[Illustration: Fig. 68.]
181. Assume the following data for an example:—
Span 100 feet.
Rise 12 feet.
Panel 10 feet.
Weight per lineal ft. 2,500 lbs.
The tension on the lower, or the compression on the upper chord, will be
(250000 × 100)/96 = 260,417 lbs.
The manner of dimensioning the chord, and of splicing, is the same as
already described for Howe’s.
SUSPENSION RODS.
The first sets of rods, A B, A′ B′, must sustain the whole weight of the
bridge and load; which is 250,000 lbs. Each side 125,000 lbs.; and each
end set of rods 62,500 lbs.; and if each set has four rods, each rod
must support 15,625 lbs.
The rod being inclined, this amount is increased by the following
proportion:—
12 (height) to 15.8 (diagonal) as 15,625 to 20,573 lbs.
This is half-way between the tabular numbers for rods of 1¼ and 1⅜
inches in diameter; 1⅜ will therefore answer. The next set of rods must
be considered as supporting the whole load, less the two end panels, and
so on as already explained for Howe’s bridge. The manner of applying the
rods to the chords is shown in fig. 68 A. The bevel block should be
connected with the block at the foot of the post, so as to prevent
crushing the chord.
[Illustration: Fig. 68 A.]
COUNTER RODS.
As both top and bottom chords are always used in this bridge, the
counter rods have only the variable load on one panel to resist. The
action is, in amount, the same as that on the counter braces in Howe’s
bridge; but acts in a different direction, and in the other diagonal.
The weight of a passing load cannot be more than two thousand pounds per
lineal foot. The panel being ten feet long, the whole weight coming on
two sets of counter rods, (one set in each side truss,) is twenty
thousand pounds; or ten thousand pounds on each set; and if there are
put three rods in each set, we have 3,333 pounds per rod, which increase
for inclination as follows:—
12 : 15.8 :: 3333 : 4389 lbs.,
requiring a rod of three fourths inch diameter.
The posts in this structure, correspond to the braces in the Howe
bridge; only being vertical, they need not be so large.
182. The following table gives all the dimensions necessary for
proportioning this truss.
┌───────┬───────┬────────┬───────┬───────┬───────┬───────┬───────┐
│ Span. │ Rise. │Chords. │ End │C post.│ End │C rod. │Counter│
│ │ │ │ post. │ │ rod. │ │ rod. │
├───────┼───────┼────────┼───────┼───────┼───────┼───────┼───────┤
│ 50│ 10│2—8 × 10│ 5 × 5│ 4^2│ 2—1⅜│ 2—1│ 1—1½│
│ 75│ 12│2—8 × 10│ 6 × 6│ 5^2│ 2—1⅝│ 2—1│ 1—1½│
│ 100│ 15│3—8 × 10│ 7 × 7│ 6^2│ 2—1¾│ 2—1│ 2—1⅛│
│ 125│ 18│3—8 × 10│ 8 × 8│ 6^2│ 3—1⅞│ 3—1│ 2—1⅜│
│ 150│ 21│4—8 × 12│ 9 × 9│ 6^2│ 3—2⅛│ 3—1│ 3—1⅛│
│ 200│ 24│4—8 × 16│10 × 10│ 6^2│ 5—1⅞│ 5—1│ 3—1⅛│
└───────┴───────┴────────┴───────┴───────┴───────┴───────┴───────┘
And the following, the sizes of counter rods, for different panels.
Length of Height of Approximate
panel. panel. diagonal of Diameter of the rod.
panel.
One in a Two per Three per
set. set. set.
10 12 16 1⅝ 1⅛ ⅞
11 13 17 1⅝ 1¼ 1⅛
12 14 18 1¾ 1¼ 1⅛
13 15 20 1¾ 1¼ 1⅛
14 16 21 1⅞ 1⅜ 1⅛
15 18 23 1⅞ 1⅜ 1⅛
16 21 26 2 1⅜ 1⅛
18 25 27 2 1⅜ 1⅛
The advantage possessed by this bridge, over Howe’s plan, is that the
panel diagonals may be adjusted by the screws; by which control is had
over the form of the truss, and of the duty done by the several parts.
Change of form cannot be had by working upon verticals. Howe’s bridge
must be adjusted by wedging the braces and the counter braces.
183. The manner of drawing the bevel block in this bridge, is shown in
fig. 68 b. The proportions of the block depend upon the proportions of
the panel; and the dimensions, upon the size of washer used.
Let C C be the centre line of the post, and A B the chord. Let _o m_,
and _o n_, be the panel diagonals, and H and _y_, the length of the
washers.
[Illustration: Fig. 68 B.]
The depth of insertion of the block into the chord, depends upon the
horizontal strain upon it. In a span of one hundred and fifty feet, with
the rods at an angle of 50°, two inches have been found ample at the end
of the truss, and one half inch at centre.
From D, perpendicular to _m m_, lay off D E; equal to H, also from E, at
right angles to _n n_, make E E′ = _y_. From E′ draw the vertical E′ L.
The strain upon the rod _o m_, being represented by _o m_; and that upon
_o n_, by _o n_, the resultant is shown, both in direction and amount,
by _o_ V. It is not necessary that this should pass through the centre
of the post, as the excess of tension on _o m_, over that on _o n_, is
absorbed by the lower chord.
NOTE.—Screwing up truss bridges, is a more scientific operation than
is generally supposed. Many builders commence at each end, and lift
the bridge from the scaffolding. By this method the greater part of
the load is often borne by a few of the end sets of rods. The better
method is to begin at the centre and work both ways towards the
ends, being sure that each set of rods does its duty before the next
is touched. The lift to be made by each set of rods, should first be
calculated, and tested while screwing up, with the level.
LATTICE BRIDGES.
184. _Town’s lattice_, consists of a simple lattice-work of plank, 3 ×
12 inches, treenailed together at an angle of forty, forty-five, or
fifty degrees. It possesses great stiffness, without by any means having
the material disposed in the best manner. Such bridges might well be
made by the mile, and cut off to order according to the span.
The improved lattice, by Hermann Haupt, Esq., C. E., avoids all of the
evils attendant upon the common lattice, and gives a very cheap, strong,
and rigid bridge. In this plan the braces are placed in pairs, with
vertical tie planks between them; by which the twisting seen in the
common lattice, is removed. The braces are also brought to the vertical,
as the point of support is reached, by which a good bearing is given to
the end sets of timbers.
To vary the size of the braces, as the strain upon them decreases, would
be both inconvenient and expensive; but the same effect may be produced
by varying the distance between them, making it greater as the centre is
approached.
S. W. HALL’S WOODEN TRUSS AND ARCH BRIDGE.
Inverting Mr. Haupt’s design for a lattice of improved construction,
(which consists of vertical ties and inclined braces,) we have the base
of the above-named bridge; where the inclined timbers are used to resist
tension, as below.
This being a very good plan, and the arrangement for building being such
as to secure the thorough execution of the work in its most minute
detail; it is thought best to extract at some length from a letter from
the inventor, dated July 31st, 1856, not however being confined to the
matter therein.
The first claim, is for a _new form of truss_, formed of posts vertical,
or nearly so, and _tension_ pieces, inclining downwards toward the
centre; thus differing from nearly all other plans. Timber resists
double as much extension as compression; and when large enough to resist
the simple tension, does not have to be increased as in resisting
compression for flexure; but requires a larger allowance for joints, as
tension tends to pull the joints apart, while compression forces them
together.
The following result was obtained, showing the superior strength of
timber work in resisting by _tension_. Two models, containing the same
amount of timber, were tested. The one built with vertical ties and
_braces_, broke by crippling the brace, under 2,400 lbs.; while that
constructed with verticals and _suspenders_, inclining towards the
centre, sustained 4,200 lbs. with no visible change of form.
The second claim is for more efficient bearings and connections than
common, and this with less cutting away of timber. The arch and arch
braces have a full, fair bearing at top and bottom. The first sets of
tension braces, (those extending from the top of the arch braces towards
the centre,) are sustained by two pins at each joint; which gives six
pin bearings, or twelve for one set of braces, of six inches each, (the
pin being two inches in diameter, and plank three inches thick,) equal
in all, to seventy-two inches of bearing surface at least, for each five
feet lineal of bridge, or one hundred and forty-four inches for ten
feet.
The third claim, is that the bearings, at joints, are _central_, and
that the shrinkage of the timber is _towards_ and not _from_ them as in
many plans.
The pin holes are bored by machinery smooth and true; the treenails when
of wood are of seasoned oak or locust, turned to a perfect fit, and when
of iron are made hollow.
These bridges, after three years, stand within an inch of their shape as
framed without exception. One indeed supporting an aqueduct, which
throws upon the truss a constant load of 2¾ tons per foot, not including
the weight of the bridge, without any apparent settling.
The connections being fast, prevent reaction and vibration from variable
loads, the strains in this case are reversed, the bridge tending to
spring up instead of settling.
The fourth claim, is for the small brace connecting the lower with the
intermediate chord; by which additional connections are obtained, and
smaller timbers rendered available.
The fifth claim, is the formation of a stronger chord than by the plan
of using a few large sticks. The chord being made of a great number of
small pieces, the strength is of course less affected at any one point,
by a joint, than when only a few pieces are used.
In the bridges built by the above engineer, are to be seen some of the
most perfect built beams in the country. The following conditions being
observed, the most uniform, and highest average strength possible is
obtained.
First. To cut but one stick in any one panel.
Second. To cut no stick at the centre of the bridge.
Third. To place every joint in the middle of the collateral piece.
The chords are cut by two rows of pins, two inches each; and if the
chord be fifteen inches, the cutting at centre, where there is no joint,
is but four fifteenths of the whole section. To resist the parting of
any two sticks, there is the resistance to shearing of ninety-six pins;
and the section of each being three square inches, the whole resistance
is two hundred and eighty-eight inches of area. If the intermediate
chord has the same number, the whole area to resist shearing, in the
lower chord, is five hundred and seventy-six square inches. The bearing
surface of each pin in the chord stick is 2 × 3 inches, or six inches;
and 96 × 6 = 576: and in both chords 1152 inches.
Sq. in.
The whole chord timber, (both chords,) is (6 × 3 × 14), 252
In the intermediate chord, (6 × 3 × 12), 216
———
Whole timber section, 468
Deduct 4 pins for both chords,
468 – (4 × 2 × 3 × 6) or 468 – 144 = 324
Deduct for joints 3 × 10 + 3 × 8 or 324 – 54, Square inches of
available area. 270
Comparing the amount thus cut away with that cut away in other plans, we
have the following figures;—
A. Hall’s Bridge, (actual bridge,) 30/100
B. Howe’s bridge, (actual bridge,) 35/100
C. Page 163, (Handbook R. R. Construction,) 39½/100
D. McCallum’s, (Susquehanna bridge,) 58/100
The sixth claim, is the peculiarly convenient form for applying an
arch,—the superiority consisting in convenience for attachment; in the
connections being less affected by shrinkage than when posts are locked
into arches; in the timbers not being weakened by cutting. The arch is
loaded with the tension timbers, inwardly, and acts as a general arch
brace, transferring at once all of the several tensions to the abutment,
thus really combining the arch with the truss.
The liability of this plan to decay, certainly appears to be less than
that of most plans of wooden bridges now in use; as will be plainly seen
by observing the position of the joints; falling rain finds a much
easier access to almost any other joint than the pin hole. The timber
work being made of plank, all the timbers are small, and are thus much
more likely to be sound.
[Illustration: Fig. 69.]
The bridges built upon this plan upon the Alleghany Valley, and upon the
Williamsport and Elmira roads, illustrate plainly the design.
185. Applying arch braces to lattice bridges, has suggested _The
Arch-brace truss bridge_, in which the whole strength lies in a series
of differently inclined braces, extending from the abutment to the head
of each post; a very light lattice being used to prevent reaction, or as
a counter-brace or stiffener. See fig. 69.
In trusses consisting of a series of triangles, when the span is large,
(150 to 200 feet,) the immense weight coming at the feet of the second
and third sets of braces, causes settling or depressing at twenty or
thirty feet off from the abutment, which can hardly be removed. The
remedy for such settling, is to transfer the load at once to the
abutment; which is completely done in the above-named bridge. Each brace
does its duty directly and well. Before the lattice-work is fastened,
the bridge should be loaded with a maximum load. Then by fastening the
diagonals, the recoil is prevented; and the effect of a passing load is
to ease the counterbracing lattice, without otherwise affecting the
truss.
NOTE.—A model of this bridge, made by the writer, of the following
dimensions:—
Length, 7 feet.
Height, 1 foot.
Width, 1 inch.
Chords, ¼ × ½ inch.
Braces, ¼ × ⅓ inch.
Lattice, ¼ × 1/16 inch.
Supported 2,500 lbs. at centre, besides a variable load of 150 lbs.
applied as a rolling weight in the most disadvantageous manner. It
represented a span of one hundred and fifty feet, and according to
Weisbach’s formula for testing a model, proved the actual structure,
(as far as can be proved by a model,) both strong and rigid to any
desired amount. The longest bridge ever built upon this principle,
was that of Schaffhausen, over the Rhine, which had a single span of
three hundred and ninety feet. This bridge was not stiff, having no
lattice, but was very strong. B. H. Latrobe, Esq. has adopted this
form upon the Baltimore and Ohio Railroad.
The calculations for the parts of this bridge are as follows:—
The Span being 150 feet,
The Rise 20 feet,
The Panel 15 feet,
Weight per foot of bridge and load 3,000 lbs.
The half number of panels is five; the diagonals of which, neglecting
fractions, are
√(20^2 + 15^2) = 25 feet,
√(20^2 + 30^2) = 37 feet,
√(20^2 + 45^2) = 49 feet,
√(20^2 + 60^2) = 64 feet,
√(20^2 + 75^2) = 78 feet.
