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TREATISE ON BRACING
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P R F F A C E.
IT is now several years since the description of bracing,
chiefly dwelt upon in the following pages, suggested itself
to the author; and he was surprised to find that a method
of such simplicity and evident excellence should have been
employed in only a few unimportant instances, and, in the
majority of these, in a mixed or not very evident way.”
* The first example that is likely to occur to the reader is that of the span-
drils of Southwark Bridge, but the arch, from its construction and depth of mate-
rial, is quite independent of additional bracing, and the wse of the lozenges of the
spandrils is merely to connect the arch with the roadway: thus, the Sunderland
bridge, which is of nearly the same span, and of greater rise, and composed of
voussoirs of less depth and inferior character for rigidity, is, nevertheless, with-
out spandril bracing.
A design somewhat similar to the above, represented by figure 19, is some-
times used for engine-shafts, &c., and its bracing power is, in such cases, brought
into action.
Another, and, though insignificant, seemingly a very clear application of the
method, is that employed to brace the transverse pipes between the main-chains
of the Menai Bridge.
But the most decided case of its employment, in its simplest form, which
the author has met with, is that figured in the March part of “The Builder” of
the present year (page 100, vol. viii.), it is the wrought-iron roof over the Stras-
tº burg Railway Station at Paris.
Town's Lattice Bridge depends, no doubt, for its strength upon the method,
but presents an example of it in a very compound state, and which might have
been adopted without a knowledge of the efficiency of the method in its simplest
form. (Figure 22.)
Y,
ſs

iv. I’REFACE.
This naturally led him to investigate its qualifications, and
it was his expectation to have had opportunities, in the
exercise of his profession, of making practical use and
exemplification of the results. Such opportunities, however,
not having as yet occurred, he adopts the less congenial"
means of the pen to place his investigations in such a posi-
tion that they may be capable of becoming useful. In these
investigations the simpler geometrical processes have been
preferred to those of a more abstruse character.
The first draft of the work was strictly confined to the
particular kind of bracing above referred to, which may be
appropriately named the double acting or triangular method;
but, upon after consideration, the plan has been somewhat
extended, as it was judged that a few moderate additions
would render the work more useful, and better fitted, though
very imperfectly, to supply an important desideratum in the
library of the engineer.
For dignifying the little work with the title of “A
Treatise on Bracing,”
it is offered as an explanation, that,
though devoted almost exclusively to one method, the re-
sults obtained may be easily applied to all other varieties,
with the exception of plate-bracing. This last kind, indeed,
is of so peculiar a nature, that a completely different process
must be employed to elucidate its action : probably, the
results of experiment must be principally depended upon to
give us correct ideas of its capabilities.
As bridges are the structures in which bracing is most
extensively employed, much will be said of them ; but, as
the intention is merely to explain its application, those parts
PREFACE V
which either influence, or are influenced by, the character of
the bracing, alone come strictly within the scope of this
treatise. In a few instances the author may have gone be-
yond this intention, in the belief that the space so employed
would not be deemed by the reader unprofitable.
In conclusion, the writer begs to say, that in launching
his little bark upon the seas of publicity, he is not expec-
tant of its passing triumphantly through the testing billows
of criticism; these may disclose many deficiencies in its
structure, with some of which, however, its architect is
already acquainted, but the first vessel built upon a new
principle, is always, in Some degree, experimental, aiding
man to construct in the future a perfect one of the same
class. Seldom, upon a new subject, was a volume ever
published that had not to undergo many changes and en-
largements ere it ranked as a standard.
EDINBURGH, November 1850.
^s.
0 0 N T E N T S.
(ſimplit 3ilitmutturi mi ºriting.
DEFINITIONs—ELEMENTARY BRACED ForMs—FAULTY STRUCTURES
#mrt fiti—Glºuti mi jià.
CHAPTER. I.
ON THE Dou BLE ACTING OF TRIANGULAR METHOD of BRACING-
Development—Efficiency—Advantages
CHAPTER II.
APPLICATION of THE TRIANGULAR METHOD To VARIous STRUC-
TURES.—New view of the action of a Bracing—Division of
Structures into Four Classes—Table of Symbols used
CHAPTER III.
THE PRESSURES THAT MAY ACT IN STRUCTUREs of CLAss FIRST.-
Process of Investigation adopted—Examples—Formulae for
the Maximum Strains in the Braces, and in the Main-Beams
CHAPTER IV.
THE PRESSURES THAT MAY ACT IN CLASS SECOND.—Strains in the
Braces, &c.—Transition into Class Third–Triangular Forms
CHAPTER, W.
THE PRESSURES THAT MAY ACT IN CLASS THIRD.—Proper Form
of Arch—Principle of Investigation—Examples—Means of
facilitating its Application—General Formulae—Table of
Formulae for the Strengths of the Braces—Pressures in the
Arch and Tie—A Form having more uniform Strains—Effects
from the Line of Pressures not lying in the Arch—The Arch
with External Bracing tº & iº
5
7
11
15
19
viii CONTENTS.
CHAPTER WI.
THE PRESSUREs TILAT MAY ACT IN CLAss FourTII
CHAPTER VII.
ON THE INCLINATION or THE BRACEs, &c.—The Proper degree
of Inclination of the Braces—Poly-systemed Bracings—The
Proper Distance between the Supported Points — Plate-
Bracing, its varieties very numerous—Comparison of the
Methods, &c.
jºurt àrium–ſulfruttiuſ.
CHAPTER, I.
WARIOUS BRACINGS REQUIRED IN DIFFERENT STRUCTURES
CHAPTER II.
CoNSTRUCTION AND APPLICATION of THE BRACEs-General Re-
marks—Wooden Braces—Ditto with Iron Sockets—Iron
Braces, Cast and Wrought
CHAPTER III.
CoNSTRUCTION of oth ER PARTs connecTED WITH THE BRACING-
Laminated Wooden Arches and Beams—Springings of Wooden
Arches—Iron Structures—Springings of Iron Arches
CHAPTER IV.
SUPPLEMENTARY.-The First Class though weaker, has many
advantages over the others—Triangular Tube Bridge—In-
ternal generally superior to External Bracing for an Arch—
Strength of Wooden Braces—The whole length of a Structure
must be Braced—Designs compounded of the First and Third
Classes—The Strains in the Longitudinals of Class First may
be spread more uniformly over them—Further observations on
the relation between the Arch and the Line of Pressures—
The Line of Tensions in the Inverted arch.
28
35
40
44
ADDENDA ET CORRIGENDA.
Fig. 12. In diagram E both lines indicating the upper horizontal beam should be
dotted ones.
Fig. 50. Letter e should be c. A.
Fig. 56. Insert letter h above the centre.
Fig. 57. The numbers 1.0, 1.3, 1.7, and 2.15, are a little out of their proper places,
they refer to the slant pieces. See the corresponding parts of figure 56.
Fig. 58. Mark the lower a thus a'.
Fig. 65. The letter c should have been placed at the crown of the dotted curve.
Page 14, line 30. For the formulae there given, substitute—
(; + sº) w tan, 0 = (N–1) w tan. 9.
And add, at the bottom of the page—
“And if n' demote the number of the bay, counted from either end, we have
the following formulae for the strains in the longitudinals (the load
being on the upper level, and the terminal braces being struts):—
n! (N–n') tan. 9, w, for the compressions in the upper beam ; and
{n' (N–n') — # + n’ tan. 6, w, for the tensile strains in the lower beam.
When the load is at the lower level (and the terminal braces are struts), the
strains will be as above, except that, for the lower beam, each will
be 3 w, tan. 4 less.
Page 31, &c. Cancel lines 6-28 of page 31, and lines 2-5 of page 32. Cancel
also the paragraph of ten lines headed, “When s is very short,” pages
37-8.
(The remarks to be cancelled are only applicable to the parts near the piers
of structures of the third class.)
7 South Gray Street, Edinburgh.
TREATISE ON BRACING.
INTRODUCTORY CHAPTER,
DEFINITIONS.
IT is necessary to state here the sense in which several
terms will be employed; as it has been found expedient to
restrict the meaning in most cases, thus rendering the words
so much the more precise.
When a construction is so framed, that, under the action
of any arrangement of forces, the angles cannot alter, nor
therefore the shape, it is described as a completely braced, or
simply a braced, structure.
When a fabric is adapted to retain its shape under the
action of a force, applied in one direction only, it is said to
be only partially braced, or braced sufficiently for the cir-
cumstances; as in the case of the simple roof, Figure 2.
When the external figure of the structure is not, of
itself, such as to ensure the retention of its form (i.e. when
not a triangle), the addition to effect this is called the
bracing, and each piece is named a strut, a tie, or a brace,
according to the office it performs.
When a part is only required and adapted to resist com-
pression, it is called a STRUT;* and in the diagrams of this
work, when it can conveniently be done, will be represented
by two lines, as shewn by a, Figure 1.
* In speaking of the parts of a structure, if it be desired to distinguish those
struts or ties used for bracing it, from others, they may be designated as bracing
struts or bracing ties.
2 DEFINITIONS.
When a part is only required and adapted to act against
a force of tension, it is called a tie-beam, tie-piece, tie-bar,
or simply a TIE ; and will be represented by a single strong
line, as b.
When a part is required and adapted to act both as a
strut and a tie, it is called a double-acting piece, complete
brace, or simply a BRACE; and will be represented by two
lines, as for a strut, with the addition of a central stronger
one, as for a tie; this is shewn by c.
When the parts of a braced figure are so long as to be
in danger of bending, and this is prevented by certain addi-
tions, these are called, collectively, the trussing ; and the
beams so stiffened are said to be trussed; the whole figure
so treated, when it does not require bracing, is called a truss.
Thus, Figure 2 represents a simple roof truss; the additional
parts cf.fd, and fö, constitute the trussing, and they truss or
stiffen the pieces a e, ec, and a c, respectively.
The term pressure, consistently with the practice of
eminent writers, is used to signify statical force, either of a
compressive or tensive character.
BRACED FORMS.
rº
As a foundation from which to commence, we assume
the following propositions:—
PROP. I. In a triangle, an angle cannot increase or
diminish, without the opposite side also increasing or dimi-
nishing.
PROP. II. When the angles of a figure are unchange-
able, the shape is unchangeable, and, therefore, the figure
is completely braced.
The converse of each of these is also true.
ELEMENTARY BRACED FORMS. 3
A triangular structure (Figure 3), having sides that are
unchangeable (by I. and II.), is a completely braced form.
