=
C the distance from the centre of the circle, of an
ordinate drawn from the axis to the point of
intersection of the radiating line with the back
of the top bed, measured on the full-sized plan
(Fig. 23).
c = the distance from the centre of the circle, of an
ordinate drawn from the axis to the point of
intersection of the radiating line with the cir-
cumference of the circle, i. e., the front of the
top bed, measured on the full-sized plan (Fig.
23).
LARGE TUNNEL SHAFTS.
33
Then
H = √ (R + e)² — C² — √˜¯R² — c²
When the point M falls beyond the soffit of the stone,
the height H must be obtained by placing the stone at its
proper angle, upon a template of the arch (Fig. 19), and
measuring from the template. (See Fig. 30.)
1)
34
THE CONSTRUCTION OF
CHAPTER VIII.
FORM AND DIMENSIONS OF SKEWBACKS ON VERTICAL SIDE
WALLS, AND THE METHOD OF WORKING THE SAME.
It is now necessary to observe that when the side
walls or abutments are vertical, and the radius of the shaft
is equal to that of the arch, a portion of the stonework
will fall below the springing line, and will therefore be
vertical on the face. Figs. 6a and 66 are elevations of the
abutment showing the position of the springing line, and
the arrangement to be adopted for the springers or
skewbacks. The two curves of sines meeting at the
springing, seen in elevation, result in two straight lines
meeting in the form of the letter V, and forming a right
angle at the apex. There are two springers or skewbacks,
analogous to the springers at the obtuse quoins of two
oblique arches, one of which skews to the right, and the
other to the left. The angles of the intrados and extrados
of the springers can be found by the foregoing rules, and
the sloping beds from which the voussoirs spring can be
worked either by the parallel and twisting rules, or by
LARGE TUNNEL SHAFTS.
35
means of two triangular zinc plates made respectively to
the angles of the intrados and extrados.
Let 0, B, and A be the angles referring to the sloping
bed of the springer; then the width of the top of each
springer on the soffit will be
R.cos ▲ = r.cos @ = 0 (formula 3),
and the total width of each springer will be
(R.cos A) + (l.cos B) = (r.cos 0) + (l.cosß).
The vertical height of the soffit above springing level
will be
R.cos Ar.cos 0
r.cos ◊ = 0 (formula 3),
and the total vertical height above springing level will be
(R + e). cos ▲ = (r + e). cos 0.
The top bed will be level.
The top of the springers will project in front of the
springing line a distance equal to
R. covers A = r. covers 0
and must be worked to a convex mould struck with radius
R, applied perpendicularly and tangent to the springing
line at dc, and ae or bf, drafts being sunk in the stone
until the mould is in contact with the same along those
lines, after which the superfluous stone must be dressed
D 2
36
THE CONSTRUCTION OF
off, until a straight-edge applied parallel to the springing
line and in contact with the backs of the drafts shall
always be in contact with the stone between them and
beyond them. The surface abc will be curved and vertical.
Having drawn the line ac or bc, by means of a mould
made from the development of the intrados (or by a
straight-edge, as they will be practically straight), and
drawn the straight line de on the side of the stone, at
right angles to the bottom bed, apply a convex mould
struck with radius r, in two or more places parallel to the
springing line, and so that it may be in contact with the
line de, and also with ac or be; also with the face of the
stone between those lines; drafts being sunk in the stone
* until this is effected. Then dress off the superfluous stone
until a straight-edge applied anywhere parallel to de, and
in contact with the backs of the drafts, shall always be in
contact between and beyond the drafts, and the face will
be complete.
The extreme length of the springer from front to back
will be
R.covers A+ e + M = r. covers 0 + e + M.
M being the distance which the springer projects beyond
the back of the arch at springing level.
be (Fig. 6b) must never be less t'an 7, the length of
the voussoirs on the soffit. In Fig. 6b, bel, and the
angle B for the sloping bed is 45°;
LARGE TUNNEL SHAFTS.
37
..tan o
=
R+ e
R
Also arc a for sloping bed = A.
360
2π R
Δ.
(formula 2),
360
.. A = α.
2π R
Ꭱ
but a =
TR-1.cos 45° TR
=
• 70717
2
and 0 = A
TR
• 70717
2
... 0 and ▲ = 360
2π R
Again O (formula 3) = r. cos 0=l. sin 45° = '70711
•70717
.. cos and ▲
A =
ԴՐ
38
THE CONSTRUCTION OF
CHAPTER IX.
