=117. The Arched Rib, of both Steel and Masonry.=—During the past ten
or fifteen years the type of bridge structure called the arched rib has
come into much use, and its merits insure for it a wider application in
the future. It partakes somewhat of the nature of both truss and arch;
or it may be considered a curved beam or girder. The ordinary beam
or truss when placed in a horizontal position and loaded vertically
yields only vertical reactions. Under the same conditions, however, the
arched rib will produce both vertical and horizontal reactions, and the
latter must either be resisted by abutments of sufficient mass, or by a
tie-rod, usually horizontal, connecting the springing points of the rib.
The arched rib may be built solid, as was done in the early days of
bridge-building in this country when engineers like Palmer, Burr, and
Wernwag introduced timber arches in combination with their wooden
trusses, or as a curved plate girder, one of the most prominent
examples of which is the Washington Bridge across the Harlem River
in the city of New York; or, again, as a braced frame or curved
truss, like the 800 feet arched rib carrying the roadway traffic and
trolley cars across the Niagara gorge, or like those used in such
great railroad train-sheds as the Grand Central Station, New York,
the Pennsylvania stations at Jersey City and Philadelphia, and the
Philadelphia and Reading station in Philadelphia. Those are all
admirable examples of steel arched ribs, and they are built to sustain
not only vertical loads but, in the case of station roofs, the normal
or horizontal wind pressures.
Within a few years, less than ten, another type of arched rib has
been brought into use and promises to be one of the most beautiful as
well as the most substantial applications of this type of structure;
that is, the arched rib of combined steel and concrete. Many examples
of this type of structure already exist both in this country and in
Europe, probably the most prominent of which in this country is that at
Topeka, Kansas, across the Kansas River.
[Illustration: =FIG. 33.=]
[Illustration: =FIG. 34.=]
[Illustration: =FIG. 35.=]
The characteristic feature of this type of structure, so far as the
stresses developed in it are concerned, is the thrust throughout its
length, more or less nearly parallel to its axis, which is combined
with the bending moments and shears similar to those found in
ordinary bridge-trusses. This thrust is the arch characteristic and
differentiates it in a measure from the ordinary bridge-truss, while
the bending moments and shears to which it is subjected differentiate
it, on the other hand, from the pure arch type or a series of blocks
in which thrust only exists. The thrust, bending moments, and shears
in arched ribs are all affected by certain principal features of
design. Those features are either fixedness of the ends of the ribs
or the presence of pin-joints at those ends or at the crown. Fig. 33
represents an arched rib with its ends _D_ and _F_ supposed to be
rigidly fixed in masonry or by other effective means.
=118. Arched Rib with Ends Fixed.=—The railroad steel arched bridge at
St. Louis, built by Captain Eads between 1868 and 1874, is a structure
of this character. The three spans (two each 537 feet 3 inches and one
552 feet 6 inches in length from centre to centre of piers) consist of
ribs the main members of which are composed of chrome steel. It was a
structure of unprecedented span when it was built, and constituted one
of the boldest pieces of engineering in its day. The chords of the ribs
are tubes made of steel staves, and their ends are rigidly anchored
to the masonry piers on which they rest. It is exceedingly difficult,
indeed impossible, to fix rigidly the ends of such a structure,
and observations in this particular instance have shown that the
extremities of the ribs are not truly fixed, for the piers themselves
yield a little, giving elastic motion under some conditions of loading.
=119. Arched Rib with Ends Jointed.=—The rib shown in Fig. 34 is
different from the preceding in that pin-joints are supplied at each
end, so that the rib may experience elastic distortion or strain by
small rotations about the pins at _A_ and _B_. In the computations for
such a design it is assumed that the ends of the rib may freely change
their inclination at those points. As a matter of fact the friction
is so great, even if no corrosion exists, as to prevent motion, but
the presence of the pins makes no bending moment possible at the end
joints, and the failure to move freely probably produces no serious
effect upon the stresses in the ribs. The presence of these pin-joints
simplifies the computations of stresses and renders them better
defined, so that there is less doubt as to the actual condition of
stress under a given load than in the type shown in Fig. 33 with ends
fixed more or less stiffly. In Fig. 34, if the horizontal force _H_
exerted by the ends of the rib against the points of support is known,
the remaining stresses in the structure can readily be computed; but
neither in Fig. 34 nor in Fig. 33 are statical equations sufficient for
the determination of stresses. Equations of condition, depending upon
the elastic properties of the material, are required before solutions
of the problems arising can be made.
=120. Arched Rib with Crown and Ends Jointed.=—The rib shown in Fig.
35 possesses one characteristic radically different from any found in
the ribs of Figs. 33 and 34, in that it is three-jointed, one pin-joint
being at the crown and one at each end. So far as the conditions of
stress are concerned, this is the simplest rib of all. Since there is a
pin-joint at the crown as well as at the ends, the bending moments must
be zero at each of those three points whatever may be the condition of
loading. The point of application of the force or thrust at the crown,
therefore, is always known, as well as the points of application at the
ends of the joints. As will presently be seen, this condition makes
equations of statical equilibrium sufficient for the determination of
all stresses in the rib, and no equations depending upon the elastic
properties of the material are required. The stresses in this class of
ribs, therefore, are more easily determined than in the other two, and
they are better defined. These qualities have insured for it a somewhat
more popular position than either of the other two classes. The ribs
of the great train-sheds of the Pennsylvania and Reading railroads in
Jersey City and in Philadelphia belong to this class, while those of
the Grand Central Station at New York City belong to the class shown
in Fig. 34, as does the arched rib across the Niagara gorge, to which
reference has already been made.