The weight upon each of these sets of braces, is the weight of the
length of one panel; which, in the present case, is 3,000 × 15 = 45,000
lbs. As there is a brace under each chord stick, and assuming four
sticks in each chord, we divide by eight, and have, in round numbers,
6,000 lbs. per brace; and correcting for inclination, as follows, we
have the numbers below.
20 : 25 :: 6000 : 10000
20 : 37 :: 6000 : 15000
20 : 49 :: 6000 : 20000
20 : 64 :: 6000 : 25000
20 : 78 :: 6000 : 30000.
The last column has the several weights coming upon the different braces
at their several inclinations; to resist which, the scantling might be
very small, for compression, but flexure requires larger dimensions.
These braces should be confined laterally and vertically, as they pass
each post, but not connected therewith; as this would not permit a free
action of the brace, without straining transversely the post.
The length of beam, therefore, in which flexure is to be checked, is the
distance between posts in any panel.
In panel No. 1, it will be 25 feet.
In panel No. 2, it will be 18 feet.
In panel No. 3, it will be 17 feet.
In panel No. 4, it will be 16 feet.
In panel No. 5, it will be 16 feet.
and applying the formula
(2240_bd^3_)/(_L^2_) = _W_
we get, in round numbers, the following dimensions, the braces being
bolted and blocked together:—
For the 1st panel, 25 feet long, 8 × 10
For the 2d panel, 37 feet long, 8 × 10
For the 3d panel, 49 feet long, 8 × 10
For the 4th panel, 64 feet long, 8 × 10
For the 5th panel, 78 feet long, 8 × 10.
For the lattice-work, a double course on each side of each truss, in
long spans, (150 to 200 feet); and a single course in shorter spans, of
3 × 6 plank, treenailed at intersections, is ample.
GENERAL TABLE OF DIMENSIONS FOR ARCH BRACE TRUSS.
┌──────────┬──────────┬──────────┬──────────┬──────────┬──────────────┐
│ Span. │ Rise. │ Chords. │ Ties. │ Braces. │ Lattice. │
├──────────┼──────────┼──────────┼──────────┼──────────┼──────────────┤
│ 50│ 10│ 2–8 × 10│ 1–8 × 10│ 2–6 × 6│2 × 9 or 3 × 6│
│ 75│ 12│ 2–8 × 10│ 1–8 × 10│ 2–6 × 6│2 × 9 or 3 × 6│
│ 100│ 15│ 3–8 × 10│ 2–8 × 10│ 3–6 × 6│2 × 9 or 3 × 6│
│ 150│ 20│ 4–8 × 12│ 3–8 × 10│ 4–6 × 8│2 × 9 or 3 × 6│
│ 200│ 25│ 4–8 × 16│ 3–8 × 10│ 4–6 × 9│2 × 9 or 3 × 6│
└──────────┴──────────┴──────────┴──────────┴──────────┴──────────────┘
[Illustration:
Fig. 69 K.
Fig. 69 A.
]
Fig. 69 A, shows the method of bringing the arch braces to the chord. To
find the dimensions of the cast-iron block, make a complete drawing of
all of the braces, at their proper angles, and then draw in the block
around the feet, as shown in fig. 69 A.
NOTE.—The centre of pressure of the braces in fig. 69 A, is not, as
might seem, at C; because the vertical components of the forces,
coming down the brace, are much less in the braces at small angles
than in those at the end of the span. The load applied to each brace
being the same, and the inclines being found, we find the centre of
pressure, or the centre of bridge seat as follows:—
The length of the brace is to the vertical height, as the applied
load to the vertical pressure. In fig. 69 A, we have the following
lengths of braces: _a_, 25; _b_, 37; _c_, 49; _d_, 64; _e_, 78; _f_,
92; and _g_, 106; and the weights corresponding thus,
_a_, 25 : 20 :: 6000 : 4800.
_b_, 37 : 20 :: 6000 : 3243.
_c_, 49 : 20 :: 6000 : 2450.
_d_, 64 : 20 :: 6000 : 1870.
_e_, 78 : 20 :: 6000 : 1540.
_f_, 92 : 20 :: 6000 : 1304.
_g_, 106 : 20 :: 6000 : 1132.
In fig. 69 A, assume the foot of the fifth brace (B) as the centre of
pressure, and adding the moments, (or products of vertical components on
the braces by their distance from B,) and we have the sum on the land
side 18,928, and on the water side 16,930; showing that the centre is
taken too far from the land side. In the same manner A will be found too
far from the land side. A third trial will give the place.
[Illustration:
Fig. 69 B.
Fig. 69 C.
]
[Illustration:
Fig. 69 D. Fig. 69 E. Fig. 69 F.
Fig. 69 G. Fig. 69 H.
]
Fig. 69 B, shows the manner of splicing the arch braces: being subjected
to compression, they are spliced in the same manner as the upper chords.
Fig. C, shows the lower chord spliced. Figs. D and E, the connection of
the posts, chord, and lattice. Figs. F, G, and H, the casting for
applying the upper end of the arch brace to the chord. Fig. 69 K, the
method of supporting the tracks at the end of the span, where the arch
braces will not allow the floor beams to bear upon the lower chord.
MCCALLUM’S PATENT RAILROAD BRIDGE.
[Illustration: Fig. 70.]
186. This bridge represents a class of structures in which the upper
chord is curved upwards (7½ feet in 200 in the Susquehanna bridge, New
York and Erie Railroad), which curved chord has the effect of
distributing an applied load at once to all of the braces _directly_, by
means of the chord, as well as _indirectly_, by means of the braces, as
in the common trusses. To this bridge is applied the arch braces A B, A
B, fig. 70, which serves to aid the 2d, 3d, and 4th pair of diagonal
braces in bearing their load.
The great distributing power of the curved chord, is shown by the fact
that a bridge of 125 feet span, actually supported a railroad train
before the diagonal bracing was introduced. The whole strain was thrown
through the curved chord and arch braces to the abutments. The bridge is
counterbraced by the pieces _d d_ and _d d_, adjustable by screws at the
ends.
The following test was applied to a span of 190 feet of this plan of
bridge. Placing the load as near as possible to the centre, the
following deflections were produced.
Load. Deflection.
41.40 tons, 0.013 feet,
95.35 tons, 0.038 feet,
140.70 tons, 0.061 feet,
187.20 tons, 0.061 feet.
Upon removing the load, the bridge entirely recovered its form.
187. As the span increases, the benefit derived from the curved chord
also augments; and though in the latter part of the present chapter its
application to small spans is shown, it may not be worth while to adopt
it.
Bridges transferring the load _directly_, from each panel to the
abutment, would not be aided, to an amount worth the increased expense,
by adopting the curved top chord.
In case of any settling at the centre of the span, the reverse effect is
seen from that produced in a truss with horizontal chords; i. e., when
the ends of the upper chords in the latter _draw in_, those of the
former _push out_; and when in such bridges, arch braces are not used,
the top chords of adjoining spans must be _wedged apart_, in place of
_tying together_ as in common plans, over the centre of the piers.
THE ARCH.
188. The arch has been applied to long spans for a great while, and when
care has been taken to prevent flexure, answers very well. The repair of
such bridges, if any of the arch timbers decay, is difficult; but is
effected, in the largest arches.
The most correct ideas on wooden arch bridge building, are to be found
in Weibeking’s _Traite d’une parte essential de construire les grandes
pents en charpente_. This engineer, (General Director of Roads and
Bridges in Bavaria,) has built a great number of wooden arches of the
best description, which show him to be master of both the science and
the art.
[Illustration: Fig. 71.]
The general plan of his bridges is shown in fig. 71. They consist of
curved ribs formed of long pieces scarfed and bolted together, from
which the road-way is supported by posts.
The bridges of Neucettringin, Freysingin, Bamberg, Scharding, Wertach,
Vilshoven, and Altenmarkt, all testify to the good judgment of this man.
The spans vary from one hundred to two hundred feet; and the width from
twenty-five to thirty-two feet. The proportions which he gives for the
ratio of rise to span, are valuable; as they are the result of his own
experience. He states, generally, that one tenth of the span is the best
rise; but that for convenience, it is better to keep it lower. The
following table shows the dimensions he has adopted in practice.
189.
┌──────────────┬─────────┬─────────┬─────────┬─────────┬─────────┐
│ Name. │ Span. │ Rise. │ Width. │ Rad. of │Scantling│
│ │ │ │ │ Arch. │of Arch. │
├──────────────┼─────────┼─────────┼─────────┼─────────┼─────────┤
│Bamberg, │ 208 │ 16.9│32 │ 422 │13½ × 15½│
│Scharding, │ 194 │ 18.8│25 │ 258 │12½ × 15½│
│Vilshoven, │ 179 │ 11.1│27 │ 378 │13½ × 15½│
│Freysingin, │ 153 │ 11.6│25 │ 246 │12½ × 14½│
│Ettringin, │ 139 │ 8.0│25 │ 305 │12½ × 15½│
│Ersingin, │ 126 │ 7.0│25 │ 285 │11½ × 14½│
│Augsberg, │ 114 │ 10.6│25½ │ 158 │12½ × 14½│
│Neucettringin,│ 103 │ 6.8│25 │ 200 │13½ × 15½│
└──────────────┴─────────┴─────────┴─────────┴─────────┴─────────┘
The last column shows the scantling of the arch timbers; these being
placed three deep, in spans of less than 150 feet; and in larger spans,
3 deep at centre, and 5 deep at ends. Mr. Weibeking’s formula for
determining the scantling of ribs, is as follows:—
((_W_ × (_S_/2)^2)/_Rn_).0011 = Scantling in sq. ft.
Where _R_ is the rise of the arch;
_n_, the number of ribs;
_W_, width of bridge;
and _S_, span of bridge.
EXAMPLE.—Required the scantling of the ribs of a bridge of 300 feet
span, 20 feet wide, and 30 feet high. The formula becomes,—
(20 × 22500)/30 × .0011 = 16½ sq. feet of
section, of all of the arches; or two parallel arches, 2½ feet wide, by
3¼ feet deep each.
From 100 to 150 feet span, he makes the rise 1/20 span,
From 150 to 200 feet span, he makes the rise 1/18 span,
From 200 to 300 feet span, he makes the rise 1/15 span,
From 300 to 400 feet span, he makes the rise 1/14 span,
From 400 to 500 feet span, he makes the rise 1/13 span.
190. The bridge built by Mr. Burr across the Delaware at Trenton, New
Jersey, is a good specimen of an arch. It is composed of white pine
planks, from thirty-five to fifty feet long, and of a scantling 4 × 12.
These planks are laid close together, breaking joint, having an entire
depth of three feet. The arches are stiffened by horizontal tie beams,
supporting the road-way, and by diagonal bracing. The spans are 160,
180, and 200 feet, and the rise twenty-seven feet.
191. The bridge over the Susquehanna, at Columbia, built in the same
manner, consists of twenty-nine arches, each two hundred feet clear
span, supported on two abutments and twenty-eight stone piers. The clear
water-way of this bridge is 5,800 feet; and the entire length, including
piers and abutments, one and one fourth miles. There are three sets of
arches, which allow of two carriage roads and one railroad, the whole
width being thirty feet.
192. An arch to support a passing railroad train must be very rigid. It
is customary to connect them with a light truss, which effectually
counter braces the arch, and prevents that change of form which would
otherwise take place; depending entirely upon the arch for strength.
Wherever the load is applied, the arch tends to sink, and a
corresponding rise takes place at the opposite point. A load placed at
E, fig. 71, settles the arch at that point and causes it to rise at C. A
load placed at the curve of the arch depresses the centre, and elevates
the haunches. To counteract these movements a light, stiffening frame
may be used, its strength being able to resist the variable load passing
over the bridge. The strain thrown by the arch upon the truss, advances
from the opposite end to meet the train, passes it at the centre, and
finally goes off from the bridge behind the load.
When the arch _is the truss_, or when a truss is made with curved
chords, the counteracting effect of the truss is not completely
obtained. We should not depend upon the curved chord _as an arch_, but
only as a member of the truss.
193. Many combinations of arch and truss have been built in America for
railroad bridges. The principle of connecting the two systems is by some
thought bad, as they can hardly be made to bear equal parts of the load;
whence each must have more than half the necessary strength of the
whole. Others maintain that by a proper arrangement of parts a perfect
adjustment may be made, by which the load may be placed more or less on
either. There seems to be no very good reason why the two systems should
be combined, as either may be made strong enough to bear the largest
loads.
Both arches and arch braces, however, are very usefully applied to
bridges which have been made too light.
194. The manner of applying arches is well shown in the bridges of the
Pennsylvania Central Railroad, built by Hermann Haupt, Esq.
These bridges are on Howe’s plan, to which have been added strong wooden
arches. The systems are connected by adjusting the counter braces
against the arch by set screws. The arrangement is simple and effectual.
The name of the builder is sufficient to warrant the stability of the
bridge.
195. However nicely we may form an arch, it will settle more or less
when the scaffolding is removed, according to its flatness; which
depression increases with time. Mr. Weibeking expresses it in inches as
follows:—
0.806_R_/_S_
Where _R_, shows the rise,
and _S_, shows the span.
To allow for this settling, the curve when laid down on the platform for
building the arch, should be made a little more convex than the
completed arch is required to be; the amount of excess being that shown
by the formula.
[Illustration: Fig. 72.]
As a bridge composed of a curved rib when the span is large yields at D,
C, and E, fig. 71, when the load is applied in the middle, the strength
must of course be increased by increasing the depth of the rib; and not
to make this too heavy, a framed or built beam should be used as in fig.
72. Here it must be remembered that the two ribs must be so framed as to
resist both tension and compression; for when a load is placed at D, the
lower rib will be extended at _d_, and compressed at _c′_, and _e_;
while the upper one will be compressed at D, and extended at C and E.
OF THE ROAD-WAY.