THE METHOD OF SETTING OUT THE SPIRALS AND
VOUSSOIRS UPON THE LAGGINGS.
THE centre having been erected, and the laggings fixed,
the face line should first be marked thereon in the follow-
ing manner.
The position of the centre of the shaft having been
decided on, let two centre lines be drawn on the laggings,
intersecting one another strictly at right angles at the
centre of the shaft, one in the direction of the axis of the
cylinder, and the other at right angles thereto. Mark
upon a flexible straight-edge the arc a for each joint
(formula 2) and its corresponding distances D (formula
5) each in its proper relative position, which transfer to
laggings on the latter centre line, on each side of the
centre of the shaft.
From the extremities of the arcs a, so transferred, set
off, strictly at right angles to the latter centre line, and
parallel to the axis of the cylinder, the ordinates O (for-
mula 3), the extremities of which ordinates give points in
LARGE TUNNEL SHAFTS.
39
the curve of the face, and also the points of intersection of
the joints with the face, on the soffit. Intermediate points
need not be set out, as they will have been already given
in the development of the intrados, and the curve can
therefore be accurately completed by means of flexible
moulds made from the development of the intrados, ap-
plied in their proper positions to the points in the curve
which have been already marked upon the laggings.
Now apply the blade of the flexible straight-edge suc-
cessively to the extremity of each distance D on the centre
line, and the point on the face line representing the joint
corresponding thereto, keeping it in close contact with the
laggings, and marking with a pencil along its edge, for the
required distance, the line thus given. These lines will
represent the spirals of which the joints on the soffit form
part. Set out upon each spiral, beyond the face line, the
distance l, and complete the delineation of the form of each
voussoir by means of the zinc plate referred to in Chapter
IV., which will check the accuracy of the work already
done. The whole of the lines should then be scored rather
deeply into the laggings by means of a pointed steel graver,
and run with red paint, when the laggings will be ready
to receive the voussoirs.
40
THE CONSTRUCTION OF
CHAPTER X.
OBSERVATIONS WITH REFERENCE TO FORM OF ARCH, ETC.
THE rules given in the preceding chapters are only applic-
able to shafts standing upon cylindrical arches, whereas
most tunnels are made of elliptical section, or an approxi-
mation thereto, formed by curves of several radii; which
is an excellent shape in most kinds of ground. In the
case of a small shaft standing on a part of the arch struck
with one radius, no alteration in the form of the arch will
be required in order that the rules here given may be
applied, and there is no other sufficient reason for altering
the form of the shaft length, but in the case of a large
shaft the form of the shaft length should be cylindrical,
not only for the sake of insuring accurate workmanship
in the base of the shaft, but also because it gives curves
for the intrados and extrados of the ring better adapted
to distribute the weight longitudinally through the arch,
whereby its strength is considerably increased. The
voussoirs are also fewer and shorter than would be re-
LARGE TUNNEL SHAFTS.
41
quired for an elliptical arch, and the general effect is more
elegant.
Figs. 26 and 27 show the arrangements of the courses
in the lengths uniting the two sections of tunnel, which
should be three or more according to circumstances. The
arches of these lengths should have the courses gauged
downwards from the crown, and marked by saw kirfs
upon what are termed 'shrouds' or projecting strips of
wood about two inches thick, attached to the end or
'leading' ribs, the tops of the shrouds being flush with
the top of the laggings, thus,
the inter-
mediate spirals being drawn on the laggings by means of
a flexible straight-edge. The form of each rib will depend
upon its longitudinal position in the arch. The ordinary
ribs used in the tunnel, with properly proportioned
packing pieces attached to them, will answer the purpose.
When the backs of the voussoirs are curved, a ring
consisting of several courses of brickwork may be first
turned, the correct summering of which can be readily
given at any part of the front or back, by means of a
curved mould made to the radius of the arch, with a
radiating blade, equal in length to the thickness of the arch,
attached to one end, and a triangular strut on one side of
the blade to keep it in an upright position (Fig. 25),
applied so that the mould may always be at right angles
42
THE CONSTRUCTION OF
to the axis of the cylinder. The strut in the position
shown will only work half the ring; for the other half
it will have to be attached to the other side of the
blade. The bricks in the ring should appear as headers
on the soffit.* This mould will also give the summering
of the radiating faces of the brickwork of the arch, for the
reception of the voussoirs, when the back of the ring is
stepped, and the 'cutting-up' of the brickwork of the
arch for the reception either of the ring of brickwork or
stonework.