If a quadralateral figure (Figure 4) alter its shape (by
neg. conv. of II.), the angles alter, and, as the sum of the
angles must be equal to four right angles, they cannot all
increase or all diminish, therefore (considering the diagonals
as third sides of triangles, by I.) the diagonals must one
increase and the other decrease. And hence—
Consequently all these

r Hi Y
In order that A diagonal must diminish: D-3 5 forms are completely
any change of º º t seale |braced, as each does not
form may take A diagonal must increase: 6 5
place in a ºpermit of an effect tak-
quadralateral & e tº º
figure, increase or diminish— ).
both must change:— }
Any diagonal must either ing place which would
Fºr
( necessarily result from
º
ja change of figure.
The above forms are the elements of all thoroughly
braced structures. They are of an insulated character. But
there are certain modifications of them which depend upon
external resistances; these may be either partially or com-
pletely braced, according as their parts are capable of acting
in one or two capacities. The figures 8–11 represent some
of these; figure 11 is an extreme of figure 8, one of the legs
becoming horizontal—it cannot transmit any vertical pres-
sure to its abutment.
There are several forms, frequently employed, which are
not sufficiently braced, though very considerable stiffness
may be produced in them, but only by the use of excessively
thick parts. To save time in describing these and their
defects, we have given three series of diagrams in Figures
12, 13, and 14. The first of each, marked A, is a sketch of
the objectionable structure; the second, B, represents, in an
exaggerated manner, the effects resulting from a weight
4 INSUFFICIENT FORMS.
placed excentrically; and the remaining diagrams illustrate
methods of remedying the defects.
The last series, Figure 14, requires particular consider-
ation. The first modification of it, C, is too often employed
as the bracing of bridges, &c. When the weight is placed
excentrically, it is supported on the principle of Figure 11.
It will be readily seen, therefore, that when the bridge is
long in proportion to its depth, and nearly one half of its
length, be, is loaded, a great force will be brought against
the point a, and a very moderate motion of a, or shortening
of ca, will produce a considerable depression of the portion
at d. Hence this cannot be admitted, as a sufficiently
braced form, for any but short spans in conjunction with
considerable depth of framing. The second modification,
D, on the contrary, is an excellent form; in it the principle
is quite different; for whereas in the form can excentric
weight is supported wholly (theoretically speaking) by one
abutment, in this, each abutment contributes support, in a
degree inversely proportional to the distance of the weight
from it. Of course, any of the completely efficient bracings,
Figures 5, 6, and 7, for a four-sided compartment, may be
introduced. The design E is suited to obviate the particular
change shewn in B, but it is not braced: other contortions
may take place, from the longitudinals approaching nearer
to one another.
£urt fit fits.
THEORY AND DATA.
OHAPTER I.
ON THE DOUBLE ACTING OR TRIANGULAR METHOD OF
BRACING.3%
To develop this method, let us take the comprehensive
case of two parallel beams requiring such bracing between
them as shall prevent any change of form in the structure,
that might result from the action of any forces whatever
thereon.
To satisfy this requisite of permanency of shape, the
most efficient methods at present in general use, are, in
carpentry, those shewn by Figures 15 and 16; and, in iron-
work those of Figures 17 and 18. Figure 19 gives the best
design for cast-iron beams, &c., when of one piece.
These are, no doubt, sufficiently strong and effective as
bracings; but there is, Figure 19 excepted, far more both
of material and workmanship used than is absolutely neces-
sary, not only causing needless expense directly, but often
also indirectly, as the excess of weight, above what is im-
peratively required, may necessitate the provision of a great
* This method is founded on the simple braced element shewn by Figure 7.
I}ut the insertion of the chapter is warranted by the further elucidation it affords
of this most important subject. ^,
6 DOUBLE-ACTING OR TRIANGULAR BRACING.
deal of additional strength in the structure, simply to bear
it; and this, in the case of bridges, would have the effect of
considerably diminishing the length of span attainable. .
Let us now consider the method of bracing shewn by
Figure 15. Here, as the forces may vary in their direction,
all the diagonals require to be strutted; but suppose that
the ends of one of these diagonal struts (say a b) are so
jointed that the strut may be enabled to act also as a tie-
bar, such being the case, the other strut (c d) of the paral-
lelogram (a cºd) is no longer required, for when it would
have been of use, the modified one (a b), by acting as a tie,
is now a sufficient substitute. Giving, then, to one strut of
each parallelogram, the power to act either as strut or tie,
and doing away with the remaining one, the braced struc-
ture becomes what is seen in Figure 20 or 21, according to
the order of procedure. In the figures, the upright pieces
are also represented as fitted with the double acting joints,
so that the iron tie-bars, or straps, seen in Figure 15, would
be superfluous. In Figure 20, the upright pieces are un-
necessary in the bracing, as the point a is fixed in position by
the braces a b and a e : doing away, therefore, with the up-
right pieces of Figure 20, our braced framing becomes what
is shewn by Figure 22, which is the method of bracing
advocated; and Figure 21 is evidently a variety of it.
Figure 22 might have been deduced from Figure 21, by
merely altering the inclinations of the braces, producing
an arrangement that would, in most situations, be more
efficient.
Figure 22, then, is an effective bracing, capable of re-
sisting a change of form under any disposition of the pres-
sures to which it can be subjected; and yet, compared with
figure 15 (which is the most generally used design), how
simple and light. The weight of material may be roughly
THE ACTION OF THE BRACING. 7
estimated at one-third (the beams braced, of course, not
being included); and as to workmanship, the saving in
that may even be greater, and susceptible of much more
perfect execution; and the parts may be made removable,
for repairs, with the greatest ease and safety, which is a
matter of primary importance when wood is the material
employed; at the same time, the liability to decay and
failure is considerably lessened by the nature of the joints
to be used, and by their diminished number.
0 H A P T E R II.
APPLICATION OF THE TRIANGULAR BRACING TO
WARIOUS STRUCTURES.
THE following view of the action of a bracing will often
be found useful, in investigating the pressures brought into
action:—
In all braced structures, though two longitudinal pieces,
at least, are necessary, yet one only of these may be con-
sidered as the primary member; all the other parts being
looked upon as the means of bracing. Thus, for example,
in Figure 23, the beam a b requires bracing; arrange, then,
as shewn in the figure, a series of pairs of braces along it;
each of the triangles a d n, n e c, &c., is thoroughly braced,
and may, therefore, be represented as solid plates. Now,
the application of weights to the beam a b will give it the
curved appearance represented in Figure 24, and this change
is necessarily accompanied by the approachment of the apices
d, e,f, g, of the braced elements to one another, and the
dotted lines, which were vertical in Figure 23, become here
8 THE ACTION OF THE BRACING.
radii of curvature. If we prevent these changes, by the
insertion of strut-pieces between the apices, as seen in
Figures 25, 26, &c., a braced structure will be produced.
And when the braces are applied beneath, or when the
upper beam is considered as the braced member, the inter-
vals between the apices, on the addition of weights, would
tend to increase, which must therefore be counteracted by
the application of ties, as seen in Figure 27. w
The action of bracing upon flea-ible arches, when viewed
in this manner, becomes very evident; and, as this is one
of its most important applications, it deserves particular
consideration. In Figure 28, P a Q represents a flexible
arch. When a load is placed on the crown, that, of course,
is flattened and depressed, and the haunches are raised
and further bent; and when the weights are placed on the
haunches, opposite motions and effects ensue. Now, the
arch may be broken in two ways, either from being so
unequally loaded that a part of it is ruptured by being too
much bent, or, when the arch is so braced as to be retained
in its proper form, it is crushed by the pressures transmitted
along it: the load required to destroy the arch in the latter
manner is very much greater than is sufficient in the former;
hence the great utility of bracing the flexible arch.
In investigating the application of bracing to a great
variety of structures, the work will be much simplified by
arranging them into different classes, though one class may,
in reality, be but the extreme of another. Adopting this
plan, and commencing with the simplest forms—
THE FIRST CLASS that claims our attention is that
exemplified by Figure 29, being two parallel beams with
CLASSIFICATION OF STRUCTURES. 9
the bracing between them, the whole acting as a girder.
This, with the triangular bracing, is an excellent design
for bridges, and also for the longitudinal framings of roofs,
steamboats, &c., and for scaffolding and other framings.
THE SECOND CLASS comprehends all structures that act
on the girder principle, but which have not the longitudinal
beams parallel. The designs shewn by Figures 30 to 34
are included in this class. In the first division of these, viz.,
30–32, though less material may be required than by figure
29, that advantage will sometimes be more than counter-
balanced by the increased difficulty of execution. The
second division, comprehending Figures 33 and 34, con-
sists of structures not requiring bracing, as that is defined
at page 1; the external form, without addition, being a
braced one; but, when large, they require trussing, and
though they do not, therefore, come strictly within the pro-
vince of this treatise, yet, as the simplest form of trussing
that can be introduced is exactly that of the simplest bracing,
with this difference, that the parts are required to act in
only one capacity, we may be excused for introducing them.
Figure 33 is a design now much used for iron roofs: the
reverse of this, shewn by Figure 34, is an excellent-design
for a rigid suspension bridge. The author has never seen the
latter employed with more than three supported points. Of
course, to secure rigidity, the sides of the triangle must be
maintained straight lines.
THE THIRD CLASS.—The structures included in this
class are those which derive their strength from a single
braced arch. Figures 35 to 42 are examples. These may
be divided into three sub-classes—first, those having the
bracing arranged along the intrados, but which have not
10 SYMBOLS EMPLOYED.
the arches tied, as 35 and 36; second, those which are
braced internally, and have the arches tied, as 37 and 38;
and, third, those which have the bracing along the extrados,
as 39 to 42.
THE FOURTH CLASS consists of arrangements of two
or more parallel, or nearly parallel, flexible arches, each
contributing its share of support, and the whole, by the
introduction of bracing between them, acting, in a great
measure, as a single deep rigid arch—Figures 43 and 44.
TABLE I.
TABLE OF SYMBOLS USED IN THE FOLLOWING PAGES.
w = Weight at each loaded point of the span.
N = Total number of weights or points.
W = N w = Total load. ---
s = Distance between the weights or points.
S = N s = Total length or span.
P = Pressure acting in a given line, a b–see Figure 45.
W = The vertical element or component of pressure
P, = b c.
H = The horizontal element of pressure P, z= a c.
0 = z a b c = The angle at which the given line, a b,
is inclined to the vertical, b c.
% = Z b a c = The angle at which the given line, a b,
is inclined to the horizontal line, a c.