196. The flooring of any system is about the same; consisting of
transverse floor beams, placed either on the top or bottom chords,
(according as the road-way is more or less elevated above the
water-way,) which support longitudinal timbers, upon which are placed
cross-ties. In some cases, two curves of diagonal plank have been placed
across the floor beams, spiked at right angles to each other, by which
the bridge is considerably stiffened laterally.
General dimensions for the floor may be thus:—
Transverse timbers, 3 feet from centre to centre, 8 × 14
Track strings, notched 2 inches to floor beams, 12 × 14
Cross-ties placed one foot apart, (clear,) 3 × 6
[Illustration: Fig. 73.]
LATERAL BRACING.
197. To prevent vibration in a horizontal direction, a system of
diagonal bracing is necessary. The chief pressure upon these braces is
caused by wind; and may be found by considering the bridge as turned
over upon the side, and loaded with a weight equal to the maximum
pressure of the wind, which may be taken as forty pounds per square
foot.
It is unnecessary to vary the size of these braces, except in very long
spans, when they should increase from the centre to the ends. For short
spans, (less than one hundred feet,) a brace 5 × 5 is large enough. For
larger spans 7 × 7 is sufficient.
198. Diagonal bracing, when it can be introduced, is a very desirable
part of a bridge. When the road is on the lower chord this cannot have
place in full, but may be applied as in fig. 74.
[Illustration: Fig. 74.]
By increasing the height of truss in any bridge, the tension and
compression on the chords is lessened; but the length of posts and rods
is increased. As a general thing, _one eighth_ of the span gives the
best results.
199. In framing a large bridge, it is customary to cut the top chord
sticks a little longer than to dimension; to allow for compression in
settling.
200. Bridges in exposed situations have been sometimes blown off from
the masonry. If a bridge _slides off_ from the masonry, the whole force
of the wind must be fifteen twenty-fourths of the whole weight of the
bridge; but if, as is generally the case, the masonry is rough, (and not
hammered,) no amount of wind will cause the bridge to _slide_.
The bridge will upset, turning about its lower edge, when the whole
pressure of the wind, multiplied by half the _height_ of truss,
overbalances the whole weight, multiplied by the half _width_. In very
exposed places the rod A D, fig. 74, answers a very good end; when the
road is upon the upper chord, and a rod from B to the masonry, when upon
the lower.
OBLIQUE BRIDGES.
201. The effect of running a train over a skew bridge, is to depress one
side before the other; as the load comes upon the centre of one truss
before it does upon the opposite one. This produces a side rocking in
the engine, dangerous alike to the bridge and to itself.
The floor timbers transferring the load to the chords should not be at
right angles to the axis of the road, but parallel to the abutment. Thus
in fig. 75, a wheel at B, throws one third of its weight upon the
abutment at E; and two thirds upon the chord at C; while in fig. 75 A,
the wheel at B, throws two thirds of the load upon C, but one third also
upon D.
[Illustration: Fig. 75. Fig. 75 A.]
202. In a very long, oblique span, the floor timbers may be arranged as
in fig. 76, that is, inclined at the entrance and exit of the bridge,
but at right angles at the middle of the span.
[Illustration: Fig. 76.]
203. The preservation of timber in wooden bridges may be accomplished by
covering with boards, whitewashing, painting, and by Kyanizing. Covering
and whitewashing are the best, if care is taken to prevent dry rot by
giving a good circulation of air about the timbers. The oil in paints
prevents the escape of moisture from within as well as the entrance of
that from without; and should not be used unless the wood is well
seasoned. The best plan is to thoroughly whitewash and cover the frame
of the bridge, and to paint the outside of the covering.
204. In framing two or more continuous spans, the chords should always
be connected over the piers; as there is thus given something for the
upper chords to pull against, and a counter thrust for the lower.
205. Bridges should never, when it can be avoided, be placed either upon
a curve or upon a grade; particularly upon the former, as the effect of
a load is thereby very much increased, in the first case causing a
lateral, and in the second a vertical shock.
PILE BRIDGING.
206. In shallow water, in marshes, and in similar situations, where an
embankment would be expensive, pile bridging is very useful. Indeed,
whenever we are at liberty to obstruct the passage beneath the road, it
is well to adopt this system, unless over twenty feet high. It is
cheaper than any other, easier to repair, the parts are quite
independent of each other, and such bridges last full as long as other
wooden structures.
[Illustration: Fig. 77.]
[Illustration: Fig. 78.]
Different plans for pile bridging are given in figs. 77 to 82. Figs. 77
to 81 show plans for temporary pile work, to be used during
construction. Nothing lighter than fig. 82, ought to be permanently
used. A pile bridge upon a curve may need stronger lateral bracing upon
the convex, than upon the concave side of the curve; and also in running
water; in which cases, such a form as fig. 81 may do good service.
[Illustration: Fig. 79.]
[Illustration: Fig. 80.]
[Illustration: Fig. 81.]
[Illustration: Fig. 82.]
[Illustration: Fig. 82 A.]
(For pile-driving, and for proper dimensions, see chap. XII.)
TRESTLING.
207. Trestling is a system of vertical posts, and of caps and braces,
used both for temporary and for permanent works; temporarily to pass a
road over low ground where embankments are to be made, and permanently
over deep, dry gorges, where the amount of earthwork or masonry would be
too great.
[Illustration: Fig. 83.]
American railroads show all sizes and arrangements of trestling, from
twenty to two hundred feet high. Figs. 83 and 84 show temporary works,
and fig. 85 permanent.
The main part of the design in trestles is to connect the several posts
and caps by well-formed triangles; the equilateral being the best.
[Illustration: Fig. 84.]
The finest example of this system of building is the Genesee high
bridge, over Genesee River near Portageville, on the Buffalo and New
York Railroad; built by H. C. Seymour, Esq. It is eight hundred feet
long, and two hundred and thirty feet above the river. It has eight
stone piers, thirty feet high, upon which are placed trestles one
hundred and ninety feet high, seventy-five feet wide at base, and
twenty-five at top. Upon the top of all is placed a bridge fourteen feet
high. To build this viaduct was used 1,500,000 feet, board measure of
timber, which covered, when standing, two hundred and fifty acres; also,
sixty tons of bolts. The whole time occupied in building was but
eighteen months, the whole cost being $140,000.
[Illustration: Fig. 85.]
DRAWBRIDGES.
208. In crossing rivers or bays open to navigation, it is required from
any companies building a bridge, to leave a free passage for shipping.
This is done by making that part of the bridge over the channel movable;
(a draw).
[Illustration: Fig. 86.]
Draws may lift up, (being counterbalanced,) may slide back upon the
fixed part of the bridge, or may turn on a pivot. Fig. 86 shows a draw
much used at present, and answering every purpose. Each half of the
movable part must be calculated as a small bridge. The rods _c c c_
support the overhanging part of the draw while open. The whole revolves
upon a centre pin and a set of rollers.
CENTRES.
209. Centres are temporary wooden frames, used in the construction of
stone arches. Their duty is to hold the masonry, while it is unable to
support itself.
For arches from five to fifteen feet span, a centre made of boards or
planks, fig. 87, is all that is necessary. For longer spans, when the
ground beneath the arch can be used, the form, fig. 88, answers well.
When there is no support but the abutments or piers, something similar
to fig. 89 must be adopted. This is the plan adopted by George Rennie,
chief engineer at the Waterloo bridge over the Thames at London.
[Illustration: Fig. 87.]
[Illustration: Fig. 88.]
Centres are strained in a different manner as the arch progresses; first
at the haunches, and last at the crown. Excess of weight at any point
causes a settling at such, and a rise takes place at some other place.
By loading the arch temporarily, such motions are checked.
[Illustration: Fig. 89.]
These frames are placed vertically upon the pieces F F, which being
connected with the braces D D by the folding wedges _c c_, admit of
adjustment of the height of the centre. The distance between the ribbed
frames depends upon the form of the arch, and the span, or upon the
weight to be supported; varying from one to four feet. The centres are
covered with a course of narrow plank, placed parallel with the axis of
the arch, upon which the voussoirs rest.
210. The method of putting a bridge upon the masonry is shown in figs.
90, and 91; the former when the road-way is upon the upper, and the
latter when upon the lower chord.
[Illustration: Fig. 90.]
[Illustration: Fig. 91.]
211. In figs. 92 to 100, are given several plans for spans from five to
seventy-five feet. Fig. 92, shows the simple beam braced beneath with
diagonal plank; the bolts passing through the ties, stringers, and
braces. The stringers are bolted to the wall plates, and when the bridge
is upon a curve notched also, by cutting the bolster. Fig. 92 A shows
the plan. This form answers for openings from five to twenty feet. From
fifteen to thirty feet, we may use figs. 94, 95, 96, and 97. From
twenty-five, to fifty and sixty feet, figs. 93, 97, and 98. And from
fifty to seventy-five feet, figs. 99 and 100.
The following tables give reliable dimensions for bridges upon the above
plans.
[Illustration:
Fig. 92. 5 to 20 feet.
Fig. 92 A.
]
Span. Bolsters. Stringers. Ties. Braces. Bolts.
5 12 × 12 12 × 12 6 × 10 2 × 10 1 inch
10 12 × 12 12 × 13 6 × 10 2 × 10 1 inch
15 14 × 14 12 × 18 6 × 10 2 × 10 1 inch
20 14 × 14 12 × 24 6 × 10 2 × 10 1 inch
The ties being notched three inches on to the stringers, without cutting
the latter.
[Illustration: Fig. 95. 15 to 30 feet.]
Span. Rise. Bolster. Stringer. Braces. Rod.
15 6 12 × 12 12 × 12 2—5 × 6 1 inch
20 7 14 × 14 12 × 12 2—5 × 8 1¼ inch
25 8 14 × 14 12 × 15 2—5 × 9 1⅜ inch
30 10 15 × 15 12 × 18 2—5 × 10 1½ inch
[Illustration: Fig. 96. 15 to 30 feet.]
Span. Rise. Stringer. Post. Rod.
15 5 12 × 12 8 × 8 1⅛
20 6 12 × 13 9 × 9 1¼
25 7 12 × 15 10 × 10 1½
30 8 12 × 18 10 × 12 1⅝
[Illustration: Fig. 94. 15 to 30 feet.]
Span. Rise. Stringer. Braces. Rods. Lattice.
15 5 2—8 × 8 2—5 × 5 1 2 × 6
20 6 2—8 × 9 2—5 × 6 1¼ 2 × 8
25 7 2—8 × 10 2—5 × 8 1⅜ 2 × 9
30 9 2—8 × 12 2—5 × 9 1½ 2 × 10
[Illustration: Fig. 97. 15 to 30 feet.]
Span. Rise. Stringer. Post. Rods. Braces.
15 5 2—8 × 8 8 × 8 1 4 × 5
20 6 2—8 × 9 9 × 9 1⅛ 4 × 6
25 7 2—8 × 10 10 × 10 1¼ 5 × 6
30 9 2—8 × 12 10 × 12 1½ 6 × 6
[Illustration: Fig. 93. 25 to 50 feet.]
Span. Rise. Chords. Braces. Posts. Rods.
25 8 2—6 × 8 6 × 6 6 × 8 1
40 10 2—7 × 9 6 × 7 8 × 8 1⅛
50 10 2—8 × 10 6 × 8 8 × 10 1¼
[Illustration: Fig. 97. 25 to 50 feet.]
Span. Rise. Chords. Posts. Braces. Rods.
25 8 2—6 × 8 8 × 8 5 × 5 1⅛ or 2—⅞
40 10 2—7 × 9 9 × 9 5 × 8 1⅜ or 2—1⅛
50 10 2—8 × 10 10 × 10 6 × 8 1¾ or 2—1¼
[Illustration: Fig. 98. 25 to 50 feet.]
Span. Rise. Chords. Posts. Braces. Rods.
25 8 2—6 × 8 6 × 8 5 × 5 2—1 or 1—1⅜
40 10 2—7 × 9 6 × 9 5 × 8 2—1⅛ or 1—1⅝
50 10 2—8 × 10 6 × 12 5 × 10 2—1¼ or 1—1¾
[Illustration: Fig. 99. 50 to 75 feet.]
Span. Rise. Chord. Posts. Braces. Lattice.
50 8 2—8 × 10 1—8 × 10 2—6 × 7 2 × 6
60 9 2—8 × 10 1—8 × 10 2—6 × 7 2 × 6
75 10 3—8 × 10 2—8 × 10 3—6 × 8 2 × 6
[Illustration: Fig. 100. 50 to 75 feet.]
Span. Rise. Chords. Posts. Braces. Centre Rods.
braces.
No. 1. No. 2.
50 8 2—8 × 10 1—8 × 10 2—6 × 7 5 × 5 1½ 1
60 9 2—8 × 10 1—8 × 10 2—6 × 7 5 × 5 1⅝ 1⅛
75 10 3—8 × 10 2—8 × 10 2—6 × 8 5 × 6 1¾ 1¼
212. In dimensioning small bridges, like the above, in estimating the
maximum load, more regard must be given to the weight of momentary loads
than (as in large bridges) the weight per lineal foot, as the weight of
the bridge itself, when under fifty or sixty feet span, is
inconsiderable. The greatest load that can come upon a single post or
rod, is that from the driving wheels of a locomotive. If the whole
engine weighs forty tons, there will be ten tons on each pair of
drivers, or five tons or 11,200 lbs. on each wheel; which, being applied
over a length of ten feet only, may be considered as at a single point,
and all parts must be able to bear such load. In large spans, where the
weight is great, if the truss is strong enough to support the bridge and
load, it will safely resist the effects of a sudden application of
passing trains.
NOTE.—_On the Static and Dynamic deflection of Bridges._
Considerable variance of opinion exists as to the relative
deflection of bridges, produced by stationary and by moving loads.
Neither experiment nor theory has exactly settled the point.
Experiments upon the Elwell bridge, (Epsom and Croydon Railway,
England).
Velocity in feet per second. Deflection in inches.
