IMPROVED METHODS OF AND APPARATUS OR DEVICES FOR BUILDING OR CONSTRUCTING ARCHES AND DOMES
Year: 1883

page_text: LETTERS PATENT to William Robert Lake of the firm of Haseltine, Lake,
& Co. Patent Agents, Southampton Buildings, London for an Invention of
"IMPROVED METHODS OF AND APPARATUS OR DEVICES FOR BUILDING OR
CONSTRUCTING ARCHES AND DOMES." A communication to him from abroad
by Thomas J Lovegrove of Philadelphia, Pennsylvania, United States of
America
COMPLETE SPECIFICATION filed by the said William Robert Lake at the
Office of the Commissioners of Patents on the 30th January 1883.
"
WILLIAM ROBERT LAKE of the firm of Haseltine Lake & Co. Patent Agents
Southampton Buildings London "IMPROVED METHODS OF AND APPARATUS OR
DEVICES FOR BUILDING OR CONSTRUCTING ARCHES AND DOMES Communicated
to me from abroad by Thomas J. Lovegrove of Philadelphia, Pennsylvania
United States of America
This invention relates to improved methods and devices or apparatus whereby
arches for conduits, sewers, bridges, and analogous structures may be constructed of
maximum strength with the minimum of labor and material.
In arches as heretofore constructed gravitation develops strains or a number of
force-resultants between the crown and the skew-backs, the direction of whose lines
of thrust tends to destroy the structure, the said lines being tangential to the
curves of the arch; consequently in order to supplement the strength of an arch of
defective structure, it is the custom to place a surplus of the material of which
it is constructed, out of line, so that the true line of equipollence shall practically
lie between the extrados and the intrados of the voussoirs. In technical works, and
particularly in those relating to the construction of arches and bridges, it is
common to see the term "equilibrium " applied to the curve of the arch, but in all
cases it is used simply in an approximate sense, as heretofore it has been impos-
sible to accurately construct an equilibrium arch supporting an unequal load. If
we consider the catenary inverted, it is not a true equilibrium arch unless we

page_text: imagine the cable without weight, and the greater the weight the greater will be
the discrepancy between the arch obtained and the true equilibrium arch.
Heretofore the best method for approximating an equilibrium arch for an uneven
load, was by considering a chain or cable hung from two points, with the variable
loads hung from the said cable along its length, and after ascertaining the curve
produced, the same was inverted and called the equilibrium arch for that unequally
distributed load; but this was only approximate, as where the greatest tension
was, now we would have the greatest compression, and it is utterly impossible to
cause all of the resultant forces to pass down through the skew-backs. All formulæ
heretofore used for the mathametical calculation and demonstration of the
equilibrium arch are based upon the above facts. An arch constructed upon the
system hereinafter explained has, both theoretically and practically, but two grand
resultants, one to the right and the other to the left from key-stone to skew-
back.
Such an arch carries its load as reliably as a plumb-column; an arch constructed
on any other principle has its counterpart in a brut-column, the tendency of which
is to spring out from under the pressure of the superincumbent load. An arch or
sewer constructed upon the principles hereinafter set forth enables me to use a
movable templet merely, instead of the usual fixed centre of support.
The said invention is illustrated in the accompanying drawing in which :—
Figure 1 is a perspective view of a conduit arch in process of construction, with a
movable templet therein.
Figure 2 is an end view thereof drawn to an enlarged scale.
Figure 3 is a diagram for reference in illustrating the system for determining the
proper form of the arch or arches to be constructed.
Figure 4 shows the movable template as applied in the construction of domes.
My foreign correspondent has discovered that in the curve of a perfect arch of
equipollence, be the superincumbent static load upon the arch what it may, the
length of the abscissa at any point in such curve is equivalent to the load at such
point multiplied by the length of leverage from the skew back to the abscissa,
divided by the maximum thrust. It is therefore requisite to find the length of the
abscissa or of a sufficient number thereof for practical purposes, their respective
distances from the skew-backs, and their respective loads, in order to be able to
obtain the proper curve of the arch in each special case.
I shall first describe the method of determining the correct form for such an arch
of even load.
Upon a base C N, Figure 3, equal in length to one half that of the span or chord
of the proposed arch, I erect a right-angled triangle A CN whose altitude A C is
equal to the rise or versed sine of the said arch. In the present illustration the
whole span is taken as twenty feet the rise at ten feet, the figure (3) being
drawn to the scale of half an inch to the foot.
The weight above and resting upon the points A A' A² to A³, say ten in all, of
the hypothenuse A N is represented by the unit one, whether one pound or one ton
is immaterial.
The
Bearing in mind the rule above stated we will now consider how the fifth
abscissa, rising from the point C³, is obtained. The entire load upon this abscissa
below the hypothenuse is ten. The maximum thrust at A is therefore five.
proportion of weight upon A5 is 7.5, which, multiplied by 5, the leverage from N
the skew-back of the proposed arch, gives a product of 37.5; this divided by 5, the
maximum thrust, gives a quotient of 7.5 feet as the correct length of the abscissa
B³ C in a true arch of equipollence of equal load.
A similar calculation starting from any point in the hypothenuse A N will
determine in like manner the length of the abscissa of the arch passing through
such point. Thus the arch Z of even load having the thrust the same at every
point therein, is laid down by drawing a line A Z N connecting the summits of the
several abscissas B' B2 to B.