The author must not be understood to give the dimen-
sions contained in this work as universally applicable.
The dimensions adopted must depend on the nature of
the ground, the height and size of the shaft, and the
materials used.
* The author has constructed shafts of nine feet diameter with small quoin
stones having their side beds and backs worked on the principles herein laid
down; their faces being of uniform height, and the whole of the top of the
stonework being worked level along lines radiating to the centre of the shaft,
the requisite strength being made up by rings of brickwork at the back and top
of the quoins as shown in cross section (Fig. 28),
LARGE TUNNEL SHAFTS.
43
CHAPTER XI.
RULES FOR ORDERING THE STONE.
Ir is in the first instance necessary to know what size to
order the stones to be sent from the quarry, and the fol-
lowing rules will be found very useful for this purpose, and
by adopting them a great waste of stone may be prevented.
By means of these rules the dimensions of every stone
can be obtained without the aid of a drawing.
The dimensions are in feet.
The length of the lowest voussoir forming a skewback as
in Figs. 3 and 3a, will be
(W-X) +
(R+e) x L
R
(50
ex sin▲ × cos▲ × sin &)+0·25′
+15
4
and A referring to each side bed; X representing the arc
A (formula 9) for the side beds, and W that for 90° (0); L
being the length of the skewback on the soffit, in the
centre. (This applies to all cases.)
The length of the highest voussoir forming the keystone
at the crown of the arch will be
L+ (r−O) + 0·25
44
THE CONSTRUCTION OF
O being the ordinate (formula 3) for each side bed; L
being the length of the keystone on the soffit, in the centre.
(This applies to all cases.)
The depth of the lowest voussoir, forming a skewback,
as in Figs. 3 and 3a, will be
W − X) +
+ (R
R+ e) × L 2
Ꭱ
2
2 (R+ e)
X representing the arc A (formula 9) for the side beds,
and W that for 90° (0); L being the length of the skewback
on the soffit, in the centre. (This applies to all cases.)
The depth of the highest voussoir, forming a keystone,
as in Figs. 3 and 3a, will be
2π . (r + L)) 2
2n
2 (R+ e)
+ e + 0.16
L being the length of the keystone on the soffit, in the
centre; n being the number of stones in the whole ring.
(This applies to all cases.)
The thickness of the lowest voussoir forming a skewback
as in Figs. 3 and 3a will be
2 (l.cos B + 0) + 0∙16
ß being the angle of intrados of each side bed: 7 referring
to each side bed; 0 being the ordinate (formula 3) for
each side bed.
LARGE TUNNEL SHAFTS.
45
The thickness of the highest voussoir forming the keystone
at the crown of the arch will be
{(R+e). 2, (r+L)}.
R n
+ 0.16
L being the length of the keystone on the soffit, in the
centre; n being the number of stones in the whole ring.
(This applies to all cases.)
The dimensions of the skewback given in Chapter VIII.,
for the case of a shaft of the same radius as the arch,
standing upon vertical side walls or abutments (as in Fig.
6b) are the neat dimensions of the stone after it has been
worked; an allowance must therefore be made for dressing.
It may be ordered to the following dimensions.
Let
M = distance from back of arch to back of stone at
springing level.
N = distance from bottom of stone to springing level
B, 0, and ▲ referring to sloping bed.
Then
Length from front to back = (R. covers A) + e + 0·16+ M
width = (r. cos 0) + (l. cos ẞ) + 0·16
height = {(R+e). cos A} + 0·16+N
(This applies to all cases.)
If the side walls are to be curved, and the bottom bed
of the skewback radiates to the centre of the circle to
46
THE CONSTRUCTION OF
which the face of the side walls is struck, the height
will be
{(R+ e). cos ▲} + 0·16 +
N × (R₂ + e + M
R2
2
N being the distance from bottom of stone to springing
level, measured on the face of side wall; R, being the
radius of face of side wall. (This applies to all cases.)