P = V sec. 9 = H sec. p = y HäTV2)
W = P + sec. 9 = P. sin. p = P cos. 9= Hitan. p.
H = P + sec. p = P sin. 0 = V tan. 9.
INVESTIGATION OF CLASS I. 11
GHAPTER III.
ON THE PRESSURES THAT MAY BE BROUGHT INTO
ACTION IN CLASS FIRST.
THE best process for arriving at the amount of the pressures
that may be brought into action in the several parts, will
be, first, to get the effects produced by each portion of the
load separately, and then, to find the maximum strains that
can be produced by any arrangement of the weights; for
the strains are not all the greatest when the whole length
of the structure is loaded, as, by removing the weights from
some of the points, the strains in some of the braces may,
it will be seen, be increased. As an example of the method
pursued in calculating for the results given in the follow-
ing Table, let us investigate the forces arising from one
weight = w, placed at the point 4 of Figure 46. As the
support contributed by each pier (from a property of the
lever) must be inversely as the distance of the weight from
it, the pier Q will support +, w, while the other pier, P,
supports the remaining portion of the weight, is w. The
vertical pressure, ## w, has therefore to be transmitted,
through the bracing, to the pier Q; the real pressure, P,
induced in the brace b will therefore be =# w. Sec. Z.j 5B,
=## w. Sec. 9, and by means of that brace, the vertical
element, W = +g w, is impressed on the point k. This point
must be borne up by the brace w, which is consequently
stretched with the strain of +} w. Sec. 9; this is next trans-
mitted through the brace c, and so on, till it is finally im-
pressed at Q. The other portion of the weight, in like
12 EXAMPLE OF THE METHOD OF INVESTIGATION.
manner, causes strains of Hºs w. Sec. 9, in all the braces from
the point 4 to the pier P, of compression in those which
are inclined in the same way as brace s, and of tension in
those inclined in the opposite direction. Proceeding in this
manner, for each weight, we obtain the quantities marked
against each brace in Figure 47, those written on the left
side of the brace denote the strains of compression, and those
on the right the strains of extension, which take place in
the brace when the bridge is uniformly loaded, each number
being the co-efficient of #. Sec. 9; of course when two oppo-
site pressures would act in a brace, the resulting strain, in
the brace, is their difference, thus in brace 5 j, the strain,
for a uniform load, is (+ 25–16). sec. 9= } sec. 9, which
is + or a compressive strain; in this way the quantities in
column second of Table II. are found. But referring again
to the brace 5 j, we see that the tensive forces are produced
by the weights 1, 2, 3, and 4; consequently, if these be re-
moved, the full compressive strain, of 25% sec. 9, will act in
the brace, which is the maximum strain of compression that
can possibly be induced in it by any disposition of the
weights: on the other hand, if all the weights, excepting
those at 1, 2, 3, and 4, be removed, there will be no com-
pression induced in the brace 5.j, and therefore the total
tensive strain of 16; sec. 8, will come into play; and this
is the maximum of tensive strain that can take place in it:
in this way the third and fifth columns of the table are cal-
culated.
The maximum strains are those we have to attend to in
fixing the strengths of the braces and of their joints: thus,
in brace w it is a tensive strain we have chiefly to guard
against, whereas the required tensive strength of the brace b is
very moderate, but it must be capable of withstanding four
times as great a compression.
FORMULAE FOR THE MAXIMUM STRAINS. 13
TABLE II.
1 2 3. 4 5 6
Strains when Maximum com. Weights re- Maximum Ten-i Weights removed
Brace all the pressive Strain moved to pro- sive Strain by to gº; the
weights are by uneven duce the Strains uneven Load- train in
OI!. ading. in Column 3. ing. Column 5.
62 + 81 + 81 |All on. Never any.
d + 63 + 64 1 — 1 |All but 1.
c + 45 —H 49 |1, 2. — 4 |All but 1, 2.
b | + 27 + 36 |1, 2, 3. — 9 |l',2',3',4', 5,4.
a + 9 + 25 |1, 2, 3, 4. — 16 |1', 2',3',4', 5.
& — 9 + 16 |1', 2',3',4, 5. — 25 |1, 2, 3, 4.
* | – 27 + 9 |1,2'3',4', 5,4. — 36 |1, 2, 3.
2) — 45 + 4 |All but 2, 1. — 49 |l, 2.
^p — 63 + 1 |All but 1. — 64 |1.


The numbers in the Table are the co-efficients of ... sec. 9.
= #. sec. 6, here, as N = 9.
From the foregoing and similar data, the following for-
mulae are deduced, the load being applied above:-
The maximum compression of the last brace (1 Q, figs.
46 and 47) is - Gº” w. Sec. 9 = }; w. Sec. 9 = } sec. 9.
The maximum tension in m 1, and compression in 2 m
= };" w. Sec. 9.
The maximum tension in l 2, and compression in 3 l
= *** w. Sec. 9.
And putting n to represent the number of the pair of
braces as counted from the pier, along the lower beam, the
(N—n)”
2N
When N is even, the formula for the central pair of
braces will be (N-S)'w, see, 0 = }w. sec. 4 = Y sec. 9.
-TN-
general formula will be w. Sec. 9.
14 STRAINS EN THE LONGITUDINAL BEAMS.
Carrying the calculation beyond the centre, we obtain
the maxima of the other strains that can act in the braces.
We see, then, that the extreme braces require to be four
times as strong against compression as is necessary for the
central ones, when N is even, and very nearly four times
when N is odd and large. And starting with the value
sec. 9, at the centre (supposing N even), the tensive strains
in the braces parallel to a, Figure 46, gradually diminish
as they approach Q, where they vanish, but increase as they
approach P, where the last amounts to (N-1); sec. 9; and
in these braces the compressive strains are also = }. sec. 9,
at the centre, but gradually increase as they approach Q,
where they amount to ¥ sec. 9, and diminish as they ap-
proach P, where the compression in the last is only ; sec. 9.
The strains in the braces parallel to s, of course, diminish
and increase in the opposite directions.
N.B.-These maxima are obtained on the supposition
that the whole value of w may be removed from any point,
but, as the weight of the bridge itself is included in W, and
cannot be altered in its distribution, the effect of uneven
loading, as given by the above formulae, will be rather exag-
gerated, particularly towards the centre of the span.
The maximum strains of the longitudinal beams take
place at the centres of their lengths, where they amount to
; :=; W, D being the depth of the framing. This amount
of strain is compressive in the upper, and tensive in the
lower beam. The strains diminish towards the ends, and at)
the last bays, they amount to G+*#) w tan. 9=(N–3)
(w tan, gof compression in the upper beam, and to * tan. 9 of
*~..., ---.” * © tº tº tº
tension in the lower, 6 being the inclination of the braces
g *. .* f " . . ; \
with the vertical, c. O sºv oſ cº-exº~ * Q_ſc{ ~~
& g \
A c \! ºf ~ P. \ KY
INVESTIGATION OF CLASS II. 15
NoTE,--Turning to figure 47, we observe that, when the weights are placed
on the upper beam, the maximum strains producible in any two braces which
meet at the lower beam, have equal vertical elements; this necessarily results
from the lower beam being horizontal: consequently, in the form of bracing
shewn by figure 21, the same equality of the vertical elements must also exist.
If the load were placed on the lower beam, the foregoing observation would
apply to each pair of braces which join at the upper beam. The formulae, &c.,
which have been given, are calculated on the supposition that the load is applied
at the upper beam, and they would therefore require to be slightly modified to
suit the case of a suspended load.
CHAPTER IV.
ON THE PRESSURES THAT MAY BE BROUGHT INTO
ACTION IN CLASS SECOND.
As a representative of this class, take figure 49. Here, the
distance between the points of support being uniform through-
out the span, the angle 0 for the braces is variable. ,
The maximum compression of the extreme braces, a c
and b l, takes place when all the weights are on, and, as
half W is upheld by each, the strain will be = }; sec. 9.
But the next braces will not be strained with so great a
tension as would take place in a bridge of class first with
the same depth, c n : being relieved from a portion of their
duty of imposing the whole of (N-1) #. on the points c and l,
by the inclined action of the upper beam. This will be
made more evident by the following explanation of Figures
50 and 51 : in these the depths at c, and the values of s,
are the same, and also, therefore, 9 for the braces. Let a g
represent in each figure the upward resistance of the pier,
= }, then will a c represent the compression induced in the
brace a c ; resolving a c into the directions, b c and c d, of
the other brace and of the upper beam, the tension produced
in the brace, b c, is=c bin Figure 50, but only=ce in Figure
16 CLASSES I. AND II. COMPAREI).
51. Further, we observe that the compressions in the upper
beams of the Figures 50 and 51, are a b and a e respectively;
the tension of the lower beam, beneath c, is the same in
each, and = g c.
The strains at the centres of the longitudinal beams in
Figure 49, are the same as for a bridge of class first having
the same depth as it at the mid-span. Let us compare the
strains at the extremity of the latter, represented by Figure
52, with the same of 49, represented by Figure 51. Make
a g the same in each, and let it represent #, then a c will
measure the compression in the extreme brace; that in 52
is the least. The tension of the lower beams is measured
by g c in each figure, and is greatest in 51. Resolving a c
into the directions c d and b c, cf is the resulting compres-
sion in the upper beam in each figure, and it is evidently
the least in 52. The resulting tension in the brace c b, is
represented by c e in each figure: the comparative amounts
are shewn in Figure 51 by c e and c v, and c e diminishes as
the angle a c d increases, vaaishing as that becomes = 180°.
The advantages, then, of a design of class second, over
a similar one of class first having the same depth at the
middle, may be stated as reducing the strains in all the
braces, except the extreme one at each end (which is rather
more strained, becoming, in Some degree, a continuation of
the upper beam)—the central ones, however, are little
altered; also as reducing the lengths of the braces towards
the extremities; and as rendering the longitudinal beams
more uniformly strained throughout their lengths, being
equally so with class first at the centre, but more so than in
it at the extremities. In fact, as will afterwards be seen,
this class is a step towards some forms of class third; it
becomes identical with them when a c, Figure 49, forms a
continuation of the curve of the upper beam (see Figures
TRUSSED TRIANGULAR FORMS. 17
37 and 38), or when a pressure is applied at c, in the man-
ner of an abutment, in the direction c d, as indicated by the
dotted arrow. (See Figure 35).
*º-º-º-º-º-º-º-º-º-º-º-mºmºm
Though the triangular forms, shewn by Figures 33 and
34, as stated in chapter second, are not braced, and conse-
quently do not demand our attention, yet, on account of their
utility, and the reasons there given, we will introduce them
here.