0 0.215
25 0.215
31 0.230
32 0.225
54 0.245
75 0.235
The bridge being a cast-iron girder of forty-eight feet span, load
thirty-nine tons.
Experiments on the Godstone bridge, (S. E. R. R. England).
Velocity in feet per second. Deflection in inches.
0 0.19
22 0.23
40 0.22
73 0.25
Cast-iron girder, thirty feet span, load thirty-three tons.
Mr. W. H. Barlow, (Eng.) observed, “that in case of a timber
viaduct, a freight train, at a low speed, produced a certain
deflection; but an extra train, with a much lighter engine, seemed
to push the bridge like a wave before it.”
The Britannia tubular bridge was depressed three fourths of an inch
by two locomotives and a train of two hundred and eighty tons
standing still; but at seventy miles per hour, the deflection was
sensibly less.
CHAPTER IX.
IRON BRIDGES.
“A little knowledge is a dangerous thing.”
213. Within the past ten years iron has been brought extensively into
use for railroad bridging; and when employed by those who understand its
chemical and mechanical nature is unequalled for strength, durability,
and elegance of appearance; but when, as is too often the case in
America, it is intrusted to men who neither know nor care for any thing
but the price they get for it, nothing can be more unsafe. No material
requires so complete a knowledge of its properties, to be safely used,
as cast-iron.
NATURE AND STRENGTH OF IRON.
214. The table below shows the properties of the several descriptions of
iron used in engineering.
┌───────────┬──────────┬───────────┬───────────┬──────────────────────┐
│ Wrought │Cast-Iron.│Iron Wire. │ Boiler │ Designation of the │
│ Iron. │ │ │ Plate. │ quality. │
├───────────┼──────────┼───────────┼───────────┼──────────────────────┤
│ 480│ 450│ ——│ 480│Weight per cubic foot │
│ │ │ │ │ in lbs. │
│ │ │ │ │Resistance to │
│ 15000│ 4500│ 25000│ 12740│ extension in lbs. │
│ │ │ │ │ per sq. inch. │
│ │ │ │ │Resistance to │
│ 11000│ 25000│ ——│ 7500│ compression in lbs. │
│ │ │ │ │ per sq. in. │
│ │ │ │ │Expansion per degree │
│ .0000066│ .00000608│ .00000685│ .0000066│ Fahrenheit in │
│ │ │ │ │ lengths. │
│.0000000424│.000000106│.0000000446│.0000000524│Extension per lb. per │
│ │ │ │ │ square inch. │
│ .000000149│.000000083│ ——│ .000000189│Compression per lb. │
│ │ │ │ │ per square inch. │
│ │ │ │ │Ratio of extensive to │
│ 90 to 66│ 20 to 111│ ——│ 127 to 75│ compressive │
│ │ │ │ │ strength. │
│ │ │ │ │Resistance to │
│ 12500│ 17500│ ——│ ——│ detrusion, or │
│ │ │ │ │ shearing. │
│ 55│ 31│ ——│ ——│Relative transverse │
│ │ │ │ │ strength. │
└───────────┴──────────┴───────────┴───────────┴──────────────────────┘
_Column four refers to boiler plate when built into tubes._
After wrought iron has become a little compressed, its power to resist a
crushing force is very much increased.
215. The tenacity of wrought iron is increased by heating. Experiments
upon thirty varieties gave the following mean result, the temperature
ranging from 500° to 700° Fahrenheit.
Strength when
Cold. Hot. Cooled.
60,000 64,000 70,000
216. Stirling’s process of toughening cast-iron, by the addition of
malleable scrap, increases the strength in the following ratio:—
The mean tensile strength of cast-iron being 18,000 lbs.
And the compressive strength being 105,000 lbs.
When Stirling-toughened the tensile strength is 23,000 lbs.
And the compressive strength 130,000 lbs.
The strength of cast-iron increases rapidly up to the twelfth or
fifteenth recasting, when it is nearly doubled; after the fifteenth
melting the strength decreases.
217. Wrought iron exposed for some time to vibration, as in the case of
railroad axles, or iron which has been wrought with light hammers, loses
its toughness and becomes “short,” (crystalline). The fibre may be
restored in such cases by reheating and cooling slowly.
218. GENERAL RATIOS OF THE STRENGTH OF IRON.
Tension. Compression. Cross Strain.
Cast, 300 1,666 31.68
Wrought, 1,000 733 55.40
OF THE STRENGTH OF BOILER PLATES.
219. The strength of rolled boiler plates is no greater in the
_direction of the fibres than crosswise_, but is more regular; whence
the length of the fibre must be placed as nearly as possible with the
direction of the force.
A mean of twelve experiments, by Mr. Fairbairn, gives the tensile
strength of wrought iron plates as 50,960 lbs. per square inch; and the
compressive strength of plates, when built into tubes, as 30,464 lbs.,
or for safe use in practice, for extension, 12,740 lbs., and for
compression, 7,500 lbs. In the remarks upon girder bridges the matter of
riveting will be considered.
CLASSIFICATION OF IRON BRIDGES.
220. Iron bridges may be classified as follows:—
Those entirely of _cast-iron_, or Arch and Girder bridge.
Those of _wrought iron_ alone, or Tubular and Girder.
Those of _iron wire_, or Suspension bridges.
Those of _cast and wrought iron_, or Trussed bridges.
The order in which these bridges may be placed as regards cost of
construction, and extent of application, is as follows:—
Number. Span. Description of bridge.
1 10 to 50 feet Cast-iron girder.
2 50 to 200 feet Cast and wrought combinations.
3 200 to 2000 feet Suspension.
4 200 to 500 feet Cast arch.
5 25 to 100 feet Boiler plate girder.
6 100 to 500 feet Tubular.
Numbers 2, 3, and 5, are the forms which are in use upon American roads.
No. 1, is very liable to failure, requires much more knowledge and care
in building, and is far more expensive than a wooden truss, or trussed
girder. No. 4, is very expensive, and causes a greater obstruction to
the water-way than any other. The enormous expense of No. 6, should, and
will prevent its adoption in the United States. Let us look at the
principles of construction of numbers 2, 3, and 5.
COMBINATIONS OF CAST AND WROUGHT IRON.
221. Under this head come all of the iron trussed frames used in this
country.
As before observed, skill in bridge construction consists in using
always that material which with the least expense is the best able to
resist the particular strain to which it may be exposed. Thus wrought
iron must always be used to resist tension, and cast-iron compression.
Posts, braces, and upper chords should always be cast, while ties and
lower chords should be made of wrought iron.
The strength of a railroad bridge must be such as to resist all extra
shocks and strains, such as are produced by derailment of engines, and
breakage of axles; also incidental strains arising from change of form
by expansion and contraction of the metal, and from high winds and
gales.
[Illustration: Fig. 101.]
Every part of a bridge not resisting some force is worse than useless,
as it adds to the weight. Lightness not only increases the economy
directly, but indirectly by removing a part of the permanent load.
222. Foremost in class number two stands Wendel Bollman’s Iron
Suspension and trussed bridge. For simplicity of construction and
directness of action, this bridge is unsurpassed. The weight at each
post is transferred at once to the abutment or pier. The upper chord is
of cast iron, hollow, octagonal without, and circular within. The posts
consist of an ¶ᕼ¶ casting, the central web cast open and the flanges
whole. The top is adjusted to the chord, and the bottom to the tension
or suspending rods. These latter are of wrought iron, rectangular in
section, joined when the length requires it by an eye bolt. Each set
after leaving the foot of the post, passes through the chair at A B,
fig. 101, and is secured by a nut. The junction of the tension rod A C,
and the counter rod B C, is attached indirectly to the foot of the post
by a pendulum or link; which serves to equalize the effect of expansion
upon the rods. Vibration and reaction are prevented by the panel
diagonal ties D H, and C E. The floor is supported by flanges at the
foot of each post. The lateral bracing consists of a system of hollow
cast-iron posts, and of wrought diagonal tie rods. A lower chord is
plainly unnecessary, its place being taken by the rods C B, F B, F A, G
A.
A bridge of this description upon the Baltimore and Ohio Railroad of the
following dimensions,
Clear span, 124 feet
Length of top chord, 128 feet
Length of panel, 15 feet
Height of truss, 17 feet
Width, 16 feet
Lbs. of cast-iron, 65,137
Lbs. of wrought iron, 33,527
Whole weight, 98,664
Weight per lineal foot, 796
was subjected to the following tests.
Three locomotives with tenders attached, and weighing in all one hundred
and twenty-two tons, (nearly one ton per foot,) were run over the bridge
at eight miles per hour, when the deflection at centre was one and three
eighths inches, and at the first post nine sixteenths of an inch. The
following tests were applied to a bridge of seventy-six feet span upon
the Washington branch of the same road:
An engine and tender weighing forty tons, caused a deflection of five
eighths of an inch. A fast passenger train deflected the bridge nine
sixteenths of an inch.
Two engines and tenders, back to back, at rest, and
weighing in all 77½ tons, caused a deflection of 11/16 inch,
The same at ten miles per hour, 13/16 inch,
Engines head to head at four miles per hour, 13/16 inch,
Engines head to head at eight miles per hour, 13/16 inch,
Engines head to head at twenty miles per hour, 14/16 inch.
The extreme expansion of the one hundred and twenty-eight feet chord
from heat, was five sixteenths of an inch at each end, or five eighths
of an inch in all, or 1/2457th of the length; and that without the
slightest derangement of masonry. The rod C B, being five times as long
as C A, expands five times as much, but at the same time the lengths D
A, D B, being so nearly proportional to C A, and C B, expand also in the
ratio of one to five; and thus no bad result is experienced.
The estimate of strains upon this bridge is extremely simple; the whole
consisting of as many separate systems as there are posts. Each set of
rods sustain a rectangle equal to one panel, i. e., the two adjacent
half panels. Thus A C, and C B, support the rectangle _m m_, _m m_, the
rods A F, F B, the rectangle _n n_, _n n_. Allowance must of course be
made for the inclination of the rods. The dimensions of the central pair
will of course be the same; but those of the other sets will vary. The
diagonals D H, and H L, prevent reaction; and must be able to resist the
action produced by the variable load upon one panel (as noticed in
Chapter VIII).
Any load, one at C D for example, gives to the posts a tendency to
revolve on A, as a centre towards the abutment; to oppose which, there
must be a force in the opposite direction. The most proper direction in
which to resist such motion is the line C K, i. e., the line of the
lower chord. In this bridge there is no lower chord, but in place of
such are put the rods A G, A K, B H, and B C; which prevent the change
of form (by the motion of the triangle) and act against the upper chord.
As an example of the estimate of strains upon this bridge take the
following.
Span, 90 feet.
Rise, 18 feet.
Panel, 15 feet.
Weight per lineal foot, 2,500 lbs.
Whole weight, 225,000 lbs.
Weight on each side truss, 112,500 lbs.
Weight on each post, 18,750 lbs.
The weight borne by each system, i. e., one post and the two supporting
rods, is 18,750 lbs. The strain to be resisted by any one rod depends
upon its inclination.
The following figures show the elements of the truss in question:—
Section
Rod. Length. Applied weight. Increased of the
strain. bar in
inches.
A B = 90.0
C D = 18.0
A C = 23.4 (18750 – 3125) = 15625 which by 23.4/18 = 20312 1⅓
A H = 35.0 (18750 × 60)/90 = 12500 which by 35/18 = 24306 1⅔
A F = 48.5 (18750 × 45)/90 = 9375 which by 48.5/18 = 25260 1⅔
A K = 62.6 (18750 × 30)/90 = 6250 which by 62.6/18 = 21736 1⅓
A G = 77.6 (18750 × 15)/90 = 3125 which by 77.6/18 = 13472 1
Column 1, gives the name of the rod; col. 2, the calculated diagonal
length; col. 4, the applied weight, (the varying weight by reason of the
varying inclination) found by multiplying the whole weight upon one
panel or post by the distance of that post from the abutment, and
dividing the product by the span. (Thus the load applied to A G is
(W × IB)/S,
that on A K is
(W × BX)/S,
and so on.) Col. 6, shows the increase found by col. 5 on account of
inclination as noticed in Chap. VIII.; and finally, col. 4 gives the
necessary sectional area of the bars or rods.
The compression on the top chord is evidently the sum of the
compressions of the separate systems; the compression from any one
system is as follows, fig. 102.
[Illustration: Fig. 102.]
Let _a d_, _c d_ be the rods, and _a b c_ the chord; also _b d_, the
post; now if _d b_ represents the weight, _e h_ shows the tension on a
lower, or the compression on an upper chord; the triangles _a c d_ and
_a b e_ are similar; as also _e b h_ and _d b c_; whence
_be_ = (_ab_ × _cd_)/(_ac_);
and
_eh_ = (_cb_ × _be_)/(_dc_) compression.
Numerically we have the following figures:—
In the first system,
_be_ = (15 × 77.6)/90 = 13,
also
_eh_ = (75 × 13)/77.6 = 12.
In the second system,
_be_ = (30 × 62.6)/90 = 21,
also
_eh_ = (60 × 20)/62.6 = 20.
In the third system,
_be_ = (45 × 48.5)/90 = 24,
also
_eh_ = (45 × 24)/48.5 = 23,
that is, the compression from the system A C B, is to the weight on the
post, as twelve is to the length of the post; or actually
18 to 12 as 18,750 to compression;
whence
compression = (18750 × 12)/18 = 12500
in system one, and in the second system
18 to 20 as 18750 to 20833.
In the central system,
18 to 23 as 18750 to 24000.
Doubling the sum of the first and second systems, and adding thereto the
central, we have
2(12500 + 20833) + 24000 = 90666 lbs.,
as the whole compression upon one side of the bridge.
As to compression only, this would require a section of about four
square inches of cast-iron, which may be obtained by a tube of four and
one half inches inside, and five inches outside diameter. We may however
need to increase this amount to resist flexure, or transverse strains;
in which case the length of tube in one panel is to be regarded as the
height of a post, or the length of a beam; and the size will be found by
the table on page 138.