The dimensions of the other voussoirs are as follows.
If the back of the voussoir is to be curved, the length
will be
1½ + (5
x
0
ex sin ▲ x cos ▲ × sin ) + 0·25
4
and ▲ referring to the upper side bed; and l, to the lower
side bed.
If the back is to be cut in steps, the length will be
1₂+ (5
e × sin ▲ × cos ▲ × sin ℗ ) + 0·25 +
4
{(W + ↳½ · sin K) − ( X + sin I)} sin I
;
and ▲ referring to the upper side bed; I being the angle
of extrados of the upper side bed, and K that of the lower
l, referring to the lower side bed : X representing the arc
A (formula 9) for the upper side bed, and W that for the
lower.
The depth will be
LARGE TUNNEL SHAFTS.
47
(W-X) + (l2. sin )) 2
2
2 (R+ e)
+e+0·16
being the angle of extrados of the lower side bed; X
representing the arc A (formula 9) for the upper side bed,
and W that for the lower; l, referring to the lower side bed.
The thickness will be
~ (W −X)² — (QR)² + {½. tan (KI)} + 0·16,
X representing the arc A (formula 9) for the upper side
bed, and W that for the lower, in the case of each voussoir
in the upper half of the ring; and the arc a (formula 2)
for the case of each voussoir in the lower half: Q and R
being their corresponding ordinates O (formula 3): ½
referring to the lower side bed: I being the angle of
extrados (4) of the upper side bed, and K that of the
lower, for the case of each voussoir in the upper half of the
ring, and I being the angle of intrados (B) of the upper
side bed, and K that of the lower, for the case of each
voussoir in the lower half.
48
THE CONSTRUCTION OF
CHAPTER XII.
ADDENDA.
To obtain the development of the line given by the
intersection of the curved surface forming the interior of
the shaft, with the extrados of the arch; which is verti-
cally above the line given by the intersection of the same
surface with the intrados, and is analogous to the line of
the extrados in the face of an oblique arch, find by pro-
portion the sines of the new angles referring to the
extrados, corresponding to the angles already given refer-
ring to the intrados; and also their arcs, thus,
sin A₂
R. sin A
R + e
to radius unity
2π. (R+ e)
α 2 = A ₂
A2·
3.0
The ordinates are the same as those found by formula 3.
Let it be required to find, by calculation, the width of
the top bed, along a line radiating to the centre of the shaft;
LARGE TUNNEL SHAFTS.
49
from any point already given, by the foregoing rules, in
the front of the extrados, i.e. the back of the top bed.
Let KDF (Fig. 29) be a quadrant of the circular
opening of which D is the centre; and the curve K B M the
line of the front of the extrados, as it would appear on
plan without the triangular extension; also B a point in
the same. From B draw BAI parallel to DM, inter-
secting the circle at the point A. Join BD by the straight
line BCD, intersecting the circle at the point C. It is
required to find the distance BC.
Join AD. Then the angle K DA is the angle 0 from
which the position of the point B was found.
Let the angle IBD = T
DI
Then tan T =
r.cos e
Ꮎ
BI
(R+ e). sin A
Also BD DI. cosec T = r. cos 0 × cosec T
==
But BC BD-CD=BD — r
=
... BC= (r. cos x cosec T) -
Also BD ✅B I² + D I²
=
... BC = √BI² + D I² − r
2
and... BC = √{ (R + e) . sin ▲ }² + (". cos 0)²
E
2
グ
​50
THE CONSTRUCTION OF
Let it be required to find the distance A C.
Let the angle ADB = S
Then S = 90° - KDA - CDF
But KDA = 0 and CDF=T
...S=900-T
Again 360 S 2πr: AC
.. AC=
2 πr × S
*
360
To draw a line at the extremity of the axis of the
curve of development of the intrados, giving the length of
the springer or skewback on the soffit, produce the axis
for the distance
1. sin ß
and from the point thus given, set off the ordinate
l.cos B
Then a line joining the extremities of the axis and the
ordinate will be the length required.
The corresponding distances upon the development of
the extrados will be
Production of axis
Ordinate
12. sin o
12. cos o
and, strictly, the back of the springer should be straight.
* The theoretical height II at the point C is
(R + e), cos ▲ − √ R² — {~.sin (0+S)}².
(See page 33.)