The weight at point 2, Figure 53, must cause equal
strains in 1 a. and 3 a, as the angles 1 a 2 and 3 a 2 are equal,
and 1 a. and a c form a straight line; so that half the weight
2 is transferred to point 3, and along with the weight at 3
is impressed on b, the total weight, *, at b is distributed,
so that ; is carried to 1, and *.* to point 4 (see note at end
of chapter), the total weight accumulated at point 4 is =3 w,
one w being received from each of the points b and d, and
this is impressed on c, and thence transmitted, equally, to 1
and 1'.
Now the rule, derived from a property of the lever, that
a weight at 2 is upheld by the piers, 1 and 1’, in the pro-
portion of 5 to 1, holds good here; for, though in the first
effect of the weight at a only ; is carried to pier 1, yet at each
succeeding point another fraction will be transmitted to 1,
by the rod c 1, until we arrive at the lowest point c, where
the remnant of #, amounting to #, will be equally divided
between 1 and 1’, so that will be the only portion of the
weight, w at point 2, which the pier 1' can receive.
After what has been said, little explanation of the
Figures 55 and 56 will be required. In these, for the sake
of having simpler numbers, the length of the side A C is
made a multiple of the central depth h C. The numbers
C
18 TRIANGULAR STRUCTURES.
(co-efficients of w) attached to the parts, are calculated on
the foregoing principles, and give the real strains or values
of P, which take place from a uniform loading. In this
form of structure, the maximum strains are produced when
all the weights are on. Of course, when, to serve for a roof
or other purpose, the design is inverted, as Figure 57, the
strains, with very trifling and evident exceptions,” are all
the same in degree, but of opposite character.
NOTE.-In Figure 54, a weight at b,-w, =ab, is distri-
buted thus:–It is resolved into b c and b d , and further
resolving b d, into a horizontal and a vertical pressure, ba:
is the latter, which is, therefore, the portion of w, or a b,
that the rod b h receives, and resolving b c similarly, we
get ce, = a ac, as the vertical, which represents the portion of
w borne by the rod b g. The triangle b a d is evidently
similar to the triangle a a f, ... ba: : a d- a a af, and alter-
nately, b a . a. a = d a a f. But triangle dafis similar to
g bh, therefore the lines a a and b a, drawn from correspond-
ing angles, and at right angles with the opposite sides,
must divide these sides similarly, therefore day: af-ga: ah,
and consequently ba: : a, a = g a a h. The fact that the
weight at b is distributed to h and g in the proportion of
g a a h might have been anticipated from the property of
the lever.
g a a h = tan g b a tan h b a b d =# w Sec. a b h.
And b c = }; w sec. ab g; the resulting horizontal pressures
are, of course, the same for each direction, and are da; and
be = b d sin a b h.
& 2.
* Arising from the weights being on the slant sides, instead of being attached
to AB, which would be the true arrangement in Figure 56 inverted, and conse-
quently relieving each perpendicular piece of 1 w.
PROPER FORM OF ARCH. 19
C H A PTE R W.
ON THE PRESSURES THAT MAY BE BROUGHT INTO
ACTION IN CLASS THIRD.
WE now come to consider the case of a braced flexible
arch. The form given to the arch should, theoretically, be
the curve of stability for the fully loaded arch (in the inves-
tigations it is assumed to be this, so that, in the internally
braced arch, no action will be induced in the bracing when
all the weights are on). This curve, in general, will be
found to lie between a catenary and a parabola; but, as the
quantity of the curve generally used is not great, a circular
arc may be adopted, as it very nearly coincides for almost
120° with the theoretical figure: When an arch of a greater
number of degrees is required, and the structure is of a light
character, it is generally advisable to use the parabolic form.
But in a braced arch we may depart considerably from the
correct curve without much evil: some of the braces might
require to be made stronger, and a trifling variation would
take place in the strains of some of the other parts.
When the load is uniformly distributed on the arch,
there is no bracing required, and consequently (if the line
of pressures remains in the arch”) there is no action induced
in the bracing. When the load is suspended from the arch
by means of the braces, of course there are tensive strains
produced in them.
As, in the investigations, the values of w and s are con-
stant, and the weights of the arch and other parts are not
considered, the curve of equilibrium will be a parabola, or
* The curious properties which occasion this restriction will be found men-
tioned at page 27.
20 INTRADOSAL BRACING.
rather will be polygonal, having its angles arranged in a
parabola.
THE ARCH WITH INTERNAL BRACING.
The following fundamental principle of investigation is
capable of very general application: it might have been
introduced in Chapter III., but the method there adopted is
a simple one deducible from it for that particular case, viz.,
where the longitudinal members are all horizontal. From
a law of the lever, we know that the vertical elements of
the pressures upon the abutments, resulting from a weight,
w, placed at the point c of Figure 59, must be inversely as
the horizontal distances of c from them; here, therefore, the
pier P will bear .75 w, while the pier Q supports the remain-
ing portion, .25 w. The vertical pressure of the weight w
at c must be resolved into the directions cl” and c Q, the por-
tions of the bridge, Pb c s and Q fet, acting as instruments
for the transmission of the respective pressures to the piers
P and Q; in acting as such, let us trace the strains produced
in the parts. The pressure in c B will be resolved into the
directions c b and cs, and the vertical elements, or the values
of V of these two pressures, since they both act downwards,
must, added together, equal .75 w; the pressure in c s is, at
s, resolved into pressures in s P and bs; that in b s is a ten-
sive strain, and must, since s P is horizontal, contain the
same vertical element as the pressure in c S. The resultant
of the two strains, in c l, and b s, conveyed to b, must have the
direction b P, and possess a vertical element = .75 w, for the
resultant of the whole, acting in the line c B, passes through
P, and as the pressure s P is horizontal and passes through
P, the remaining component pressure, b B, must also pass
through P, and must contain the whole vertical element.
The pressure b P, is resolved into ba and bºr, and b r is re-
INTRADOSAL BRACING. 21
solved into r P and a r, the resultant of the compression in
ba, and the tension in a r, must have the direction a P, and
contain a value of W = .75 w. The final resultant of the
pressure in a P, and the horizontal pressures, arising at the
points s and r, must have the direction c P, and its vertical
element = .75 w. Returning to the point c ; the pressure
in c Q is resolved into the directions c d and c t, and since
cd acts upwards, the vertical element of c t must eaceed V
of c d by .25 w, as that is the downward vertical element of
the resultant to the pier Q: the pressure in ct is resolved
into tG) and dt. The resultant of the two pressures conveyed
to d, by the compression in c d and the tension in dt, must
have the direction d0, and contain a vertical element=.25w.
The resultant in d Q is resolved into de and du ; but we
need not proceed further in tracing the transmission of the
pressures to Q; sufficient has been said to explain the
principle. We see that (when the lower longitudinal mem-
ber is horizontal) the resultants at the various points of the
arch, at either side of the weight, are directed to the pier or
spring of the arch at that side; and all those at one side of
the weight have the same vertical element or value of W,
which is eventually imposed on the pier at that side.
We will now give, in considerable detail, the investiga-
tion of a simple example having five points or weights,
shewn by figure 58. Let a weight be placed at the point
a, and represented by the line w a = 1, this will be resolved
in the directions of the piers P and Q, and the amount of
the pressures produced in these directions will be = a fand
a m = a o, the vertical elements of these will, as given by
the lever rule, be = 0.9 and 0.1 or = fºn and n m, the hori-
zontal element or value of H is the same for each, and
= n a. The resultant a o is resolved into a l and a t , a t =
a'g into g w and r g; and so on throughout.
22
COMPRESSIVE AND TENSIVE MAXIMA
Table III, gives the results for the various parts, from
the action of one excentric weight at a. -
TABLE III.
7 art-i. Real Strains or
Yºº viº=
w a = vertical weight at a 1.000 1,000
a f = resultant to pier P . .900 I,260
m a = a o = result. to pier Q .100 .900
or a = value of H of each resultant *- .895
Pressure in a b (measured by a l) * .300 .582
,, in a g (measured by a t) .400 .563
9 / . s e * .580
, in g b, tensive — .400 — .4364
Resultant b Q .100 .333
b c .050 ,255
b h. .150 .165
h v ºmºm- .125
h c — .150 — .161
Resultant c Q .100 .216
c d | .034 .173
c .066 ,071
7 w * ,055
d — .066 — ,0726
Resultant d Q ...100 .168
de .075 .1455
dj .025 .0273
j w smºs- .036
je — .025 — .035
Resultant e Q ..100 .140
Q k tº- .099
Q y .100 ..I.00
Horizontal thrust = n a = y2 = 9 u + h v -H iw -- ja + Qk.
Ditto =-895 = .580 + .125 + .055 –H .036 -- .099. ,

Tables, similar to this, being constructed, one for each
weight, it is easy to discover by what arrangement of the
OF A BRACE ARE EQUAL. 23
weights a maximum strain in any part is produced, and the
amount of that maximum. Thus taking the brace b h, we
have produced in it—
TABLE IV.”
Real sº or Values Values of V. º
By weight a + .165 —H .150 \
, , b | + .495 }=+,660 + .450 } = + .600
,, , c | — .3674 * — .334
,, , d — .2200 }=-w — .200 } = — .600
e — .0726 — .066

25 22
From Table IV. it is evident that when the weights a
and b alone are on, there is a maximum compression pro-
duced in the brace b h, and when these are removed, and the
other weights c, d, and e, are placed on the arch, a maximum
strain of tension takes place of exactly the same amount.
When all the weights are on (agreeably with a rule pre-
viously given, since the contour of the arch is such as to
render it equilibrated under the whole load), the compres-
sive and tensive strains neutralise each other, and no effect
is produced in the brace.
The maacimum compression for any brace is equal to the
maazimum tension for the same: hence, calculating the com-
pression is sufficient; as the result is the measure of the
necessary strength to be given to the brace, to resist both
compression and extension.
At any point of the arch (c, figure 59) we can easily
* These values are taken from tables similar to table third, but it would be
useless to give them here. The sign + indicates a compressive, and — a tensive
strain.
*--~-
**J`-l
º
&
§§+}
S$&
Sº,
t
s
;
j
$- *>3-.