Each post must bear 18,750 lbs., and these being of cast-iron, to resist
flexure, by the same table above referred to, should, if made as a
hollow cylinder, be a little over four inches in diameter, and one half
inch thick; and if of + or ¶ᕼ¶ section, should have a square of nearly
five inches.
The flooring will be dimensioned by the rules given in Chapter VIII. for
single beams.
There is nothing about this bridge to burn, in case of fire, except the
floor; and that might easily be made of iron.
To use the words of the inventor, “The permanent principle in bridge
building sustained throughout this mode of structure, and in which there
is such gain in competition with any other, namely, the direct transfer
of weight to the abutment, renders the calculation simple, the expense
certain, and facilitates the erection of secure, economical, and durable
structures.”
WHIPPLE’S IRON BRIDGES.
223. The bridges built by the above-named engineer are in all respects
well proportioned, rigid, safe, and durable. Cast-iron is used as a top
chord, and wrought iron is employed to resist the tensile forces. The
plan put up upon the New York and Erie Railroad, consists of a hollow
cast-iron top chord, circular in section. Lower chords of wrought iron
rods. Posts cast cruciform in section. Diagonal tension rods, as in
Pratt’s bridge, (Chapter VIII.). The whole structure is in iron exactly
what the above-named bridges are in wood; and the method of calculation
is the same. For spans not exceeding one hundred feet, this form answers
every purpose as a railroad bridge. It is open to the same objection in
larger spans as are all trusses transferring the load by a series of
triangles through which the weight passes successively, namely, the
effect of an enormous pressure at the feet of the second and third pairs
of braces, which should be taken up by arch braces, as in fig. 69; or by
rods from the top of the abutment pillars to the feet of the second and
third sets of posts.
A span of this plan, upon the New York and Erie Railroad, of forty feet,
and which weighed only three tons, supported a load of fifteen hundred
pounds per lineal foot for two days; when the bridge had settled nearly
one half inch. A load of rails weighing 1318 lbs. per foot (of bridge)
was then rolled over, upon a truck without springs, thus making the
whole load upwards of 2,800 lbs. per foot, when the whole deflection was
three fourths of an inch. Upon removing the load the bridge returned to
its original position, within one fourth of an inch.
SUSPENSION BRIDGES.
224. Suspension bridges of large span have been generally considered as
entirely unfit for railroad purposes; but John A. Roebling has proved
the contrary by erecting a wire suspension railroad bridge of eight
hundred feet clear span across Niagara River; which with heavy loads and
violent gales has shown itself to be both stiff and strong to any
desired amount. The construction of a bridge upon any other plan would
have been hardly possible at the site of Mr. Roebling’s Niagara bridge,
there being no opportunity for scaffolding or for piers, pontoons or
hydraulic presses.
The simple road-way supported by cables, possesses great strength with
very little stiffness. It must be accompanied by stays and trusses to
check vibration.
No bridge involves more simple calculations, and in none can we proceed
with more absolute safety, than in the wire suspension. European
suspension bridges are generally formed of cables made by linking bars
of wrought iron together. This method is more expensive and more liable
to failure than the American plan of forming cables of iron wire. An
apparently good bar may be defective inside, while we are sure of every
component fibre of the cable; indeed it is very little trouble to test
each wire as it is laid into the cable.
The parts to be considered in proportioning a suspension bridge are
The anchoring masonry,
The anchor chains,
The towers and plate,
The suspension cables,
The suspending rods,
The stiffening arrangement,
The road-way.
The data given in the construction of a bridge of this description are
The span,
The load to be supported.
The assumed data
The versed sine of the cable,
The width.
And the required elements
The length of cable,
Lengths of suspending rods,
Angle of tangent of cable at point of suspension with axis of tower,
Tension upon the cables,
Section of the anchor irons,
Amount of anchoring masonry,
Size of the towers,
Dimensions of trussing and of road-way.
OF THE CABLES.
225. The curve formed by the cable of a suspension bridge lies between
the parabola and the catenary. When loaded the curve is nearly the
former, and when unloaded the latter.
_Problem 1._
Given the horizontal distance between the points of suspension and the
versed-sine, to find the length of the cable, fig. 103.
[Illustration: Fig. 103.]
Represent C E by _b_, and E F by _a_, and the length of the semi-curve
is
_L_ = _b_[1 + ⅔(_a_/_b_)^2].
Let the half span be five hundred feet, and the versed-sine or
deflection eighty feet, the formula becomes
_L_ = 500[1 + ⅔(80/500)^2] = 500 × 1.0171 = 508.55 feet,
which is the half length of cables between towers.
_Problem 2._
226. To find the length of the suspending rods. Calling E the horizontal
distance between the vertical suspenders, we have the formula
_X_ = (_Y^2_)/(_b^2_) × _a_,
in which we place E, 2E, 3E, etc., in place of Y, thus calling the rods
one hundred feet apart, we have
┌───────────────────────┬───────────────────────┬───────────────────────┐
│ Centre. │ Rod 1. │ Rod 2. │
├───────────────────────┼───────────────────────┼───────────────────────┤
│ 0 │ (_E_^2)/(_b_^2) × _a_ │(4_E_^2)/(_b_^2) × _a_ │
│ 0 │ (100^2)/(500^2) × 80 │ (200^2)/(500^2) × 80 │
│ 0 │ 3.20 │ 12.80 │
└───────────────────────┴───────────────────────┴───────────────────────┘
┌───────────────────────┬───────────────────────┬───────────────────────┐
│ Rod 3. │ Rod 4. │ Rod 5. │
├───────────────────────┼───────────────────────┼───────────────────────┤
│(9_E_^2)/(_b_^2) × _a_ │(16_E_^2)/(_b_^2) × _a_│(25_E_^2)/(_b_^2) × _a_│
│ (300^2)/(500^2) × 80 │ (400^2)/(500^2) × 80 │ (500^2)/(500^2) × 80 │
│ 28.80 │ 51.20 │ 80.00 │
└───────────────────────┴───────────────────────┴───────────────────────┘
_Problem 3._
227. To find the angle E C G, fig. 103. The formula for the angle
between the axis of the tower, and the tangent to the curve of the cable
at the point of suspension is
tang _a_ = E C G = (2_a_)/_b_.
Span being one thousand feet, _b_ is five hundred; and _a_ being eighty
feet, we have
tang E C G = 160/500 = log 160 – log 500:
or 2.204120 – 2.698970 = tang 9.505150 = 17° 45′ = E C G.
Also, 90° – 17° 45′ = 72° 15′ = angle G C A, or A C H.
When the points of suspension are not at the same elevation, we proceed
in the same manner: only using G L, G E, in place of F L, F C, in fig.
103 A.
[Illustration: Fig. 103 A.]
That the resultant of the forces acting upon the top of the tower may be
vertical, the angles G C A, and A C H, fig. 103, must be equal; if not,
the masonry must be so arranged as to cause the resultant to pass
through the centre of gravity. When more than one span is used, and the
openings are unequal, that the intermediate pier or piers shall not be
pulled over, the cable of the largest, and consequently heaviest span,
must have a greater inclination from the horizontal than that of the
shorter span; the product of the tensions by their respective
inclinations must be equal. Mr. Roebling’s plan in connecting several
spans, is to attach the cables of adjacent spans to a pendulum upon the
pier, by which arrangement the difference in tension upon the different
cables swings the pendulums, without racking the masonry.
_Problem 4._
228. Given the weight per foot of bridge and load, to find the tension
at the lowest point of the curve. The formula for the minimum tension,
that at the vertex F of the curve, is
_T_ = (_ph^2_)/(2_f_);
where _p_ is the weight per foot of bridge and load, _h_ the half
distance between the points of suspension, and _f_ the versed-sine. Thus
the span being one thousand feet, the versed-sine eighty feet, and the
load per lineal foot six thousand lbs., the formula becomes
_T_ = (6000 × 500^2)/160 = 9375000 lbs. or 4185 tons.
The maximum tension is at the points of support, and is expressed by the
formula
_T_ = ((_ph_)/(2_f_))[_h^2_ + 4_f^2_]^½:
which, in the case before us, becomes
_T_ = ((6000 × 500)/160)[500^2 + 4 × 80^2]^½ = 4395 tons.
229. The object of the anchoring is to connect the cable with a
resistance upon the land side, which shall more than balance the weight
and momentum of the bridge and load upon the opposite side. The
anchoring of the Niagara bridge consists of an iron chain made of flat
links, 7 feet long, 7 inches wide, and 1.4 inches thick; the chain links
consist alternately of six and of seven of these bars; see fig. 104.
[Illustration: Fig. 104.]
In the Fribourg bridge (Switzerland) the anchorage is made as in fig.
105, (see p. 220,) by a cable in place of the chain. In M. Navier’s
suspension bridge at Paris, over the Seine, the anchorage depended
somewhat upon the natural cohesion of the earth forming the bank of the
river, and this being destroyed by the bursting of a water-pipe in the
vicinity, the bridge fell. When there is no natural rock for an
anchorage, the masonry of the shaft must, by its own weight, resist the
tension.
230. The height of the towers must be at least as much as the
versed-sine of the cable. Their duty is to support the whole bridge and
load. The breadth and thickness of these columns must be determined more
with a view to opposing lateral, than downward strains. The former
result from the horizontal vibrations of the bridge caused by the action
of the wind. Tremor and vibration caused by a passing load, tend to pull
the towers into the river. The section for weight only might be very
small. From the practice of the best builders, a mean section of one
fifth of the height seems to give the best results; thus, if a tower is
sixty feet high, the mean thickness should be twelve feet; or the top
being 8 × 8 feet, the bottom should be 16 × 16 feet.
If the bridge is so little braced laterally as to swing, a dangerous
momentum will be generated which would very much increase the strain,
both upon the masonry and upon the cables.
231. The object of the stiffening truss is to transfer the weight
applied at any one point over a considerable length, and to prevent
vibration. Its dimensions should, therefore, be those of the
counterbracing in an ordinary truss.
Any applied load produces a certain depression in the bridge: to use the
words of Mr. Roebling, “every train that passes over the bridge causes
an actual elongation of the cables, and consequently produces a
depression. If the train is long, and covers nearly the whole length of
the bridge, and is uniformly loaded, the depression will be uniform. If
the train is short, and covers only a part of the floor, the depression
will be less _general_ and more _local_; and will be the joint result of
an elongation of the cables, and of a disturbance of the equilibrium.
Depressions will be in direct proportion to the loads, and indirectly as
the length of train.” The amount of depression depends on the elongation
of the cables; the elongation upon the length. The depression is shown
by the formula
_D_ = √(¾[_V^2_ – _d^2_]) – √(¾[_l^2_ – _d^2_]).
where _D_ = depression,
_l_ = half length of curve before elongation,
_V_ = half length of curve after elongation,
_d_ = half distance between points of suspension.”
The effect of heat, by expanding the cables, is also to depress the
road-way; the amount being shown by the expression
_D_ = √(¾[_V^2_ – _d^2_]) – √(¾[_l^2_ – _d^2_]).
_V_ being the length of semi-curve as elongated by heat instead of by
tension; the elongations, both by heat and tension, being found by table
on page 193.
Upon the top of the towers is placed a pair of cast-iron plates
separated by rollers; the upper plate (the saddle) is thus enabled to
move over the lower one when pulled either way by the movement of the
cables.
The length of the half cable between towers being generally greater than
the distance from the top of the tower to the anchoring, expands more,
when the saddle moves towards the land side. The dimensions of these
castings must be sufficient to resist the whole weight of bridge and
load.
232. As an example of the preceding formula, take the following:—
Assume the span as 1,000 feet.
Height of towers 100 feet.
Deflection of cables 90 feet.
Weight per foot (lineal) of bridge 2,500 lbs.
Weight per foot (lineal) of load 2,000 lbs.
Whole weight per foot 4,500 lbs.
Total weight 4,500,000 lbs.
CABLES.
The formula for the half length of cables between tops of towers is
_L_ = _b_[1 + ⅔(_a_/_b_)^2],
which becomes
_L_ = 500[1 + ⅔(90/500)^2] = 510.80,
which doubled, is 1021.38. To this add double the distance from the top
of tower to the anchorage, (see page 206,) which is found as follows:—
tang E C G = (2_a_)/_b_.
Also, tang E C G = log 2_a_ – log _b_, or 2.255273 – 2.698970 = tang
9.556303 of which the angle is 19° 48′ and 90° – 19° 48′ is 70° 12′ =
angle G C A or A C H.
The height of the tower being one hundred feet, and the angle at the
tower 70° 12′, we have
Sin 19° 48′ 9,529,864
Sin 90° 00′ 10,000,000
log height (100) 2,000,000
log distance (295.2) 2,470,136
which double, and we have 590.4; finally, add twice the breadth of the
tops of the towers, and the whole length of cable is, from anchorage to
anchorage,
1021.38 + 590.40 + 16 = 1627.78 feet.
The formula for the maximum tension, (that at the point of suspension,)
is
_T_ = ((_ph_)/(2_f_))√(_h^2_ + 4_f^2_),
which becomes
_T_ = ((4500 × 500)/180)√(250000 + 32400) = 2966 tons.
Number 10 iron wire (20 feet per lb.) will support 1,648 lbs. per
strand; this is the ultimate strength; the maximum load for safety is
400 lbs. per strand; whence 2,966 tons, or 6,642,500 lbs. will require
16,606 strands; and if we use two cables, each must have 8,303 wires; or
four cables of 4,151 each. The permanent load on suspension bridges
should never be more than _one sixth_ of the ultimate strength; _one
eighth_ is a good standard. The accidental load should never exceed one
fifth of the whole strength of the cables. The permanent weight
supported by the Niagara bridge is only one twelfth of the ultimate
strength of the cables.
ANCHOR CHAINS.