LARGE TUNNEL SHAFTS.
51
The length of the springer on the soffit, along its centre
line, will be
1. sin ß+M
M being any required addition.
The back of the springer will radiate to the centre of
the arch, and its length on the centre line of the extrados
will therefore be
(R+e). (l. sin ẞ+M)
R
The width of the top bed, in the centre, will be
e. sin ▲
and the height of the face in the centre will be
e.cos A
THE END.
INDEX.
Addenda
Circular opening, Plan of
""
Curve of sines
""
""
""
""
Tangent to plan of
Construction of
Tangent to.
Joint at extremity of axis of
Ordinates to.
Abscissæ to
Cutting up' of brickwork
Extrados, Angle of
""
Arcs of.
""
""
33
""
39
""
""
Development of
•
Ordinates for development of
•
PAGE
48-51
1, 30, 49
2
3, 8
3
3, 4
8
3
3
41
•
7-9, 13, 37
11-14
11–13, 50, 51
12
11, 12, 14
12, 14
Directions of joints upon development of
Spirals upon plan of
Length of stones upon
Table of dimensions for.
Backs of stones upon, cut in steps
Triangular extension of stones
Plan of
Line of, on face
Harmonic Curve, Construction of
12, 13, 50, 51
13
14
14, 49, 50
12, 14, 29, 30, 49
48
2
2-4
""
Tangent to
Ordinates to
""
99
Intrados, Angle of..
Abscisse to.
""
Development of
""
Tangent to development of
""
""
Spirals upon plan of
""
39
Directions of joints upon development of
Lengths of stones upon
Backs of stones upon, cut in steps .
Plan of
""
""
Arcs of
2 2
2
6, 8, 9, 15, 18, 36
2, 5, 6, 9, 10, 24
2
4, 6, 10, 14
4, 10, 11
10, 50, 51
10
10, 14
1, 2, 5, 37, 38
54
INDEX.
Intrados, Ordinates for development of
Table of dimensions for
""
PAGE
5, 9, 37
9
38, 39
17, 20, 21, 23-26, 28, 30, 31, 36, 39, 41
Laggings, Setting out spirals and voussoirs upon
Moulds
Observations with reference to form of arch, &c.
Ordering the stone, Rules for .
Parallel rules, Dimensions of
Ribs
Rods for parallel and twisting rules
Section, Cross .
99
Longitudinal
Segmental pieces
Shrouds
Side walls, Vertical
""
Curved
Square
Straight-edge.
Summering of brickwork .
Table of leading dimensions
""
""
of dimensions for intrados
of dimensions for extrados
•
40-42
43-47
7, 16, 17
41
17
14
14
18, 19
41
34-37, 45
8, 45, 46
20
11, 16, 19, 25, 36, 38, 39, 41
of dimensions for full-sized plan of top bed
Template to complement of angle of intrados
41
9
9
13
30
15, 17-19, 21
of arch
""
Acute
""
Obtuse
""
for working the top bed
Twisting rules, Dimensions of .
Voussoirs, Side beds of
Faces of .
>>
""
Backs of
Soffits of
Top beds of
23, 33
23-25
21, 26-28
30-32
7, 9, 14, 16, 17
15-19
14, 23—28, 32, 33, 50, 51
20-22
15-19
29-33, 48-51
LONDON: PRINTED BY
SPOTTISWOODE AND (O., NEW-STREET Squarb
AND PARLIAMENT STREET
Jor M
JH W. BUCK, DEL
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0
Tho" Kell Lithographer
40, King St Covent Garden
Fig. 6a
d
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e
Level
f
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Dor M
JH W. BUCK, DEL.
Fig. 4.
PLAN OF INTRADOS
Fig. 4a
PLAN OF EXTRADOS AND TOP BED.
Fig. 5.
1
CROSS SECTION.
}
Shewing Shaft
With Shaft removed
Fig. 6b
d
Springing
نا
Level
f
Fig. 3.
Development of Intrados
With back
cut in steps
Fig. 3ª
Development of Extrados
Fig. 6
LONGITUDINAL SECTION.
With back of ring
airved
With back of ring
cut in steps-
Scale, 8 Feet - 1 Inch.
Inches 12 59
5
70
16
20 Feet
Crosby Lockwood and C 7, Stationers' Hall Court. London.