$§i&
24 FORMULAE FOR THE MAXIMUM
find, by the lever, rule, the value of V for each resultant
from that point to one of the piers (Q), caused by each of
the weights (c, b, a) situated at, and to the other side of
the point (it is .25, .15, and .05); now, the greatest com-
pound resultant that can act in the same line, will evidently
have the value of V in it equal to the sum of the values of
V above mentioned (thus, .25 + .15 + .05 = .45), and its
amount will be V multiplied by the secant of its angle with
the vertical (P = .45. Sec. 9). Having thus the compound
resultant, in value and direction, all we have to do in order
to find the maximum strain (= 1.097) in the first brace (ct)
from the point, is to resolve it into the directions of the
chord of the arch (c d) and of the brace. But the following
formulae offer very ready means of calculating the necessary
strengths of the braces, as they give at once the value of
V for the maximum strain in each. -
The formula for the end pairs of braces (as a r, rb, and
jz, 2 i, of figure 59) is ; (N-1) = ** w = W.
The general formula, putting n to denote the number of
the pair as counted from the pier, is
º (N-n) = **w-v. -
The second column of the following table gives the for-
mula for each of the first six pairs of any bridge. The last
two columns contain the results for the case of figure 59,
wherein N = 10.
The values of V for the braces are not altered by a
change of the rise or span of the arch, while N and w
remain the same : hence having calculated the vertical ele-
ments for the given value of N (as is done in figures, 60
and 61), it only remains to multiply each by the particular
value of w. Sec. 9, in order to obtain the strains, or values
of P.
STRAINS IN THE BRACEs, ETC. 25
TABLE W.
ºf ºv, lºgº,'º'
the Pair of a max. in the ... ºig. 5: ...Fig. 59, wherein the
Braces. Braces. * * |Rise to span=1:475.
1 (N-1) a r = 0.702 op
1. =#N* * 0.45 w {: b = 0.496 w
2 (N–2) b s = 0.882 w
2. *** * | 0.80 w ł. .I.
3. 3 (N–3). 1.0 c - t = 1.097 w
==N* * ** |} . Ii.
4 (N–4) (s d w = 1.240 w
4. -à N- 20 1.20 w w e = 1.235 w
5. 5 (N–5). 1.25 e v = 1.286 w
===N* * * |is fei.286,
6. 6 (N-6), a 1.20 w f w = 1.235 w
2 NT w g = 1.240 w
n. * (N·m). wo
2 N

Each weight produces compression throughout the arch,
therefore the compression will be a maximum when all the
weights are on. The vertical element of the compression at
the springing or foot will evidently be the half load; and
for any point, it will be half the load above the level of that
point. Of course, the real compression at any point in the
arch is = the vertical element multiplied by the secant of 9
there.
It is scarcely necessary to mention the well-known fact
that the horizontal element is, for an equilibrating load,
uniform throughout;* and that the horizontal thrust for
* When the load is not an equilibrating one, the braces come into action, and
transmit parts of the horizontal pressures to the piers, but if a vertical section be
26 ADVANTAGES OF FIGURE 49.
the wholly loaded arch, is the measure of the maximum
compression of the arch at the crown.
The strain in the lower beam, when it acts as a tie to
the arch, is greatest when all the weights are on; the ten-
sion is then uniform throughout the length, and equal to
the horizontal thrust of the arch, or to the compression in
the arch at the crown.
When the lower beam does not act as a tie” to the arch,
but simply as an adjunct to the bracing, it undergoes,
principally, compressive strains, and these are rendered a
maximum for one half length of the beam, when the weights
above that half are removed. Under such a loading, the
compressive pressures in the lower beam increase as they
approach the pier at the unloaded side, where the compres-
sion may amount to about one half the value of the hori-
zontal thrust for the wholly loaded arch. (See NOTE.)
SCHOLIUM.–Reviewing what has been said, we observe
that in class first the strains in the braces increase towards
the extremities, and that there, for the same brace, the ten-
sive and compressive maxima are very different in value;
whereas in the form treated of above (figures 59, &c.), the
central braces are those most strained, and every brace has
its compressive and tensive maxima equal. Now, there is
an intermediate form between these two, and its parts will
made at any part of the span through the arch, bracing, &c., then the sum of the
horizontal elements, in the various parts of the section, is a constant quantity for
any one loading. From this property, the line, which in the subsequent pages is
called the line of pressures, has been named, by Mr. W. H. Barlow, the line of
equal horizontal thrust; it is the resultant of all the pressures acting in the struc-
ture, with the exception, when the arch is a tied one, of the strains occasioned, by
the horizontal thrust of the arch, in the tie beam.
* A tied arch of this form is evidently the extreme of the change of class first
(figure 46) into figure 49 of class second. It might, indeed, without impropriety,
have been included as a variety of the latter class.
^
EXTRADOSAL BRACING. 27
undergo strains of a medium character: sueh is figure 49 of
the second class; and it is possible so to modify that form,
as to secure a very equable distribution of the strains, both
in the braces and in the longitudinal members.
NOTE.-In the form figure 59, with the lower beam not
acting as a tie—When all the loading is on, the line of
pressures, of course, corresponds with the line of the arch.
But, when the load is only placed on a space at the crown,
the line of pressures resulting is shewn by the dotted line
in figure, 62. Where the line rises above the arch, the
lower beam, immediately beneath that part, acts negatively
or as a tie; and when the line falls within the arch, the
lower beam, at that part of the span, acts positively, or as
a strut, conveying part of the compression to the abutment.
These observations upon the relation between the position
of the line of pressures, with regard to the arch, and the
nature of the pressures induced in the lower beam, apply
universally. Figures 63 and 64 give an idea of the posi-
tions the line may assume, when half the length of the arch
is loaded.
THE ARCH WITH EXTERNAL BRACING.
In this variety (exemplified by figure 65) the line of
pressures for the fully loaded arch does not necessarily coin-
cide with the arch; for it has been demonstrated by Mr.
W. H. Barlow (in an excellent and valuable paper read at
the I.C.E., July 1847, a report of which is given in the
Civil Engineer and Architects’ Journal, vol. x., p. 211),
that the real curve of action is that which produces the
least horizontal thrust, namely that curve (parabola here)
which rises to the neighbourhood of the extrados, which is
here the upper beam, as shewn by the dotted line.*
* The curve or line of pressures may extend beyond the structure, when that
28 ARCHES WITH
Nevertheless, it will be proper to give sufficient stability
to the abutments, and strength to the arch and bracing, to
withstand the pressures produced when the central line of
action coincides with the arch, as such a coincidence would
take place if the upper part, c, should become injured or
materially shortened. If the part c become slightly short-
ened (from the great compression), the crown of the line of
pressures will descend proportionately.
Unlike the foregoing case of figure 37, of internal
bracing, the values of V for the braces, when N is given,
are not Constant.
(; H A P T E R W I.
ON THE PRESSURES THAT MAY BE BROUGHT INTO
ACTION IN CLASS FOURTH.
THE introduction of bracing between the parallel, or nearly
parallel, flexible arches of this class, renders the whole
structure in its action a single rigid arch. The dotted line
in figure 66 shews, as stated in last chapter, the line of
pressures (W being all on) as long as the structure remains
perfectly without change; but the centre of the upper arch
and the ends of the lower one, from their compressibility,
will become somewhat shortened, and from these changes
the line of pressures will sink at the crown, and not descend
So far at the springings, becoming, in fact, more nearly
identical with the midway line; it approaches this as the
pressures become more nearly equal to the ultimate strength
is braced; but if the parabola rise above the upper beam, the action will partake,
in a proportionate degree, of the nature of that of the girder—the arch beneath
the part being then in a state of tension.
INTERMEDIAL BRACING. 29
of the compound arch; and if one arch be stronger than the
other, the line will lie nearer to it, as it recedes from an
arch in proportion to the compressibleness thereof.
Such are the deductions in theory; but to ensure these,
we must have a structure far more perfectly framed than is
practicable. Let us see what the shortcomings of practice
may entail.
If, at the time of the erection of the structure, or from
after causes, such as decay or other damage, the abutments,
at the different footings, be not all applied with the same
accuracy, disturbing causes will be introduced; and when
the faultiness of an abutment is so great that the arch there
is relieved from pressure, it becomes only a part of the
bracing of the other arch, through the spring of which the
line of pressures must necessarily then pass. From this it
is evidently proper that one arch should be made sufficiently
strong to carry the whole load, unaided by the other as
such.
(; H A PTE R W II.
WE will now investigate the effect of inclination, or the
value of 9, of the braces, on economy. Let figures 67 and
68 represent two structures, having the same span and depth
of framing, and loaded to the same amount. Referring
back to Chapter III., we find that the strains in, and there-
fore, the necessary strengths of, the braces, are proportion-
ate to W. Sec. 6; and as W is here constant, the requisite
strengths are proportionate to sec. 9. The length of a brace in
each figure, since the depths are alike, may be taken to repre-
sent sec. 9. Let the values of sec. 9 be equal to b and b.
30 INCLINATION OF THE BRACES.
The requisite sections* of the braces may be assumed
to be proportional with the strains, and therefore also with
sec. 9, or as b : b. The total lengths of the bracings are
evidently as 16 b : 8b.
Now, the total weights of the bracings are to one another
in the same proportion as the total lengths multiplied by
the respective sections, or as 16 b x b : 8 b x b = 16 b : 8 bº
= weight of bracing of figure 67 : weight of bracing of
figure 68. Or, as a general rule, the weight of a bracing is
proportional with 2 N. (sec. 8)", but N=s+. which is pro-
portionate to #7. Substituting this for N, we have the
weight of the bracings proportionate to #:
mum when 0 = 45°.
, which is a mini-
Though the minimum of weight of
TABLE VI. bracing is arrived at, when 0 is made =
45°, yet we may vary this angle con-
Vº *: siderably without much increase of the
tan. 4. weight, as will be seen from Table VI.
O But when, with 9=45°, the dis-
10 292.40 e
15 200.00 tances between the supported points are

20 155.60 very much too great, the difficulty is not
; ; to be overcome by lessening the value
35 | 106.40 of 0, but by means of the intermediate
: lº props shewn in figure 69, or by using
50 iois; two or more systems or ranges of bracings,
55 | 106.40 as shewn in figures 70, 86, and 87. We
60 115.50 ; c. x. •
65 130.55 need only make this remark, at present,
70 155.60 upon the theory of systems, that as the
75 200.00 value of w is reduced as much as the
80 292.40 • *
number of systems is increased, the total
weight of the bracing (supposing the section to represent
* Of course this is not strictly correct with regard to compressive strains (and
therefore 4 will be a very little less than deduced). But, while the number of
LEAST LENGTH OF S. • 31
the strength of a brace), remains the same as for a single
system; but the quantity of workmanship will be much
greater.