The maximum tension being 6,642,500 lbs. the whole section of the four
anchorings will need to be
6642500/15000 = 443 inches,
or 111 square inches for each shaft; which is obtained by eleven links
ten inches wide and one inch thick. If we so attach the anchor chains to
the masonry as to reduce the tension one fourth at the first arch, (see
Fribourg anchoring,) we may fasten three bars of the chain at that
point, and descend from the first to the second arch with eight bars;
and leaving two bars at that point, proceed to the bottom with the
remaining six.
Where there is no natural rock to build the masonry into or against,
enough artificial stone must be put down to balance the bridge and load.
ANCHORING MASONRY.
The entire weight of the bridge and load being 4,500,000 lbs. and the
whole tension, as above found, 6,642,500 lbs., or upon each tower
3,321,250 lbs.; this is the tension tending to draw the masonry out of
each shaft. This tension must be reduced on account of the inclination
of the pulling force. The tower is one hundred feet high. The distance
on the line of tension from the top of the tower to the anchoring, as
already found, is 295.2 feet; whence the actual effort to move the
anchor masonry, is thus,
295.2 to 100 as 1,660,625 to the effort or 562,542 lbs. If rock
weighs 160 lbs. per cubic foot, which is resisted by a column of
masonry of 3,321,250/160 = 20,758 cubic feet, or 20 × 20 × 52 feet,
or by a mass 15 × 15 × 91 feet.
TOWERS.
The height of towers being one hundred feet, and the mean thickness
being one fifth of the height, we have mean section 20 × 20; or top 12 ×
12, and base 28 × 28.
SUSPENDING RODS.
Assuming the horizontal distances between the centres of the vertical
suspenders as five feet, their lengths, then, will be found by formula
_X_ = (_Y^2_)/(_b^2_) × _a_;
and placing for _Y^2_ the distances 5, 10, 15, 20, etc., we have,
commencing at the centre,
┌───────┬──────────────┬───────────────┬───────────────┐
│Centre.│ 5 │ 10 │ 15 │
├───────┼──────────────┼───────────────┼───────────────┤
│ 0 │5^2/500^2 × 90│10^2/500^2 × 90│15^2/500^2 × 90│
│ 0 │ .009 │ .036 │ .081 │
└───────┴──────────────┴───────────────┴───────────────┘
┌───────┬───────────────┬───────────────┬───────────────┐
│Centre.│ 20 │ 25 │ 30 │
├───────┼───────────────┼───────────────┼───────────────┤
│ 0 │20^2/500^2 × 90│25^2/500^2 × 90│30^2/500^2 × 90│
│ 0 │ .144 │ .225 │ .360 │
└───────┴───────────────┴───────────────┴───────────────┘
┌───────┬───────────────┬───────────────┬───────────────┐
│Centre.│ 35 │ 40 │ 45 │
├───────┼───────────────┼───────────────┼───────────────┤
│ 0 │35^2/500^2 × 90│40^2/500^2 × 90│45^2/500^2 × 90│
│ 0 │ .490 │ .576 │ .729 │
└───────┴───────────────┴───────────────┴───────────────┘
┌───────┬───────────────┬─────────────────────┐
│Centre.│ 50 │ _d_ │
├───────┼───────────────┼─────────────────────┤
│ 0 │50^2/500^2 × 90│(_d_^2)/(_b_^2) × _a_│
│ 0 │ .900 │ _h_ │
└───────┴───────────────┴─────────────────────┘
and so on, until we arrive at the tower. Whatever distance above or
below the vertex of the curve the road-way is placed, is of course
constant, to be added to or taken from the above lengths.
The manner of putting in any camber is simple both in theory and
practice. The strain upon the suspenders is merely the direct weight of
the road-way and load. If this is 3,500 lbs. per foot, the five feet
supported by two rods (one each side) will weigh 17,500 lbs.; each rod
or wire rope must hold 8,750 lbs.; this can be done by a section of one
half inch area. For extra strains, however, on so large a span as 1,000
feet, one inch of area is not too large.
OF THE STIFFENING TRUSSES, GIRDERS, AND STAYS.
The object of the girders supporting the rails is to diffuse the applied
weight; these girders may be made of a Howe truss four or five feet
deep, by trussed girders, only simply deep and stiffly framed track
strings. They should be able to distribute the load applied at one point
at least fifteen or twenty feet. The side trusses transfer to a still
greater extent any applied load. Mr. Roebling estimates the combined
effect of trusses and girders in the Niagara bridge as transferring the
weight of a locomotive over a length of two hundred feet. This
transferring counteracts the _local_ depression. The Niagara truss is
formed by a system of vertical posts, five feet apart, and diagonal rods
passing from the top of the first post to the foot of the fifth; the
inclination being 45°, spreads the weight placed upon any one pair of
posts over twice the height of the truss, or about forty feet. As to the
actual dimensions of the girders supporting the rails, if we intend them
to spread an applied weight over forty feet, they must be as stiff as a
bridge of forty feet span. And as regards the truss, if we would
effectually distribute the applied weight and check vibration, the
trussing should be as strong as the counterbracing in a large span upon
the ordinary plans. The principle of trussing a suspension bridge may be
thus explained. See fig. 106. Suppose that in place of supporting the
three trusses D _s w_, _s m m′_, and _m′ m d_, upon piers at _w_ and
_m′_, we suspend these points from the cable A _c_ B. The cable is
flexible, and when we apply a load at _m_, the truss will assume the
position D _s c n d_, but between D and _s_, _s_ and _n_, _n_ and _d_,
the truss will be quite stiff. What we require, then, is to make the
figure _o p m m′_, incapable of changing its form, which is done by
diagonal bracing.
[Illustration: Fig. 106.]
233. Undoubtedly the finest specimen of a bridge of large span upon the
suspension principle, or indeed upon any principle, is that built by
John A. Roebling, across the Niagara River, a short distance below the
falls. The dimensions below of this admirable structure are from the
final report of the above-named engineer.
Length of bridge from centre to centre of tower 821′ 4″
Length of floor between towers 800 ft.
Number of wire cables 4
Diameter of each 10″
Solid wire section of each cable 60.40 sq. in.
Total section of four cables 241.60 sq. in.
Whole section of lower links of anchor irons 276 sq. in.
Whole section of upper links of anchor irons 372 sq. in.
Ultimate strength of chains 11,904 tons.
Whole number of wires in cables 14,560
Average strength of a wire 1,648 lbs.
Ultimate strength of four cables 12,000 tons.
Permanent weight supported by cables 1,000 tons.
Resulting tension 1,810 tons.
Length of anchor chains 66 ft.
Length of upper cables 1,261 ft.
Length of lower cables 1,193 ft.
Deflection of upper cables (mean temperature) 54 ft.
Deflection of lower cables (mean temperature) 64 ft.
Number of suspenders 624
Aggregate strength of suspenders 18,720 tons.
Number of over-floor stays 64
Aggregate strength 1,920 tons.
Number of river stays 56
Aggregate strength 1,680 tons.
Elevation of grade above mean water 245 ft.
Depth of river 200 ft.
Cost of the bridge $400,000.
231. The following items are extracted from the report above referred
to:—
“The trains of the New York Central, and Canada Great Western Railroads
have crossed regularly at the rate of thirty trips per day for five
months. (At present over two years.)
“A load of forty-seven tons caused a depression at the centre of five
and a half inches.
“An engine of twenty-three tons weight, with four driving wheels,
depressed the bridge at the centre 0.3 feet. The depression immediately
under the engine was one inch; the effect of which extended one hundred
feet.
“The depression caused by an engine and train of cars is so much
diffused as scarcely to be noticed.
“A load of three hundred and twenty-six tons produced a deflection of
0.82 feet only. The Conway tubular bridge deflects 0.25 feet under three
hundred tons; the span being only one half that of the Niagara bridge.
“The specified test for the wire was, that a strand stretched over two
posts four hundred feet apart should not break at a greater deflection
than nine inches; also, that it should withstand bending square and
rebending over a pair of pliers without rupture. This test corresponds
to a tensile strain of 90,000 lbs. per square inch, or I,300 lbs. per
wire of twenty feet per pound.”
The wire is preserved from oxidation by coating with linseed oil and
paint. Upon the durability of wire cables employed for suspension
bridges the following fact came to light: Upon taking down the cables of
the footbridge, put up in 1848, by Mr. Ellet, the wire was found so
little impaired that Mr. Roebling did not hesitate to work it into the
new cables; also, the original oil was found to be still soft and in
good condition, having been up six years.
That iron-work lying under ground has been completely covered with
cement grout, as this is found by the above-named engineer to be an
effectual guard against oxidation.
Engineers wishing to study the details of the Niagara bridge, will find
the final report of Mr. Roebling full of valuable matter, both as
regards the making of cables, anchoring, stiffening, and the effect of
passing trains.
NOTE.—This engineer is at present engaged upon a still greater work,
namely, a railroad suspension bridge across Kentucky River, of 1,224
feet span, 300 feet above the water. There is no lower road-way in
this bridge, the cross section being a triangle base upwards.
235. NOTE.—The Britannia tubular bridge, across the Menai Straits,
is doubtless a great work, and also an enormously extravagant one.
If no other structure were possible it would be admissible; but it
is equalled in strength and by far surpassed in economy by Mr.
Roebling’s system of trussed suspension bridges. The cost of
material alone in one span of the Britannia bridge, of 460 feet,
exceeds the entire cost of the Niagara bridge of 800 feet span; add
to this that we are sure of the strength of wire cables, but not of
tubes, and that the 800 feet span of the Niagara bridge weighs only
1,000 tons in itself against 1,400 in a 460 feet span of tube, and
it will not be difficult to prove the superiority of the suspension
over the tubular system; thus,
A suspension bridge of 800 feet span costs $400,000.
A tubular bridge of 460 feet span costs $500,000.
When we double the linear dimensions we increase the weight by the
cube; and the cost of a tube is very nearly as the weight; whence a
tubular bridge of 800 feet span will cost 2 × 2 × 2, or eight times
500,000, or $4,000,000 against $400,000. Thus,
Suspension 400,000 1
Tubular 4,000,000 10
[Illustration: Fig. 105.]
Fig. 105, shows the anchoring of the Fribourg bridge.
[Illustration: Fig. 107.]
[Illustration: Fig. 108.]
Fig. 107, the manner of fastening the ground stays of the Niagara
bridge.
Fig. 108, connection between cable and suspender.
[Illustration: Fig. 109.]
[Illustration: Fig. 109 A.]
Figs. 109, 109 A, another method of effecting the same.
[Illustration: Fig. 110.]
[Illustration: Fig. 111. Fig. 112.]
Fig. 110, floor beam attachment to suspender.
Figs. 111, 112, floor beam attachment in Niagara bridge.
[Illustration: Fig. 113.]
Fig. 113, connection of land and water cables in Fribourg bridge.
[Illustration:
Fig. 114.
Fig. 114 A.
]
Figs. 114, 114 A, fastening of cables at G, (fig. 105).
[Illustration: Fig. 115.]
Fig. 115, Mr. Roebling’s pendulum connection for the cables of two
adjacent spans.
BOILER PLATE BRIDGES.
Spans from 25 to 100 feet.
236. These structures fulfil every requirement of safe, durable, and
rigid bridges; being open however to the contingency attendant upon all
similar structures of wrought iron, namely, the becoming crystalline
when exposed to vibration. Time only will show whether this is a
sufficient cause for their non-adoption.
Each side truss consists as it were of a top and bottom chord connected
by a vertical web. The whole being of wrought iron, requires that the
section of the upper chord should be to that of the lower, as ninety to
sixty-six.
The general plan of such bridges is shown in fig. 116. This is the
patent wrought iron girder bridge of Mr. Fairbairn. The upper chord is
formed by connecting the four plates _a a a a_, by angle irons. The web
is formed either by a single or a double plate, stiffened laterally by T
iron placed at the vertical plate joints, as shown generally at B, and
detailed at C and D; or by a pair of plates separated by a space as at
B′, thus forming a rectangular tube. The lower chord is made by bending
horizontally the lower part of the web, and to the flanges thus formed
riveting the plate _m m_. The suspending rod _f_ is applied to the upper
chord by a washer as at E.
[Illustration: Fig. 116.]
The central connecting web, acting as do the braces and ties in a wooden
truss, should be more stiff at the ends of the span than at the centre.
This is easily effected by joining the web plates towards the end by
stronger T irons than at the centre. The joints for the rib, or the
vertical plates, either single or double, are shown in figs. C and D.
An example of the need of such increased stiffness towards the ends, was
given to the experimenters upon the Britannia model tube, which (tube)
was found to yield by buckling near the ends of the span sooner than
elsewhere. Thus advised, the vertical plates were made thicker as the
end of the span was approached. Examination of the principles of
proportioning a common wooden truss would have shown this without
experiment.
The tensile and compressive strength of rolled boiler plates (by the
table on page 193,) is, extension 12,740 lbs. per square inch,
compression 7,500 lbs. The strength of such work depends in a very great
measure upon the size and disposition of rivets. In plates exposed to
compression, the strength is not so much affected by riveting as in
those subjected to tensile strains; as to whatever amount the plate is
cut away, by the same amount is the resistance to tension reduced.
