Essay on the Theory and History of Cohesive Construction, Applied Especially to the Timbrel Vault

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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 •«"«