THE DISTANCE BETWEEN THE POINTS OF SUPPORT.
In order to determine the proper distance between the
points of support afforded directly by the bracing, or the
best value of s, we require to know, not only the amount of
W, but also the greatest weight that may, at any time, be
concentrated within a small length of the span (as from the
wheel of a locomotive steam-engine); let this latter quantity
be represented by e. The weight e, then, at certain moments
during its passage over the bridge, will be wholly borne by
one point of support, but that point is only calculated to
carry w =\, this must not, therefore, be less than e, that
is, N must not be more than Y; or j = the smallest distance
allowable between the points of support =%, where f=\:
When the weights and lengths are given in tons and feet
respectively, fexpresses the rate of loading, in tons, per foot
of span. As an illustrative example, let f= 1 ton (dead
weight test) and e = 7 tons (the pressure from an engine-
wheel, allowing for the effect of motion), then the smallest
value of s by formula =}=7 feet.
If, instead of a double system, in large structures, the
design with props, figure 69, be used, the weight e that
may be concentrated at one point is = 2 w. But, gene-
rally, in this kind of bracing the distance between the
points need not be made so short that w will not be as great
systems remains the same, the disparity will not be important. And in the case
of an increase in the number of systems, the more significant error that might
take place, will be moderated in practice by the braces being rivetted, bolted, or
tremailed together at the crossings.
32, - PLATE BRACING.
as any probable amount of e, and, further, an amount e
will often be found at the same time imposed on each of
two consecutive points; therefore, the above quality of the
props cannot be admitted as of much practical importance.
When the number of systems is considerable, as from
four to six or more, we have the design known as the
Lattice Bridge, a particular construction of which has been
patented in America by Mr. Town.
Let the number of systems be supposed to become ex-
ceedingly numerous, and the braces at the intersections
imagined to be united (see footnote, page 31), the bracing
may still retain considerable lateral thickness; but, in the
longitudinal direction, each brace will be exceedingly thin,
retaining, however, considerable strength against compres-
sion, as it is stiffened, in the direction it most requires it,
by being connected at all its crossings with the other braces.
Now, if we further imagine the whole bracing to undergo a
lateral contraction, and the parts to become solidified, the
resulting bracing will be in the form of a sheet or plate,
and may be appropriately named
PLATE BRACING.
But it would be difficult to define the limits of this
description of bracing. In one direction it includes the
simple beam; in another, the cylindrical tube, and even
hollow spherical forms cannot be excepted as varieties of
it. Among the more obvious cases of its employment, we
may mention the feather on a casting, the sides of a box,
the hull of a ship, and the floor, walls, and roof of the
majestic tube of Stephenson and his coadjutors. It is the
bracing employed in nature, and that in all its varieties.
In the plate and shell form, we see it in the beak of a bird,
FLATE BRACING. 33
in the bones of the cranium, and in animals with an ex-
termal skeleton, as the crab, &c.; as a feather strengthening
a plate, we view it, most conspicuously, in the keel of the
breast-bone of the falcon tribe, and in the spine of the
scapula of the land mammalia; and in the form of the
tube, we witness it universally adapted to the wants of the
animal and vegetable kingdoms;–the bones of the ex-
tremities, the quills of the wing, the humbler grasses, and
the lofty bamboo, may be cited as familiar examples. We
shall, however, confine ourselves to the consideration of it
in its simplest form, and endeavour to compare it with the
brace or linear method.
We have observed above, that, to derive the plate from
the linear bracing, the lateral width must shrink into the
thickness of the resulting plate; consequently it has com-
paratively little lateral stiffness, and would readily warp or
pucker wherever the compressive pressures were consider-
able; fortunately, however, this can be remedied with a
moderate addition of material, in the form of featherings,
and of easy application in the case of the material (malleable
iron) which would generally be employed for this kind of
bracing. *-
At those parts where the tensive strains considerably
predominate, we may expect to find the plate stronger than
the linear form, from the following considerations:—Let
figure 71 represent a bracing of numerous series, and let a
tensive strain act in a b : now the strain is only borne by
the parts lying in the direction of a b, but when the systems
become infinite and the whole solid, there are, as it were,
two plates—one resulting from the braces lying in one
direction, and the other from those in the opposite one; and
each of these plates serves to bear the strain in a b. And,
ID *
34 PLATE BRACING.
further, the direction of a strain is not confined to a particu-
lar angle.
The comparison, then, may be enunciated thus, “The
plate bracing possesses much greater strength against ten-
sive strains, but less against compressions, than the linear.”
And, as a corollary, we must infer that, generally, the plate
form is better suited for shallow than for deep bracings.
The general distribution of material in the plate bracing
must be subject to the laws regulating it in the other
methods; thus, in the first class, the plate must be made
more substantial towards the extremities; and in figure 37
of the third class, the central part must receive the greater
share of material. Figures 72 and 73 give an idea of the
strains in class first; the lines shew the general action of
the pressures, and the degree of shade produced is an index
to the strength required. But these diagrams are, of course,
made much simpler than would really occur, and they are
only adapted to illustrate the action of a uniform loading.
In figure 72, at the pier P, the perpendicular lines indicate
that the pressures impressed at a by the lines of suspension,
shewn in figure 73, are supported from beneath; but at pier
Q these are supposed to be received by a higher part of the
pier from the edge b, as is done by means of the ball-rollers
in the Conway and Britannia tubes.
part fit śtrum.
CONSTRUCTION.
(JHAPTER I.
THE WARIOUS BRACINGS REQUIRED IN
DIFFERENT STRUCTURES.
BESIDES the longitudinal vertical bracings, upon which
depends its strength to support the loading, a bridge re-
quires other bracings, and to the consideration of these we
now proceed.
Let figure 74 represent the skeleton of a bridge, com-
posed of two vertical frames; that these may act with effect
they must be retained in straight lines, but there is the
pressure of the wind, &c., tending to bend them laterally;
this, therefore, must be sedulously guarded against: let us
add, then, a series of horizontal braces arranged between
the two upper beams, as in figure 75. The upper beams
are now rendered permanent in their position, and a road-
way may be carried along them; the lower beams will be
less steady, they may be moved by a lateral blast, but, as
they undergo a tensive strain, they are in a state of stable
equilibrium, and consequently return to their places. This
wriggling motion would, however, very soon injure the
structure, by loosening the joints of the vertical bracings to
the upper beams: such a design is, therefore, inadmissible
36 METHODS OF HORIZONTAL BRACING.
for a permanent structure; for such, the lower beams must
be retained perpendicularly beneath the upper ones, which
may be effected, either by the introduction of bracing be-
tween them, similar to that used for the upper ones, or by
the application of oblique transverse braces, as shown in
figures 76 and 77. The first of these methods will be
sufficient, without addition, for bridges having, moderate
spans compared with their widths; while the second may
be trusted to in any case, but, of course, in it the horizontal
bracing employed in the upper part will require to be of
double strength, as it must resist the whole pressure of the
wind. The common practice is to employ both of these
methods in the same structure, and in long spans, if it do
not interfere with the roadway; the practice is to be com-
mended. Figure 77.
The principal recommendation of the first method, when
used alone, is that it permits of the roadway being placed
at the level of the lower beams, which is of the utmost im-
portance in situations where the bridge spans over roads or
navigable waters, and sufficient height could not otherwise
be given for the passage of carriages or shipping beneath.
Further, it requires little expense for parapets, and offers
great facilities for the addition of a roof, or even of a second
roadway, along the upper beams. But particular attention
must be paid to carrying up the abutments, so that they
may offer sufficient resistance to the outsides of the upper
beams. gº
In the design, figure 38, when the rise of the arch is
very small, and the span moderate, and when the arches
have a considerable width of section, (which may be con-
veniently given to them if built up with deals, or constructed
as tubes with wrought iron, as figures 80 and 144), then a
single series of horizontal braces along the lower part may
PRACTICAL VALUE OF S. 37
be sufficient, and the roadway may be on that level. (This
also applies to class first, when the depth of the framing is
small). But when the structure does not comply with the
above conditions, it will be necessary to have a series of
braces between the arches also ; for, as they are in a highly
compressed state, a very moderate deviation from a horizon-
tally straight line might bring about their rupture. But,
horizontal bracing being introduced between the arches, the
road cannot be carried along the chord level, it must be
raised to the level of the crown of the arch,” and the design
then becomes that of figure 37, and the roadway will also
require bracing.
Most of the other structures of class third will be secure
with two lines of horizontal bracings, and of these figures
35 and 42 admit the roadway at the lower level. But
when figures 36 and 41 are of considerable span, it will
be prudent to bestow three series on them.
In structures belonging to class fourth, each arch will
require to be braced horizontally, and, generally, the road-
way also.
when s is/vKRY SHORT. J.- 4, . A
2.6.(A ºccle/2–
We have/seen, in chapter seventh of the previous part,
that, theoretically, there is a º the diminution of s in
as it is here supposed to be constructed. The
gion, howeyér, permits º: modification in
practice; thus,’assuming th t s is considerably too short,
* Unless some new arrangement be made in the design; as in the following
structures.—Figures 78 and 79, which are examples of the use of plate-bracing,
may be used when the rise is not great.
Figure 81 is an arrangement suitable for linear bracings, and there is less
waste in it than would at first sight appear. The inside arches are kept perpen-
dicularly over their chords by the transverse bracing, which occupies the central
compartment throughout the whole length of the bridge; and cross connecting


38 PLATE BRACING.
of e equally over an extent of the span not less than ; or so
as to reduce the pressure received by any one of \be poi ts
to w—we do away with the danger: to produce this spread-
ing of e we must interpose a very rigið body.
as in º Lattice Rridge, if we can spread the pressure
PLATE BRACING. $
;
The forms of bridges to which the plate bracing is more
especially adapted, are the tubular and the segmental.
|Figures 29, 35, 38, &c.
In the first there is this important source of economy,
that each side of the tube is useful in two capacities, first, as
the bracing against pressures acting in its plane; and
second, as a longitudinal member against pressures acting at
right angles to its face. In the form, figure 35, the strains
in the plate are more uniform throughout.
In the tubular form let us consider, for an instant, the
action under a load, confining ourselves, for the present, to
the consideration of the bottom plate; this is stretched, but
it will evidently be more so immediately below the vertical
bracings, and rupture thereof would commence by the tear-
ing of the edges. The same quantity of material collected
into tie-beams, and attached to the bottoms of the vertical
bracings, would offer a greater resistance, but then, on the
other hand, an additional horizontal bracing would become
necessary: similar reasoning applies, and with greater force,
pieces or braces, throughout as great a length at the crowns as can be permitted,
retain the outer arches parallel with the inner ones: in this design the rise must
necessarily be considerable.