PLATE 2.
Tho Kell. Lithographer
40. King St Covent Garden
LONGITUDINAL SECTION.
CROSS SECTION.
Uork
J.H. W. BUCK, DEL.
X
Z
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Fig. 10
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Fig. 9.
Fig. 8.
Back
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Extrados
Intrados
Front of Eatrados and Back of Top Bed
Face
PLAN.
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CROSS`\\\SECTION.
Spiral of Intrados
Spiral
-
Fig. 7.
Back
Back
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of
DEVELOPMENTS.
Extrados
Intrados
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Front of Extrades and Back
Front
of
Intrados
Spiral
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Back of Top
of Top Bed
b
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ENLARGED PLAN.
a
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Crosby Lockwood and C° 7, Stationers' Hall Court, London .
Line of Face
PLATE 3.
t
Spiral of Intrados
Radiating line
Spiral of Extrados
ENLARGED CROSS SECTION.
Fig. 11.
Radiating
line.
Radiating line
Tho" Kell. Lithographer
40, King St Covent Garden
A
Fig. 12.
Soffit or Intrados
Az
Fig. 13.
B
A2
B2
F
D
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Fig. 14.
D
Bk
Extrados
Fig. 15
D
B
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F
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Fig. 18.
C
B
J.H.W.BUCK, DEL .
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Crosby Lockwood and C° 7, Stationers' Hall Court, Londou.
D
Fig. 16.
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B
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Fig.17.
B.
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PLATE 4.
Tho Kall, Lithographor.
40, King St Cavent Garden
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0
Fig. 24.
SIDE VIEW.
J.H.W.BUCK, DEL.
Fig. 19.
Ο
B
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Fig. 21.
R
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B
PLAN.
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0
Fig. 22.
A
Fig. 20a
a
m
•
Fig. 25.
SIDE VIEW.
END VIEW.
B
D
Fig. 23.
Fig. 20.
E
0
D
PLATE 5.
Soale to Fig. 23.
10 Feet
Int 12 8 0 1 2 3 4 4 5
سيليا
Crosby Lockwood and Cº 7, Stationers Hall Court. London.
Hidpers Hall
Tho" Kell. Lithographer.
40, King St Covent Garden
Uor M
J.H.W. BUCK, DEL.
Fig 26.
1260 7
2.
E
Scale.
3
A
10 Feet.
Brickwork
Stone
Fig. 28.
Crosby Lockwood and CO 7, Stationers' Hall Court. London.
Fig. 27.
PLATE 6.
Tho Kell. Lithographer
40, Kins St Covent Curlin
K
Fig. 29.
D:
a
T
F
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}
1
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J H.W BUCK, DEL
Crosby Isword and C7
Crosby Lockwood and C7, Stanoners' Hall Court, London
PLATE 7
Tho Kell Lithographer
40, King St Covent Garden
A
A
·B
PLATE 8.
GEOMETRICAL METHOD OF OBTAINING THE WIDTH OF THE TOP BED AND ITS
HEIGHT ABOVE ANY POINT IN THE SOFFIT OF THE FACE.
TOGETHER WITH THE FORMS OF THE TEMPLATES.
-
1
B
B
D
Fig. 30.
A.A Height of top bed.
BB_Width of top bed.
A.A.C_Acute Template.
A.A.D_Obtuse Template.
-----===
B
B
B
Scale.
Inches 12 9 6 3 0
1
2
3
4
5 Feet
سيليا
ليلي
Uor M
JH W. BUCK, DEL.
Crosby Lockwood and C° 7, Stationers' Hall Court, London.
Tho Kell. Lithographer
40. King St Covent Garden
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With 43 Plates and Woodcuts. Originally Edited by EDWARD DOBSON, Architect.
New Edition by E. WYNDHAM TARN, M.A. 8vo. 10s. 6d. cloth.
THE ELEMENTARY PRINCIPLES OF CARPENTRY.
By THOMAS
TREDGOLD, C.E. Edited by PETER BARLOW, F.R.S. Fifth Edition, corrected and
enlarged. With 64 Plates (11 of which now first appear in this edition), Portrait of
the Author, and several Woodcuts. In 1 vol. 4to. Published at £2. 2s., reduced to
£1. 5s. cloth.