page_text: I shall now proceed to show the application of my said method in the determi-
nation of the form of an arch Y of equipollence in the case of unequal load, the
chord and rise of the arch being assumed as of the same length respectively as
before.
The space contained by the straight lines F, A, N, E, Figure 3 is presumed to be
filled in with earth or other material of the specific gravity of that which shall be
over the arch after it is built, say for instance, that this mass of material below
the line F E of the road or street, supported by the hypothenuse A N weighs
150 pounds to the cubic foot, this would give 5000 pounds as the theoretical
maximum thrust at the point A, now taking for illustration the weight or load
sustained at the point A which is found to be 8125.75 and multiplying the same
by the leverage, five feet, the product 40628.75 results, and this divided by the
maximum thrust 5000 gives 8.12 feet, the proper length of the abscissa D* C³ of
the true arch Y in this case of equipollence. By repeating the foregoing calcu-
lation with suitable change of factors, required by the rule hereinbefore stated,
based from as many other points in the hypothen use A N as may be thought
advisable, the respective abscissa lengths may be obtained.
It is obvious the greater number of abscissa lengths thus found, the more nearly
the curve will approach the theoretically perfect arch of equipollence for the special
case.
Ordinarily it is necessary to determine the lengths of but a few of such
abscissas However if the work to be constructed is of much magnitude or
importance, it is desirable to attain a greater degree of accuracy in the curve of
the arch, by calculating the lengths of a comparatively large number of abscissa.
The curve X Figure 3 is that of a semi circular arch of the same span and rise
as the true arch Y under the same circumstances. The difference between these
curves X and Y in the present instance indicates obviously a large quantity of
material of the arch built upon the former out of line. In long sewers or conduits
the economy in labor and material secured in the construction of arches according
to this invention over the semi-circular or Roman arch is very considerable.
The curves of equipollence of the arch under the special circumstances having
been thus substantially determined, I now construct a corresponding templet T of
wood or other suitable material such as is shown in figures 1 and 2. This templet
is mounted upon a bed or frame L and is provided with a series of platform steps P
upon which the masons are supported while building the arch H. The templet is
adapted to be easily moved forward horizontally, as desired, upon wheels W fitted
to run upon rails R which are suitably laid in the line of the intended conduit.
In the present instance these rest upon a cross piece O thrown across the lower
section of the conduit. These wheels and rails may however be dispensed with,
and the device be made to slide upon ways or guides fixed at the sides or upon
the bottom, of the interior of the conduit, or otherwise.
In building long arches as for sewers, conduits or the like the arch bricks or
voussoirs are racked up, as shewn in Figure 1.
At the beginning of the arch H, for a short distance, the templet T performs the
function of a centre or support for the material being laid; that is to say, until
the bricks are racked up on each side, near to the crown of the arch, and the crown-
bricks (or keystones) have been driven in place, whereupon the arch springs slightly
outwards sufficient to free the templet from it, as shown in Figure 2, and allows
the latter to be moved forward. Thenceforth the arch is self supporting, until its
entire length is finished; the bricks being laid racked in close proximity to the
templet, not against it. Thus, with a templet having its outline formed according
to the system hereinbefore explained, and such templet adapted to be shifted as
described, there is no need, as has been heretofore the case, of using the usual rigid
centres, which are left in place until the mortar or cement between the voussoirs of
the superstructure shall have hardened, and the earth or other material be filled in
above the arch. Although such arch be designed to carry an uneven load, and be