237. Mr. Fairbairn found that to obtain the maximum strength of riveted
plates, the section of the rivets should equal that of the plates, that
is, in a plate four inches wide, if there are two rivets, the area of
each must be one inch; or the diameter 1⅛ inches; thus leaving a section
of
4 – 2¼ = 1¾ inches,
which divided by four gives seven sixteenths of an inch as the distance
from the edge of the plate to the side of the first rivet; and seven
eighths of an inch between rivets. If the bolt yields by shearing, the
rim is destroyed by _detrusion_, or crushing across the fibres. That the
rivets and plates may be equally strong, their products of area of
section by the actual strength per unit of area must be equal. The
detrusive strength of wrought iron (see page 193) is 12,500 lbs. per
inch, whence the proportion
12,500 : 15,000 :: 1 : _d_
where 1 is the resisting length of the plate at right angles to tension,
and _d_, the sum of rivet diameters. Thus suppose we have a plate 13.2
inches wide, to be fastened with nine rivets of 0.8 inch diameter; we
have
9 × 0.8 = 7.2 = _d_,
and the above proportion becomes
15,000 : 12,000 :: 7.2 to 6 inches,
which is the length of plate section at right angles to tension. As
there are nine rivets, there will be eight spaces between them, and one
space at each edge of the plate, half as large as those between; or
reducing all to the same size,
8 × 2 = 16, 16 + 2 = 18;
and as the whole plate section after punching is six inches,
6/18 = .33 or ⅓ inch
for the edge space, and two thirds inch between rivets. Proceeding thus,
the result compares with the practice of Mr. Fairbairn as follows:—
Diameter of rivet. Distance between rivets.
Mr. Fairbairn ⅝ inch, or 0.625 inch, 0.8
Handbook ⅔ inch, or 0.666 inch, 0.8
The difference between the results, or 0.041 inch, less than one
sixteenth inch, will be partially absorbed by the remark of Mr.
Fairbairn that the area of the rivet should be _nearly_ as much as that
of the plate, and partly by the difference in results showing the
detrusional force of iron.
[Illustration: Fig. 117.]
238. In experimenting to determine the resistance of rivets, Mr.
Fairbairn found that by the common plan of riveting, fig. 117, the
strength of plates when whole, single, and double riveted, was as
follows, the section of the punched plate being in each case equal to
that of the whole one.
Whole plate, 100.
Single riveted, 56.
Double riveted, 70.
This loss of strength made him fearful of the ability of the tension
plates of the Britannia bridge to do their duty; and he was led to adopt
what he terms “chain riveting,” which consists in placing the rivets as
in fig. 118, or _in the same line of tension_. The strength of plates
thus made he considers as great at the joints as elsewhere.
239. As to the diameter of rivets, we have the following results of the
practice of the best English engineers.
Thickness of plate, ¼, 5/16, ⅜, 7/16, ½, 9/16, ⅝, 11/16, ¾.
Diameter of rivet, ⅝, 6/8, ⅞, 1, 1, 1⅛, 1¼, 1⅜, 1½.
240. As to the distance _in the direction of the force_ from rivet to
rivet, also from the first rivet to the plate end, we gather the
following from the best executed works in boiler plate. See fig. 118.
Plates exposed to compression,
_cb_ = 2 diam., _df_ = 1½ diam.
Plates exposed to extension,
_cb_ = 2½ diam., _df_ = 2 diam.
the diameter being that of the rivet.
The distance at right angles to the force has already been given.
241. If we knew the lateral adhesion of rolled plates, that is, the
resistance of the fibres to sliding horizontally past each other; we
should determine the distance of rivets in the direction of tension as
follows:—
Let _R_, equal the resistance per unit of area for _detrusion_ or
shearing, _R′_ the lateral adhesion of rolled plates, and we should have
_R_ × _a_ = _R′_ × (2 _d_ × _t_);
whence _d_ = (_R_ × _a_)/(2_R′_ × _t_);
where _a_ = area of rivet,
_d_ = distance,
_t_ = plate thickness.
also
2_d′_ + _d_ = (_R_ × _d_)/(_R′_);
and
2_d′_ = (_R_ × _d_)/(_R′_) – _d_;
whence
_d′_ = [(_R_ × _d_)/(_R′_) – _d_]/2
and finally
_d′_ = ½[(_R_ × _d_)/(_R′_) – _d_]
supposing the piece 1, 2, 3, 4, fig. 118, to split out.
The diameter of the semi-spherical head of the rivet should be three
times the thickness of the plate to be riveted; that of the conical head
four times; and the height of both of the heads, one and one half the
plate thickness.
242. Examples of the application of the preceding remarks.
Suppose we wish to build a boiler plate bridge of one hundred feet span,
twelve feet rise, weight of bridge and load 3300 lbs. per lineal foot.
The tension by formula
_T_ = (_WS_)/(8_h_) (see Chap. VIII.)
becomes
33000000/96 = 343,750 lbs.
Each side truss will bear one half of this or 171,875 lbs., and as
wrought iron resists eleven thousand pounds of compression per square
inch, the required section of the top chord will be
171875/7500 = 22.9 square inches.
Also the lower chord resisting fifteen thousand pounds per square inch,
must have
171875/12740 = 13.5 square inches
of area nearly.
If we make the tube at top of one fourth inch iron, and 8 × 10 inches;
fastening the plates by one fourth inch angle iron, four inches on the
side, the section becomes
One top plate 10 × ¼ = 2½ square inches.
One bottom plate 10 × ¼ = 2½ square inches.
Two side plates 8 × ¼ = 4 square inches.
Four angle irons ¼ × 8 = 8 square inches.
—
In all, 17 square inches.
In the lower chord, if we bend the web plates (of ⅜ inch) so as to form
a flange of eighteen inches in width, and to that rivet a bottom plate
18 × ¼, we shall have
In the flanges, 18 × ⅜ = 6¾
Bottom plate, 18 × ¼ = 4½
———
In all, 11¼
The web acting as both ties and braces, must be able to support the
following load.
Whole weight of bridge and load is, in round numbers, 344,000 lbs.
One half, 172,000 lbs.
And upon each end of the truss, 86,000 lbs.
to resist which, at eleven thousand pounds per square inch, requires
eight inches nearly, regarding the plate as a brace.
Now the side of the bridge being one hundred feet long, and twelve feet
wide, will contain any system of bracing that we choose to draw thereon.
Suppose, for example, that we chalk a line upon the erected bridge
representing an arch-brace, extending from the end to the centre. Such a
brace has actual existence in the bridge; and the same idea holds good
for any system of braces that may be assumed. We ought, therefore, to
take the most disadvantageous system that can have place, and giving
such a good bearing upon the abutment, estimate its width and thickness.
Suppose that we draw a natural size representation of Howe’s bridge, the
end braces must support a load of eighty-six thousand pounds, which at
eleven thousand lbs. per inch, requires a section of nearly eight
inches; and if the plate is one half inch thick, the brace must be
sixteen inches deep. The manner, however, in which the plate would yield
is by bulging laterally; which is to be checked by the before-mentioned
T connecting irons at the sides. It may be thought that the above method
of considering the plates as braces, would give very little thickness by
assuming very wide plates. The answer to this is, that the side plates
must not be so thin as to need more stiffening angle irons by weight,
than a thicker plate with less stiffening. Of course the weight should
be minimum.
243. As an actual example of this plan we have the following, built by
Mr. Fairbairn for the Blackburn and Bolton Railroad, across the Leeds
and Liverpool Canal.
Span, 60 feet,
Length, 66 feet,
End bearings, each, 3 feet,
Rise, 5 feet,
Width, 28 feet,
for a double track. Top chord of three eighths iron, web of five
sixteenths, lower flange of three eighths, and vertical web plates
stiffened by T irons.
This bridge was tested as follows:—
Three engines, weighing twenty tons each, running from five to
twenty-five miles per hour, deflected the bridge .025 feet. Two wedges,
one inch high, being placed upon the rails, and the engines being
dropped from that height, the bridge was deflected at the centre .035
ft.; with wedges of one and one half inches the deflection was .045 ft.
The cost of this bridge (in England) was estimated by Mr. Fairbairn at
$4,500, while that of a cast-iron bridge of the same span was $7,150.
244. _Example 2._—Manchester, Sheffield, and Lincolnshire Railroad
(England) Bridge, at Gainsborough, on the river Trent. Two spans, each
one hundred and fifty-four feet. Rise twelve feet. Top chord, double
rectangular tube, 36¾ × 16 inches, vertical web as before, and
horizontal plate for the lower chord. The floor beams are wrought iron
girders, cruciform in section, ten inches wide, and one foot three
inches (15″) deep, placed four feet apart.
245. _Example 3._—Fifty-five feet boiler plate bridge, built by James
Millholland, in 1847, for the Baltimore and Susquehanna Railroad
Company. Each truss consists of two vertical plates 55 × 6 feet, formed
of plates thirty-eight inches wide by six feet deep, the plates being
fastened together by bolts passing through cast-iron sockets. The lower
chord is formed by riveting two bars 5 × ¾ inches to each side of each
truss plate; making in all eight. Top chord—one bar of the same size on
each side of each plate, compression being made up by a wooden chord
between the plates. Height of bridge, six feet; length, fifty-five feet;
width, six feet; weight, fourteen tons; cost, $2,200, or forty dollars
per foot. The inventor thinks thirty dollars per foot enough when a
considerable amount of such bridging is wanted.
NOTE.—White, buff, or some light color should be used in painting
iron bridges, as such throw off, and do not absorb heat from the
sun.
CHAPTER X.
STONE BRIDGES.
A complete treatise on stone bridging would be of little practical value
to the American engineer, and would occupy too much of the necessarily
small space allowed here. The object in the present chapter is to give
the manner of dimensioning stone arches of from ten to sixty feet span,
and of proportioning retaining walls, piers, and abutments.
CONTRACTION OF THE WATER-WAY.
246. In building a bridge across a stream, we must be careful not to
obstruct the water-way so as to prevent free passage to the highest
floods. Regard must be had to this in fixing the size of the spans, and
the thickness and number of the piers. By contracting the width of the
stream the velocity is increased beneath the arches, the same amount of
water being obliged to pass through a smaller space, and when the bottom
is of such a nature as to yield to this action, there is danger of the
foundation being undermined. If the form and size of the piers be so
arranged as not to increase the velocity, such danger will be avoided
and floods will pass without harm. In bridges crossing navigable
streams, if the bottom is not destroyed the velocity may be made so
great as to impede navigation.
247. The following table is from Gauthey, _Construction des Ponts_,
showing the velocities which are just in equilibrium with the material
composing the bottom of the stream.
State of the water. Velocity in feet per second. Nature of bottom.
Torrents, 10′ 0″ Large rocks.
Floods, 3′ 3″ Loose rocks.
Common, 3′ 0″ Gravel and stones.
Regular, 2′ 0″ Fine gravel.
Moderate, 1′ 0″ Sand.
Slow, 0′ 6″ Clay.
Very slow, 0′ 3″ Common earth.
248. If _b_ represents the width of the natural water-way; _c_, that as
reduced by the structure; _V_, the velocity of the stream in the natural
state; then the augmented velocity is expressed by
_W_ = _mV_(_b_/_c_);
and _c_ = (_mbV_)/_W_;
where _m_ is a constant quantity expressing the contraction which takes
place in passing the narrow place, which, according to Du-Buat, is 1.09;
but depending somewhat upon the form of the bridge piers; adopting which
value, we have
_W_ = 1.09_V_(_b_/_c_);
and _c_ = (1.09_bV_)/_W_.
_Example._—Let the bottom be gravel, the width of the natural water-way
one hundred feet, the velocity one foot per second: now for a gravel
bottom the velocity must not exceed two feet per second, whence
_c_ = (1.09 × 100 × 1)/2 = 54½ feet;
which is the width of the contracted water-way; and 100 – 54½, or 45½
feet may be occupied by piers or other obstructions.
The amount of fall which the water suffers in passing the pier is found
by the following formula, the notation being the same,
fall = ((_V^2__m^2__b^2_) – _c^2_)/(64_c^2_)
Thus the velocity being one foot per second, _m_ being 1.09 and _b_ =
100; also, _c_ = 54½, we have
(1 × 1 × 1.09 × 1.09 × 100 × 100) – (54½ × 54½)
fall = ——————————————————————————————————————————————— = 0.047 ft.
64 × 54½ × 54½
The velocity of a river is greatest at its surface and at the centre of
the stream. In the same river the velocity is nearly as the square root
of the depth; thus the surface velocity being known, that for any other
depth may be easily found. The velocity of streams should always be
noted at the times of the highest floods. For measuring the velocity of
running water a bottle enough filled with water to maintain an upright
position, with a small rod placed through the stopper having a red flag
upon the upper end, answers very well. Velocities of undercurrents may
also be measured by so loading the bottle as to cause it to float two,
four, six, or ten feet below the surface.
OF THE FORM OF THE ARCH.
249. There are three general forms which may be given to the intrados of
a stone arch.
_Semicircular_, or one hundred and eighty degrees.
_Segmental_, less than one hundred and eighty degrees.
_Basket handle_, nearly elliptical, being formed by a number of
circular curves.
Full centre (semicircular) arches offer the advantages of great solidity
and ease of construction; but unless the springing lines are high,
contract considerably the water-way.
Segmental arches give the freest passage to the water, are easily built,
but throw a great horizontal strain upon the abutments.
The basket handle gives free passage to the water, when not too flat are
very strong, are easily adjustable to different ratios between the span
and the distance between grade and the spring line, and except making
the centres, are easily built. Whatever the form of the arch, the line
of arch springing should not be below high water.
The manner of tracing the full centre and segmental curves is too simple
to need remark.
250. In tracing the basket handle curve, the following conditions must
be observed:—
_The tangents at springing must be vertical._
_The summit tangent must be horizontal._
_The curve at springing must inclose the ellipse._
_The radius of summit must not be greater than the span._
The number of arches composing the curve must not be less than three,
nor more than eleven; and must be uneven. Perronet’s fine bridge of
Neuilly, over the Seine at Paris, has eleven centres. In spans of sixty
feet and under, it is unnecessary to use more than five centres.
[Illustration: Fig. 119.]
251. The three centred curve is described as follows, fig. 119:—
Let A B represent the span, and _c_ D the rise, with _c_ as a centre and
_c_ A as radius, describe the quadrant A F E; make the angle A C F 60°.
Parallel to F E draw D G, and parallel to F C draw G K. H is the centre,
and A G the arc of the springing curve; also GD is the arc, and K the
centre of the summit curve.
THE FIVE CENTRED CURVE.