Figures 82, 83, and 84, show varieties of another arrangement, it is a compound
of the first and third classes. The arch has the roadway suspended from it by
iron rods, and, in 84, it is stiffened by separate braced frames, placed one on
either side of it, bolts are passed through the whole wherever the arch crosses
a part of the frames; the rise is made sufficient for the roadway to pass beneath
the horizontal bracing between the upper beams.

PLATE BRACING. 39
to the top plate; and on the determination of which of these
arrangements is the most economical, -depends, in a great
measure, the question of the respective merits, for this case,
of the plate and linear bracings when constructed of wrought-
iron : for other materials the plate form of bracing is not
suited.*
The want of uniformity between the strains at the edges
and those at the middle line of the plate, when it acts as a
longitudinal member, may be obviated by giving it a slight
convexity outwards, as in figure 85. This may be done to
the top and bottom plates, as, though it weakens them as
bracings, sufficient strength, as such, will remain to coun-
teract any lateral disturbing influences. But this must be
looked upon rather as a réfinement in theory than a practical
improvement.
In the form of figure 38, the plate-bracing may be
modified so as to give lateral stability to the arch ; this is
shewn more particularly in the design figure 78, wherein
the plate is double; the deep featherings in figure 79 are
designed to effect the same object.
The featherings alluded to at page 33, may be arranged
on two plans;—first, in perpendicular pieces, as in the Britan-
nia tubes, and, second, radiating from each pier as do the
lines of compression in figure 72; in this last arrangement
they require to be of larger section to be equally effective,
but besides stiffening the plate they carry a portion of the
compressions to the piers. Perhaps, from various considera–
tions, a combination of the two plans would be the best, the
featherings being applied in the one manner to the inside,
and in the other way to the outside of the plate.
* Cast-iron may, indeed, be used for plate-bracing; and, since that form of
bracing demands comparatively little tensive strength in the material, probably
with good effect, at least where heaviness would not be a defect. Considerable
skill must, however, be exercised, when the bracing is composed of many parts.
40 TIII. BRACES.
CHAPTER II.
ON THE CONSTRUCTION AND APPLICATION
OF THE BRACES.
}
WHEN the bracing is poly-systemed, the braces may be
arranged on different plans; thus—All the braces of the
same system, as shown by figure 86, may be in the same
plane, and the systems so near that the braces touch, notch,
or halve into one another at the crossings. Or, as seen in
figure 87, all the braces lying in the same direction may be
in the same plane, and those inclined the contrary way, in
an adjacent plane.”
In designing the joints of the braces, the following ob-
servation on the action of the bracing is of such great im-
portance, that, though indicated in the previous part, we
again place it before the reader. That part of a pressure,
in a brace, which is resolvable at right-angles to the longi-
tudinal member, is, when that member is straight, wholly
transferred to the other brace; and when the longitudinal
beam is curved, the part transferred is slightly diminished
or increased, according as the curvature is concave or convex
towards the braces; when the longitudinal is compressed;
and when it is stretched, the reverse is the case. And the
only pressure resulting in the longitudinal is in the direc-
tion of its length.
WOODEN BRACES,
1. Those requiring little or no iron.—Braces of this
character would, generally, be used in those situations only
* Or, as a variety of this second arrangement, suitable to a very broad longi-
tudinal member, there may be several layers of parallel braces, inclined alternately
in opposite directions.
WOODEN BRACES. 41
where iron-work might be difficult to procure. The simplest
kind of jointing for them is that by means of trenails, as in
figure 88, or by dovetail notching, as in figures 89 and 90.
These methods are very suitable when the braces are thin,
or plank-shaped, as in Town's Lattice Bridge. But when a
brace is liable to undergo great compression, its section
should be nearly square or circular. For braces of a stouter
character, the figures 92 to 95 illustrate methods of jointing,
adapted to a longitudinal member constructed in a similar
way to that shewn by figure 95. Modifications of these
joints might be easily constructed, capable of correcting the
length of the brace; but such a property is unnecessary, as
there is no difficulty in the way of preparing the braces
with the requisite accuracy.
2. Those jointed with iron.—The most appropriate way
of applying wrought iron is in the form of straps, of which
several varieties are shewn by figures 96 to 101. But gen-
erally, the method, illustrated by the succeeding figures 102
to 117, of jointing the braces by means of CAST-IRON
SOCKETS, is recommended by the author. These sockets are
of two kinds, the first, including figures 102 to 109, requires
the end of the brace to be of a dovetail shape; the second,
shewn by figures 110 to 117, to be square-headed. The
first is best suited to braces formed of a wood that is easily
split, the second for those of a tougher material.
The disadvantages attending the braces with the dove-
tail joints are, that they require stronger sockets in order
to resist their wedge action; and that they might be sensi-
bly lengthened by the compressibility of the dovetails al-
lowing the ends to draw a little, unless they be wedged
very tightly in, which would increase the former evil.
The only remark, since the subject is so fully illustrated
by figures, that it is necessary to make upon the construc-
\
2
tion of the square-headed braces is this:—if the joint were
formed as in figure 116, there would be great danger of the
piece a being wrenched off by any slight wriggling motion
that might take place in the brace before the structure was
perfected; it is therefore constructed, as shown by figure 117,
so that there is no angle at b, which, by acting as the tooth
of a crowbar, could split off the piece a , but the slant of
b c must be slight, otherwise there would be the wedge
action objected to above.
Figures 109, 114, and 115, are forms of double sockets
for the case, mentioned in the beginning of the chapter, of
the braces of the same system being arranged in two planes,
see figure 87. And figures 102 to 106, and 110 to 112, are
forms of single sockets, which may sometimes be found use-
ful in transverse and other bracings. The other forms are
for the case of the braces of one system being all in the
same plane, as seen in 86.
BRACE SOCKETS,
The braces are retained in the sockets by short screw-
bolts, either with or without covers.
The head of the brace and the inside of the socket
should be coated with some preservative substance, such as
a heated mixture of tar and pitch.
One great advantage attending the use of the socket-
joint, and also participated in by most of the previous forms,
is that if a brace becomes injured, by decay or accident, it
can readily be removed, and replaced by a sound one.
The designs given are calculated for braces of moderate
dimensions.
IRON BRACES. ---
To braces of iron, the forms that may be given are very
numerous; but if we exclude those which do not admit of
being individually removed, the number will be very con-
IRON BRACES. 43.
siderably reduced; and the best forms of sockets, such as
figures 118 and 119, are liable to this objection.
1. Cast-iron braces may have various forms of section,
either feathered or tubular, but those of the latter kind are
not removable. The design 124 is an excellent one; it
possesses every desirable property, especially that of the
pair of braces interchanging their pressures without the in-
tervention of the longitudinal, the advantages of which
were pointed at in the second paragraph of this chapter.
But it cannot, without modification, be applied to a longi-
tudinal member of the tubular class, as the bolts must not
be inserted from the inside of the tube, for should one
break, it could not be easily replaced.
When tubular forms of the longitudinals are used,
whether of cast or wrought iron, they must be furnished
with sufficient featherings, single or double, to which the
bracings are to be attached: the featherings may be placed
either longitudinally or transversely.
Figures 125 and 126 show modifications of figure 124
when it is connected with a tube. But when braces are
applied to the sides of a longitudinal feather, the designs
shown by figures 132 and 134, which are suitable for the
arrangement shown in figure 87 of a poly-systemed bracing,
possess considerable advantages.
2. Wrought-iron braces may be of round or rectangular
section, or rolled with a section possessing greater strength
against compression, or they may be constructed of thin
plates, strengthened by angle iron, or by being formed into
pipes. Braces with solid ends may be jointed, as shewn in
figures 131 to 134, for cast ones; but those with thin and
broad ends will be most effectively joined to the main beams
by being rivetted to a projecting feather thereon (figures 135
and 136).
44 WOODEN BEAMS AND ARCHES
On the subject of the respective merits of cast and
wrought iron as the material of the bracing, the only general
remark we shall make is, that for very long braces the
former, and for very short ones, the latter kind of iron, is to
be preferred.*
(; H A P T E R III,
ON THE CONSTRUCTION OF OTHER PARTS OF BRIDGES.
As the contents of this chapter are in a great measure super-
erogatory, the meagreness of the account of such important
subjects needs no apology.
WOODEN STRUCTURES.
The longitudinal members.--When these are too long, or
too much curved to be formed of one piece, the best con-
struction is to build them up with planks, using compara-
tively thin ones when the part is to act as a tie, or is an arch
of moderate radius, and thicker planks or deals when the
part is straight and only subject to compression. In the
latter case, care must be taken to bring the butting ends
into forcible contact; indeed, it might even be advisable to
place in the joints, thin wedges of hard wood or iron, which
could, when needful, be tightened up to the proper degree.
But, in arches formed of thin planks, if care be taken in the
construction, the wedges may be dispensed with ; the pur-
* Wrought-iron may, however, be very properly employed for great lengths,
if formed into tubes, or otherwise braced, especially when lightness is a considera-
tion,
TORMED OF PIANKS. 45
ſº
pose will be effected, in a great measure, by the great fric-
tion, and this may be further increased by coating the planks
with a hot mixture of pitch and tar, or by placing in all the
joints a layer of strong brown paper dipped in boiling tar
(as used by Messrs. J. and B. Green in their laminated
timber arches); but the most important benefit arising from
this treatment is its preservative effect by the exclusion of
moisture.* The friction here mentioned is also of great im-
portance from its stiffening effect, rendering the arch, when
sufficient, equivalent to one of whole timber: it primarily
depends upon the compression induced by the screw-bolts,
straps, &c., that act upon the arch in the direction of its ra-
dius; these, therefore, should be numerous and well tightened
up: when brace-sockets are employed, the bolts connecting
them with the arch contribute to this end, and, of course,
effect a corresponding saving; these bolts, with the addition
of screw-bolts, such as shown in figures 138 to 140, would
constitute a good arrangement. Trenailing forms a cheap
additional means of stiffening the arch, but renders the
removal of a plank, in the event of its becoming decayed,
an almost impossible operation.
Figures 141 and 142 show an American method of
drawing the planks together: by the use of these dovetail-
gibs and wedges, and trenails, a stout arch may be formed
without the aid of iron.