page_text: built before the load actually comes upon it, yet it will stand, under almost all
circumstances without additional support, for the reason that practically the true
curve of the arch of equipollence of even load will generally lie somewhere between
the intrados and the back of such arch of uneven load. If however, as in exceptional
cases, the weight of the superincumbent load which is to rest upon the completed
arch, is such as to demand a curve so far from that of the arch of equipollence of
even load, that there may be danger of the unloaded arch falling in, I place upon
the main arch H reinforcing ribs or courses J at suitable distances apart. I also
employ re-inforcing ribs in long arches such as for sewers, in order to counteract
the relative weakening due to the length of such structures: for the longer they
are, the weaker they are with the same thickness of material throughout.
I prefer to locate one of the said ribs, which need be but two or three feet
broad under ordinary circumstances, opposite each house along the route of the
sewer, so that the connection between the latter and the house mains may be made
through them, thus avoiding or counteracting the weakening of the main arch,
which would otherwise result. In constructing domes, I first determine the proper
outline of a suitable templet according to the principles of the system hereinbefore
explained, the effect of the lantern, if any, which is to crown the dome being taken
into account in arriving at the several abscissa-lengths of the arch of the dome.
I then construct a templet, such as shewn in Figure 4. The said templet is provided
with a rib G of the desired contour on its exterior edge, and is secured upon a
frame or platform I having wheels W, whereby the templet may be rotated upon a
centre K around the circle of the dome, in process of construction, from time to
time as may be necessary as the work of building proceeds. Instead of a single
templet several such templets supported upon the frame I may be used, where a
number of masons are at work, at as many points in the circumference of the circle
of the dome M as desired. In lieu of such templet, a revolving finger, devised so
that it may be swept around and give the proper line upon which to build the
dome, may be employed. A dome constructed according to the principles of this
system needs no support while being built.
Having thus fully described the said invention as communicated to me by my
foreign correspondent, and the manner of performing the same, I wish it understood
that I claim :-
First The improved method of constructing an arch, which method consists in
building the said arch to a curve, the length of each and every abscissa of which
at any point in such curve is equal to the static load at such point multiplied by
the leverage thereof, and divided by the maximum thrust.
Second-In the construction of an arched structure, substantially as described,
the employment of a templet formed to a curve which is identical with the line of
an arch, the length of each and every abscissa whereof is equal to the estimated static
load which is to come upon such point in the arch when built, multiplied by the
leverage of such load and divided by the estimated maximum thrust of the said
arch.
Third-The method, substantially as hereinbefore described, of building arches
or domes without the use of a centre or support.
Fourth In the construction of arches, substantially as described, the employ-
ment of a movable templet as, and for the purpose, set forth.
Fifth The movable templet provided with platform steps.
Sixth-The combination, with the arch H, constructed substantially as set forth,
of the reinforcing ribs J.
Seventh-An arch constructed for a given ununiform load having but two
resultants, which coincide with the line of curvilinear pressure, in combination with
the said ununiform load.
Eighth An arch in combination with an unevenly distributed static load, the
.said arch having but two resultants; coinciding with the line of curvilinear

page_text: pressure, the abscissa at any point of the said arch being equal to the gravity
pressure or static load multiplied by the leverage thereof and divided by the
curvilinear pressure or estimated maximum thrust as set forth.
In witness whereof I the said William Robert Lake have hereunto set my
hand and seal this Thirtieth day of January in the year of our Lord one
thousand eight hundred and eighty three.
W. ROBT. LAKE (L.S.)