'Ought to be in every architect's and every builder's library, and those who do not already possess it ought
to avail themselves of the new issue.'—Builder.
TRAMWAYS: THEIR CONSTRUCTION AND WORKING, with special
reference to the Tramways of the United Kingdom. By D. K. CLARK, M.I.C.E. Large
crown 8vo, with numerous Illustrations and thirteen Folding Plates. 18s. cloth.
'All interested in tramways must refer to it.'-The Engineer.
'Mr. Clark's book is indispensable for the students of the subject.'—The Builder.
LONDON: CROSBY LOCKWOOD AND CO.
WORKS BY WILLIAM HUMBER,
ASSOC. MEM. INST. C.E., MEM. INST. M.E., ETC.
A COMPREHENSIVE TREATISE ON THE WATER SUPPLY OF
CITIES AND TOWNS. Imperial 4to. illustrated with Fifty Double Plates, Coloured
Frontispiece, and upwards of 250 Woodcuts, and containing 400 pages of text,
£6. 6s.; elegantly and substantially half-bound in morocco.
'The most systematic and valuable work upon water supply hitherto produced in English, or in any other
language.'-Engineer.
'We can congratulate Mr. Humber on having been able to give so large an amount of information on a
subject so important as the water supply of cities and towns. The plates, fifty in number, are mostly
drawings of executed works, and alone would have commanded the attention of every engineer whose practice
may lie in this branch of the profession.'-Builder.
CAST AND WROUGHT-IRON BRIDGE CONSTRUCTION, INCLUDING
IRON FOUNDATIONS-A Complete and Practical Treatise on. In Three Parts-
Theoretical, Practical, and Descriptive. Third Edition, revised and much improved,
with 115 Double Plates (20 of which now first appear in this edition), and numerous
Additions to the Text. In 2 vols. Imp. 4to. £6. 16s. 6d. half-bound in morocco.
'Mr. Humber's stately volumes-in which most important bridges, under the direction of our most eminent
engineers, are drawn and specified in great detail.'-Engineer.
‘A book—and particularly a large and costly treatise like Mr. Humber's-which has reached its third edition
may certainly be said to have established its own reputation.'—Engineering.
A RECORD OF THE PROGRESS OF MODERN ENGINEERING,
comprising Civil, Mechanical, Marine, Hydraulic, Railway, Bridge, and other Engineering
Works. In 4 vols. Imperial 4to. Price of each volume, £3. 3s. in half-morocco.
FIRST VOLUME (1863). With 36 Double Plates, drawn to a large scale,
and Photographic Portrait of John Hawkshaw, C.E., F.R.S., &c.
SECOND VOLUME (1864). With 36 Double Plates, drawn to a large
scale, and Photographic Portrait of Robert Stephenson, C.E., F.R.S., &c.
THIRD VOLUME (1865). With 40 Double Plates, drawn to a large
scale, and Photographic Portrait of J. R. M'Clean, Esq.
FOURTH VOLUME (1866). With 36 Double Plates, drawn to a large
scale, and Photographic Portrait of John Fowler, Esq.
'Mr. Humber's works fill a void occupied by no other branch of literature. The drawings have a con-
stantly increasing value. Sooner or later the works will be found a necessity.—Engineering.
'Handsomely lithographed and printed. Will find favour with many who desire to preserve in a permanent
form copies of the plans and specifications prepared for the guidance of the contractors for many important
engineering works.'-Engineer, First Notice.
'A résumé of all the more interesting and important works lately completed in Great Britain; and con-
taining, as it does, carefully executed drawings, witn full working details, it will be found a valuable accessory
to the profession at large.'-Engineer, Second Notice.
A HANDY BOOK FOR THE CALCULATION OF STRAINS IN
GIRDERS and similar Structures, and their Strength; consisting of Formulæ and
Corresponding Diagrams, with numerous Details for Practical Application, &c. Illustrated
with nearly 100 Woodcuts and 3 Plates. Third Edition. Crown 8vo. 7s. 6d. cloth.
'Mr. Humber has rendered a great service to the architect and engineer by producing a work especially
treating on the methods of delineating the strains on iron beams, roofs, and bridges by means of diagrams.'
Builder.
LONDON: CROSBY LOCKWOOD AND CO.
UNIVERSITY OF MICHIGAN
3 9015 07504 2740
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