For the case of figure 38, the arch should have a broad
section, and then it might be expedient to use more than
one system of braces; a section of such an arrangement àS
is meant is shown by figure 144 (see also footnote, page 40).
THE SPRINGING of a tied arch is a part that demands
great attention. The most perfect arrangement is that
* Or, the planks being dressed with the plane, white lead paint may be used,
as in building large masts.
46 THE SPRINGING OF THE ARCH.
which distributes the thrust of the arch equally over the
section of the tie. Hence, the method of figure 146 is per-
haps the best for an arch and tie of the nature shown: each
of the iron steps is fixed to its plank of the tie by numerous
screw-nails. Figures 148 and 149 are methods for the case
of a laminated arch with a tie beam of a broad and thin
section. - - - -- '
IRON STRUCTURES.
The lower beam should be constructed of wrought iron;
the upper one may be of either kind, and, generally speak-
ing, the tubular will be the best form that can be given to
it, for which form, wrought iron, as a material, is perhaps
the more preferable; but it is the more difficult of the two
to manage when the member is of the arched form, as the
whole must be put together on the ground and thence
raised with powerful tackle, or the rivetting must be per-
formed, with great inconvenience, on the centring.
The springing of a tied wrought-iron arch may be
formed, as shown in figure 150, by surrounding the end
with strong angle iron, which is then rivetted to the flat
end of the tie; or by inserting the foot of the arch into a
cast-iron socket, as is done with the woodon one in figure
148. A cast-iron arch may have its terminal pieces formed,
as in figure 151, with solid expanded soles, bolted down to
the tie. *
For untied arches a common cast-iron socket, with its
bearing against the abutment expanded and normal to the
arch, will generally be sufficient.
THE FIRST CLASS. 47
(; H A P T E R I W.
SUPPLEMENTARY.
IT is not here intended to institute a comparison of the
merits of the various designs, as that would demand a very
large space, depending so much as the excellence of a
structure does, upon its adaptation to the particular com-
bination of the requirements and circumstances of situation,
in which it may be assumed to be placed; such an extended
inquiry would be more appropriate to a Treatise on Bridges.
But we shall recapitulate a little, and add anything that
may suggest itself as closely connected with the subject of
the work, but which came not conveniently under any of
the foregoing heads.
Though the first class may be surpassed in strength by
those wherein the arch is used, it possesses many advan-
tages in other respects; foremost of these is its simplicity
offering easy construction: there is great uniformity of parts,
which, especially when cast-iron is employed, may be a
source of economy: it admits of a roadway at either level,
without interfering with the horizontal bracings, of which
only two series are necessary : it exerts no horizontal thrust:
and, as practised by Mr. Stephenson, with the tubular
bridges, it may be put together at Some convenient spot, and
thence removed in a nearly complete state to its permanent
site: or if erected there only a very simple and inexpensive
scaffolding will be necessary.
The form, of which figure 152 is a transverse section,
may be considered as a variety of the first class: in it the
48 INTRADOSAL BRACING PREFERABLE.
main braces require to be longer: the horizontal braces are
also necessarily longer, but this renders them more power-
ful: its advantages are, that by concentrating the upper
beams into one, their joint strength against compression
will be increased; that one series of horizontal braces is
sufficient, and that it may be roofed in at very little
expense.
Speaking generally, the internally braced arches are
superior in strength to those with external bracing : the ad-
vantage attends the deepening of the bracing towards the
mid-span.
When a structure of class fourth has the arches parallel,
it may possess the recommendation, when cast-iron is em-
ployed for the bracing, of uniformity of the castings.
The method of connecting the braces to the main beams
by means of trenailing, may appear very rude, but it is
capable of producing considerable strength; and, as it may
in some situations be a convenient one, the reader is referred,
if unacquainted with better data, to a table of the strengths
of trenails, given in the last edition of the Encyclopædia
Britannica, art. “Shipbuilding.”
We are not aware of any satisfactory experiments having
been made to determine that strength of timber which arises
from the resistance to severance by the sliding longitudi-
nally of a part of the material, upon which primarily de-
pends the strength of the socketed wooden braces against
a tensive strain. This strength or resistance to severance
will be increased in the dovetail form by the lateral com-
pression, but it may be lessened in the square-headed
variety, if care be not taken to obviate any splitting action,
that is, any force acting so as to wrench the fibres separate.
The following observations are made in order to shew
the applicability of the formulae to the forms of bracing
THE WHOLE LENGTH MUST BE BRACED. 49
seen in figures 15, 16 or 17, and 18. In figure 15, if a
weight be placed at c, the portion of it which must be im-
posed on the pier to the left side will not influence cb or ba,
as b a cannot act the part of a tie; it is therefore conveyed
by c d, da, &c.; and through these parts would, in like
manner, pass the strains arising from all the weights placed
to the right of c, which would ultimately arrive at the pier
to the left side. When all such weights are alone on, the
strains in cd and da are maxima. A similar course would
be pursued by the strains in the bracing of figure 21. The
effect of the portion of the weight at c, which is to be sup-
ported by the pier to the right, must flow through the strut,
and not through c b. Thus it is readily seen by what load-
ing the strain in a part is rendered a maximum, and also
the formula by which its amount will be given, after sub-
stituting the proper values for the symbols.
The bracing must be continued throughout the length
of the structure: this will be clearly seen by inspecting
figures 153 and 154, which shew changes that may take
place when only a part is braced.
An arch may be employed without having the bracing
arranged along its intrados or extrados; it may be stiffened
by being connected with a braced structure of class first.
Figures 82, 83, 84, 155, and 156, are illustrative of this
compound arrangement. In the arch, the radius of curva-
ture may be made to diminish more rapidly in approaching
the crown than if the curve were a parabola, as it is princi-
pally required to aid the class-first portion at the mid-span.
The maximum strains in the parts of the class-first portion
will be very much reduced by the addition of the arch.
Figure 156 is a good design for very large structures, as it
permits of the roadway being at the lower level, and, as then
the horizontal bracing between the upper beams would be
E
50 CHANGE FROM ARCH TO GIRDER ACTION.
sufficiently high : the arch and each of the straight longitu-
dinals will require horizontal bracing.
In class first a saving of half the scantling at the centre
of the main beams might be effected, if a pressure of eacactly
... were made to act tensively at each end of the upper
'beam, and compressively at the ends of the lower one; for
then one-half of the load would be borne up on the prin-
ciple of the girder supported at both ends, while the other
half would be upheld on the principle of a projecting beam
fixed at one end and unsupported at the other. The strains
in the longitudinals would be subject to a peculiar arrange-
ment. Their maxima would occur at the mid-span and at
the extremities, and would be = ..., or half what would
otherwise take place at the centres, and the strains at the
extremities would be tensive should that at the centre be
compressive, and vice versa ; and at the points midway be-
tween the mid-span and the extremities, where the strains
change their character, the longitudinals would of course be
free from all action. Advantage is taken, to some extent,
of this compound action in the Britannia Bridge, by the
method pursued in uniting the tubes over the central pier or "
to Wer.
The following observations are an extension of a prin-
ciple employed in the latter part of Chapter W., Part First.
Let figure 64; represent a portion of a voussoired arch, now,
if the line of pressures, from not being capable of conform-
ing to the curve of the arch, rise above it at a, as shewn
by the dotted line, the joint a b will open at b. But let us
suppose the joint at b to become solid, or that it is made
good there by means of bolting, the line may then rise
above a or any joint sufficiently united at its lower point;
and might also sink below a part of the arch, were the
upper points of the joints tied. This is the transition con-
INVERTED LINE OF PRESSURES. 51
dition between an arch and a girder action: it is that in
which a cast-iron (or other continuous rigid) arch may be
placed, by an uneven loading too severe for its depth of sec-
tion. When strengthened by bracing, the joints cannot
open, as the voussoirs are then prevented changing their
proper radial directions.
The strains in the several parts of figure 31, will be
exactly the same in amount, but of opposite character, with
those in the corresponding parts of figure 38. And gene-
rally, whatever has been said with regard to the comport-
ment of the line of pressures in the braced erect arch, applies,
in a reversed state, to the curve or line of tensions in the
braced inverted arch, or rigid suspension bridge.
And here the writer would almost be induced to specu-
late upon the possible extent to which bracing might be
employed, imagining structures wherein each brace would
offer in itself no contemptible example of braced construc-
tion, but he restrains himself, deeming the opportunity ob-
jectionable for such suggestions.
A PPEND I X.
WHEN the printer had made considerable progress, and
consequently when it would have been very inconvenient
to notice it at the proper places, I observed, at page 391 of
the Civil Engineer and Architect's Journal for the present
month, an account of the failure of a girder bridge in which
the triangular method of bracing was employed. I was not
previously aware of a patent having been granted for any
structure employing the method, however imperfectly.
The instance of failure of the structure of which I speak,
offers an opportunity of employing the results obtained in
Chapter III., Part First. I will, however, merely outline
the application, as it is a very simple matter:—The braces,
on approaching the piers, ought to increase in strength, as
seen in figure 47. In that figure 1 m will represent the
fractured brace: it had to undergo a very great tensive strain,
but was made of cast-iron of no greater section than the cen-
tral braces; it, and we cannot wonder at it, broke. The
great virtual loading impressed on the point 2, by the brace
!2, being thus left to be upheld by the transverse strength
of the upper longitudinal part, that part became fractured
near the said point. The remaining fracture probably re-
sulted from the direct action of the load on the portion left
projecting. It is not stated that any further breakage took
54 APPENDIX.
place; if none did, the circumstance may be explained by
the peculiar construction of the “girder.” The lower longi-
tudinal part, which was composed of wrought-iron chain-
work, though it could not give aid in supporting the point
m, while it continued quite horizontal, offered, after the
failure, sufficient to uphold the wreck—and the terminal
braces of what remained as a “girder” proved sufficient for
the diminished load and span.
The writer of the article suggests that the accident
would have been averted if the lower longitudinal part had
been of cast-iron, and all cast in one piece. The longitu-
dinals require very great strength at the mid-span, conse-
quently, if of cast-iron, the lower one would demand a very
large section there to resist the great tensive strain. But
let it not be supposed that I approve of the mixture of cast
and wrought iron as managed here. The danger of the
combination lies in this:—That the elasticity of the wrought-
iron tie may be so great as to allow the points (say j and k,
figure 47) of the braced elements to separate further than
the elasticity of the upper cast-iron longitudinal can permit
without fracture (at point 4). If, however, the upper longi-
tudinal member be jointed at each of the points, 2, 3, 4, 5,
&c., the latter danger will be provided against.
DECEMBER 1850.
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