252. The common construction of the five centred curve leaves the radii
of the extreme curves to be assumed. The following method fixes all of
the dimensions when the span and rise are given:—
Let _c_ B be half the span and _c_ D the rise.
Join D B.
Draw _n_ K through _n_ perpendicular to D B.
Make B _a_ equal to _c_ D.
Also _c e_ to _c a_.
Draw _e_ K′ _o_ and K _a m_.
K H′ and K′ are the centres, and H′ _m_ and H′ _o_ the lines
separating the several curves.
For spans of from twenty-five to one hundred feet, the five centred arch
answers every purpose; making a strong and well proportioned structure.
THICKNESS OF VOUSSOIRS, (RING STONES).
253. The thickness of the voussoir, or arch stone, depends upon the form
and size of the arch, the nature of the masonry, and the character of
the stone. No authority gives more reliable results than Gauthey, who,
for stone of average quality, with hammer dressed beds, laid in cement,
gives the following proportions between the span and depth of key:—
For spans _under six feet_ the depth should be _thirteen inches_.
From six to fifty feet, 13 inches plus 1/48 of the span.
From fifty to one hundred feet, 1/24 of the span.
For over one hundred feet, 1/24 of 100 plus 1/48 of the remainder.
Thus for a span of one hundred and ninety-six feet we have
(100 × 12)/24 + (96 × 12)/48,
or, 50 + 24 equal in all to seventy-four inches, or six feet and two
inches; whence the following table:—
Span of arch in feet. Thickness of voussoir in inches.
6 13 + 0 = 13
8 13 + 2 = 15
10 13 + 3 = 16
12 13 + 3 = 16
15 13 + 4 = 17
18 13 + 5 = 18
20 13 + 6 = 19
25 13 + 7 = 20
30 13 + 8 = 21
35 13 + 9 = 22
40 13 + 10 = 23
45 13 + 11 = 24
50 13 + 13 = 26
60 = 30 inches.
THICKNESS AND FORM OF ABUTMENTS.
254. The above depend upon the span and form of the arch, the height of
the abutment, and character of the masonry.
Different methods of determining the thickness of an abutment have from
time to time been given; several very correct rules have been arrived
at, but difficult of application. The most simple rule is given by
Hutton in the course of mathematics edited by Rutherford; it is as
follows:—
[Illustration: Fig. 120.]
Let A B, C D, fig. 120, be one half of the arch and A G F the abutment.
From the centre of gravity K of the arch, draw the vertical K L; then
the weight of the arch in the direction K L will be to the horizontal
thrust, as K L to L A. For the weight of the arch in the direction K L,
the horizontal thrust L A, and the thrust K A will be as the three sides
of the triangle K L, L A, K A; so that if m denotes the weight of the
arch,
(_LA_)/(_KL_) × _m_,
will be its force in the direction L A, and
(_LA_)/(_KL_) × _GA_ × _m_
its effect on the lever G A to overturn the wall, or cause it to revolve
about the point F.
Again, the weight or area of the pier is as EF × FG, and therefore EF ×
FG × ½FG, or ½FG^2 × EF, is its effect upon the lever ½FG, to resist an
overthrow. Now that the abutment and the arch shall be in equilibrium
these two effects must be equal to each other; whence we must have
½_FG^2_ × _EF_ = (_LA_)/(_RL_) × _GA_ × _m_;
whence
_FG_ = √((2_GA_ × _LA_)/(_EF_ × _RL_) × _m_).
The following table has been calculated for the use of builders and
engineers, giving the thickness of abutments for different spans and
heights.
255. THICKNESS OF RECTANGULAR ABUTMENTS.
│ Semicircular arch. │ Basket-handle arch.
│ The height being.
Span. │ 5 8 10 15 │ 5 8 10 15
6│ 3 3 3¼ 3½ │ 3½ 4 4 5
8│ 3½ 4 4¼ 4½ │ 4 4½ 5¼ 6
10│ 4 5 5 5 │ 4½ 5¾ 6¼ 7
15│ 4½ 5¼ 5½ 6 │ 5 6½ 7¼ 8
20│ 5 5½ 6 7 │ 6 7¼ 8 9
25│ 5½ 6 6½ 7¼ │ 7 7½ 8¼ 9½
30│ 6 7 8 8½ │ 8 8½ 9¼ 10
35│ 6½ 7 8¼ 9 │ 9 9¼ 10 11
40│ 7 7½ 8¾ 9½ │ 9½ 10 11 12
45│ 7½ 8¼ 9¼ 10 │ 10 10¾ 11½ 12½
50│ 8 9 10 11 │ 10¼ 11½ 12¼ 13
[Illustration:
Fig. 121.
Fig. 121 B.
Fig. 121 A.
]
256. The form of a bridge abutment will depend upon the locality and
upon the use to which the bridge is to be put, whether used for a
railroad, or for common travel; whether near a large city, or in a
location where appearance need not be regarded. Where a river acts
dangerously upon a shore, wing walls will be necessary. These wings may
be curved or straight, and may be simply the abutment produced, or may
be swung around into the bank at any required angle, until the winged
abutment, as in figs. 121, 121 A, 121 B, becomes the U abutment, fig.
124; or by moving the walls, W and W, parallel to themselves, takes the
form of the T abutment, fig. 122.
[Illustration: Fig. 122.]
The curved wing, in fig. 121, being arched, requires a little less
thickness, but at the same time is longer. B B, show the bridge seats.
The slope of the wings may be battered with an inclined coping, or
off-setted at each course. Wing walls, subjected to special strains or
to particular currents of water, require positions and forms
accordingly. In skew bridges, as in Chap. V., the wing, at the acute
angle, is longer and inclines less from the face of the abutment than
that at the obtuse angle. The more the wing departs from the face line
and swings round into the slope, the greater the thrust becomes upon it,
as the centre of pressure is raised; the thrust becomes a maximum when
the wing is inclined from forty-five to seventy degrees from the face of
the abutment. The body of an abutment, as well as any other retaining
wall, may be much stronger by giving it a trapezoidal instead of a
rectangular section, as the resisting leverage is thereby much
increased. Abutments may be to advantage buttressed in order to resist
special strains, as in case of the arches or braces of wooden bridges.
[Illustration: Fig. 123.]
[Illustration: Fig. 124.]
257. Railroad abutments except for a double track, require but little
breadth on top, except where the truss itself rests. The common T
abutment originated by Captain John Childe, and now in very extensive
use, seems to fulfil any requirement of a good abutment, see fig. 122,
page 242. B B is the bridge seat, and the mass T T takes the place of
wings. The difference of level of the top and of the bridge seat depends
upon the difference between the height of the bearing of the lower chord
of the bridge, and grade. The line of contact between the earth and the
wall is shown by _s s′ s″ s‴_. The length of the top of the masonry is
found thus. Suppose the slope to be one and one half to one, and the
whole height thirty feet, the whole horizontal length of slope is then
forty-five feet; from this we take the sum of the horizontal distances,
_s s′_ and _s′ s″_, and suppose these to be, respectively, six and eight
feet, we have the whole operation thus:—
30 × 1½ – 6 + 8 = 45 – 14 = 31 feet.
It may be advisable in very high abutments to lighten the masonry by an
arched opening as in fig. 123. The walls, also, of the U abutment (see
fig. 124), when large, may be pierced with arches to save masonry.
Probably the cheapest mode of bringing a bridge to the embankment is
that shown in fig. 125; A being the bridge seat for the main truss, and
B that for the trussed girder.
[Illustration: Fig. 125.]
PIERS.
258. The thickness of a pier may be considered either as depending upon
the weight of the superstructure, or as resisting the thrust of arches
or braces. For the first requirement, very little thickness would
suffice; for the second, it may require to be considerable. The
objection to thick piers is the expense, and the contracting too much
the water-way; the benefit, a large bearing surface, and in stone
bridges where there are several continuous spans, a saving of centring;
as where the piers are not able to resist the thrust of the arches, they
must all be carried up at once.
259. Piers supporting truss bridges, require very little thickness
provided a good foundation is obtained. The following table shows the
sufficient dimensions for the piers of wooden or iron trussed bridges,
when the masonry is good. (See First Class Masonry, specification, Chap.
IV.) From ten to twenty feet in height the batter is assumed at one
twelfth; from twenty to fifty feet in height at one twenty-fourth.
Span. Length of bridge seat. Width of seat.
20 to 40 feet, 20 feet, 4 feet,
40 to 60 feet, 20 feet, 4½ feet,
60 to 80 feet, 22 feet, 5 feet,
80 to 100 feet, 23 feet, 5½ feet,
100 to 125 feet, 23 feet, 6 feet,
125 to 150 feet, 24 feet, 6½ feet,
150 to 200 feet, 24 feet, 7 feet.
260. Upon the form of the up-stream end of the pier, or the starling,
depends, in a considerable degree, the contraction of the water-way. In
sluggish water the form is not of much importance, but in swift flowing
rivers a great deal depends upon the choice. The forms in use are the
rectangle, the rectangle terminated by right-lined triangles, and the
same terminated by curved-lined triangles, and finally the ellipse.
The latter is that which causes the least disturbance to the water, but
is also the most costly.
The effect of gyration at the shoulder, deserves notice, as it may be
the cause of the ruin of the foundation when the bottom is of yielding
material.
River beds being porous, springs work up through them with a force equal
to the whole depth of water; and whenever there is a means of escape for
such, its pressure will act upwards against any structure that comes
within its reach; and if four or five feet deep, is capable of moving
enormous weights. Such springs gave a great deal of trouble at the
foundation of the United States Dry Dock, at Brooklyn, N. Y. When
checked in one place they burst up in another, and to proceed with the
work it was necessary to allow them a passage through which to flow.
[Illustration: Fig. 126.]
261. However proper it may be to give to piers the proper form to cause
as little contraction as possible to the water, it is no less necessary
to give them strength to oppose the shocks to which they are subject
from floating ice, timber and shipping. The best method of breaking up
ice, when it comes in large masses, is by inclining the front of the
pier, as shown in fig. 126. The angle of the front being inclined from
30° to 50°. The ice running up this slope breaks by its own weight, and
falls off on either side.
The foundations of piers may be protected by sheet piling, (see chap.
XII.,) or the bottom, if soft, may be dredged out for a few feet and
filled in with loose rock.
The form of the down-stream end is not of so much importance as of the
upper one, but deserves consideration; as when the water is swift or the
bottom soft and yielding, the eddies caused by sharp angles wear upon
the soil in a dangerous manner.
CHAPTER XI.
MASONRY.
STONES.
262. The varieties of this material most commonly used in engineering
operations are granites, limestones, sandstones, slates, brick, and
artificial stones; the latter being made by compounding clays, limes,
and cements.
Rock taken from the surface, which has been exposed to the atmosphere,
is of an inferior quality to that found at a depth where it has been
exposed to a strong pressure; and is consequently denser. Therefore, in
opening a quarry it is advisable to excavate upon a hill-side and come
at once to the sound stone. Rock is generally found in beds, divided by
joints or seams, at which the natural adhesion is broken and the layers
are easily separated. When the quarry shows no natural line of
separation, one may be produced by drilling a line of holes at equal
distances from each other, into which conical steel pins are driven, and
the stone splits; the pins being placed in the plane of the required
seam.
263. Stone is used almost entirely to resist a compressive strain; as in
the voussoirs of an arch, or in the courses of a pier. The resistance of
stone to crushing, is as follows:—
Pounds per square inch.
Granite 10,000 to 16,000
Limestone 12,000 to 14,000
Sandstone 10,000
Marble 9,000 to 14,000
Firm, hard burned brick 2,600
Yellow burned brick 1,500
Red brick 1,200
Pale-red brick 900
Chalk 750
264. When stone cannot be found, brick forms an excellent substitute;
being made from clay earths, which can be found in almost any locality.
Bricks are well fitted for nice work, are cheap, and easy of transport.
The French, at Algiers, have used concrete, rammed in boxes so as to
make large cubes and other shapes. The structures built of this material
are found to be very nearly if not quite as strong as those of natural
rock.
LIMES, CEMENTS, MORTARS, AND CONCRETES.
265. Nothing is more important in the construction of masonry than good
cement; and generally, no part of construction is intrusted to more
ignorant persons. Under the above head are to be considered limes,
cements, sands, common mortar, hydraulic mortar, and concrete.
266. Lime is obtained by burning off the carbonic acid from the pure
limestones; when it is put up in air tight barrels and is unslacked
lime. Natural cements are composed of pure lime mixed with argil,
magnesia, iron, and manganese. Artificial cements are prepared by mixing
with pure lime, calcined clay, forge scales, powdered bricks which are
underburnt, and other materials of like nature. Cements made thus
artificially, are as good as those naturally hydraulic.
Lime is termed rich, poor, hydraulic, and eminently hydraulic, according
to its properties.
Rich or fat limes are those which double their volume in slacking and
dissolve in fresh water to the last particle. They absorb about 300 per
cent. of their weight of water.
Poor limes do not much increase their volume, do not dissolve
completely, and absorb 200 per cent. of water.
Hydraulic limes set in fifteen or twenty days after immersion, and
continue to harden as they grow older. After one year their consistency
is about that of hard soap.
Eminently hydraulic limes set in five or six days, and continue to
harden.
Limes are said to set when they will bear, without depression, a rod of
1/20 of an inch diameter loaded with ten or twelve ounces.
NOTE.—The following test was applied to every tenth cask of
Rosendale cement used upon the masonry of the United States Dry Dock
at the Brooklyn (N. Y.) Navy Yard. Cakes two inches in diameter and
three fourths of an inch in thickness, after being immersed five
days, were required to bear a rod of one twenty-fourth of an inch
diameter loaded with fifty lbs. Two bricks united with the cement
and immersed five days, were required to resist one hundred lbs.
before separating. The following shows the progress of hardening.
The force required to thrust a rod one twenty-fourth of an inch in
diameter through a cake three fourths of an inch in thickness, was,
after