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English Pages 166 [164] Year 1893
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^
jttv
*s. per sq. in.
=
COHESIVE CONSTRUCTION. No. 4872, June lbs.
34
6,
59
1887, in plaster-of-Paris, 2,450
per sq.
=
in.
GENERAL FORMULA FOR SEGMENTAL ARCHES.
TC
(i;4) ^
^ LS — 8 r _
(The explanation given
distributed load.
(i) for
of these formulas will
be
later.)
L = Load
in
pounds including material. (L is X span and X load in
always 12" in length lbs.
per superficial foot, including material.)
R = Resistance in middle of arch, or T X C. C = Coefficient for compression = 2,060 breaking load. per per C' = Coefficient tension = 300 breaking per C" = Coefficient transverse = 90
lbs.
sq. in.,
sq. in.,
lbs.
load.
lbs.
sq. in.,
breaking load.
T = Area of
cross-section, in superficial inches,
in the middle of the arch.
be 12 12
X
will
always
= thickness.)
r^ Rise of S
(T
thickness, or area represented by
= Span in
arch in
feet.
feet.
We
use the formula (i) to get the thickness necessary at the centre of the arch with a distributed load, including the weight of the (55)
arch
itself.
After that
we
find the line of the
60
COHESIVE CONSTRUCTION.
extrados of the arch in a graphical
manner,
derived from the formula given by Dejardin for tracing the equilibrium profile of the extrados for the vaults, giving the section of the arch in
the skewbacks, or base of the arch on each side. (56)
This formula
is (2)
V == X COS.
a
and
is
the general one for any semicircular or seg-
mental arch, but making for the
first
case
M'
Fig
a
Fig.
16,
= 60° for the
segment
of an arch
;
18.
a equals
the degrees corresponding to one-third of the
segment
in
extrados
ON
X c
is
which
V is
the radius vector of the
(Fig. 18).
the radius of the intrados
O
the thickness in the centre, or
M.
A
B.
a the angle that any radius makes with the vertical O A.
.
COHESIVE CONSTRUCTION.
Hence
the complete formula
L (
\
8rx
^^^
61
is
S 12
C
which represents the thickness
of the
centre
of the arch in inches, or
area of
i
foot arch in length, 12
and ^4^
87^X
Y^C +
(V
-
(X
+
B A)
for the
thickness of the spring of the arch in inches. (57) The graphic procedure is as follows (Fig. i8):
Take the thickness T, or, say A B, and lay O H equal to it draw H' H parallel with the chord O K, draw O N through the point M, which is one-third of B M' Lay off M N O G, N gives us one point off
;
—
=
of the curve of the extrados.
N M
is
safely put
which
See Fig.
the weakest part of the arch, the
same thickness
at
gives us the third point that
As we can
i8.
the spring is
necessary
to trace our curve. (58)
With the same formula we can
find the
thickness of the arch necessary for a single load at the middle, but
We
now come
any point
of
making 4^ instead to the
an arch.
problem
of 8
;-.
of a load
on
;;
COHESIVE CONSTRUCTION.
62 (59)
The remark has been made
that these
kind of arches cannot be used for a moving or concentrated load. We know that if an arch is with the condition that the curve of pres-
built
sure
is
inside of the middle third, the arch
starting with this
we apply
for finding the thickness in the centre,
and we
trace graphically the curve of pressure as in Fig. 19, as
if
safe
is
the general formula
shown
the load was resting on point
1 1. That gives a lower line of pressure than any other point for one side of the arch, and when the load is on a corresponding position on the opposite side, the same curve reversed gives us the whole arch.
Now it is necessary to put half the thickness given by the formula on either side of the line of
pressure O, O', O".
lines
?
X, X', X", Z,
Z',
Fig.
19,
forming the
Z", that represents the
total thickness of the arch.
With
we
this, as
have said, we have the thickness of the arch necessary for the lowest line of pressure required for any position of the load.
When
the load
is
on
1 1,
the pressure
sing through the imaginary lines
when
1 1
is
C and
pas1 1
e
on 10, the pressure is passing through the imaginary lines 10 a, 10 e' etc. Consequently it is inside of this area, between the level of the floor and the line of lowest the load
is
,
COHESIVE CONSTRUCTION.
63
lines of the arch (Fig. 19)
where the curve of pressure rests that means that if we fill up solid, or in ;
way
such a
as to take the
place of solid materials, as, for example, in tubular
we
girders,
are sure that
the curve of pressure will
always
the
be inside of
arch.
We
(59)
a
way
as
say, or in such
take the place
to
of solid materials, because in practice
avoid
it
better to
is
enormous
the
weight of this unnecessary mass of material, and, besides, to avoid the condensation
mass lates
such
that
a
accumuceiling, by
of material in
the
building the lower part of
the arch X', X", X'", and Z',
Z", Z'",
and over that
building bridges at a cient
distance
two feet •«"«