=121. Relative Stiffness of Arch Ribs.=—Obviously the three-hinged
ribs are less stiff than the two-hinged ribs or those with fixed
ends. This is a matter of less consequence for station roofs than for
structures carrying railroad loads. The joints of the two-hinged rib
being at the ends of the structure, there is but little difference in
stiffness between that class of ribs and those with ends fixed. Indeed
the difference is so slight, and the uncertainty as to the degree of
fixedness of the fixed ends of the rib is so great, that the latter
type of rib possesses no real advantage over that with hinged ends.
=122. General Conditions of Analysis of Arched Ribs.=—In each of the
three types of arched ribs shown in Figs. 33, 34, and 35 it is supposed
that all external forces act in the vertical planes which contain the
centre lines of the various members of the rib. There are, therefore,
the three conditions of statical equilibrium expressed by the three
equations (35), (36), and (37). In practically all cases, except those
of arched ribs employed in roof construction, all the external loads
are vertical. In such cases the equations of statical equilibrium of
the entire structure may be reduced to two only, viz., equations (36)
and (37). These features of the problems connected with the design
of arched ribs will always make necessary, except in the case of the
three-hinged rib (Fig. 35), equations of condition depending upon the
elastic properties of the structure.
The rib represented by Fig. 33 is supposed to have its ends so fixed
that the inclinations of the centre line at _F_ and _D_ will never
change whatever may be the loading or the variation of temperature.
This requires the application at each of those points of a couple whose
moment varies in value, but which is always equal and opposite to the
bending moment at the same point produced by the loads imposed on the
rib. It is also to be observed that the loads resting upon the rib are
not divided between the points of support _F_ and _D_ in accordance
with the law of the lever, since the conditions of fixedness at the
ends are equivalent to continuity. There are then to be found, as
acting external to the rib, the two vertical reactions and the two
moments at _F_ and _D_, as well as the horizontal thrust exerted at
the ends of the structure, which is sometimes resisted by the tie-rod,
making five unknown quantities. Inasmuch as all external loading is
supposed to be vertical, equations (36) and (37) are the only statical
equations available, and three others, depending upon the elastic
properties of the structure, must be supplied in order to obtain the
total of five equations of condition to determine the five unknown
quantities. Inasmuch as the end inclinations remain unchanged, the
total extension or compression of the material at any given constant
distance from the axis of the rib taken between the two end sections
_F_ and _D_ must be equal to zero. Similarly, whatever may be the
amount or condition of loading, the vertical and horizontal deflections
of either of the ends _F_ or _D_ in relation to the other must be zero,
since no relative motion between these two points can take place. It
is not necessary in these lectures to give the demonstration of the
equations which express the three preceding elastic conditions, but if
_M_ is the general value of the bending moment for any point of the
rib, and if _x_ and _y_ are the horizontal and vertical coordinates of
the centre line of the rib, taking the central point of the section at
either _F_ or _D_ as an origin, those equations, taken in the order in
which the elastic conditions have been named, will be the following,
in which _n_ represents a short length of rib within which the bending
moment _M_ is supposed to remain unchanged.
F F F
⎲ ⎲ ⎲
⎳ _nM_ = 0; ⎳ _nMx_ = 0; ⎳ _nMy_ = 0. (57)
D D D
The second and third of these equations express the condition that
the vertical and horizontal deflections respectively of the two ends
in reference to each other shall be zero. The conditions expressed by
equation (57) are constantly used in engineering practice to determine
the bending moments and stresses which exist in the arched rib with
fixed ends. The graphical method is ordinarily used for that purpose,
as its employment is a comparatively simple procedure for a rib whose
curvature is any whatever.
If the rib has hinged joints at the ends, as in Fig. 34, obviously
there can be no bending moment at either of those two points, and hence
the two equations of condition which were required in connection with
Fig. 33 to determine them will not be needed. There is, therefore,
no restriction as to the angle of inclination of the centre line of
the rib at those two points. Again, it is obvious that either end _A_
or _B_ may have vertical movement, i.e., deflection in reference to
the other, without affecting the condition of stress in any member of
the rib; but it is equally obvious that neither _A_ nor _B_ can be
moved horizontally, i.e., deflected in reference to the other, without
producing bending in the rib and developing stresses in the various
members. The unknown quantities in this case are, therefore, only the
horizontal thrust _H_ exerted at the two springing points _A_ and
_B_, and the two vertical reactions, making a total of three unknown
quantities, equations for two of which will be given by equations
(36) and (37). The other equation required is the third expression in
equation (57), expressing the condition that the horizontal deflection
of either of the points _A_ or _B_ in respect to the other is zero,
since the span _AB_ is supposed to remain unchanged. By the application
of the graphical method to this case, as to the preceding, the
employment of equations (36), (37), and (58) will afford an easy and
quick determination of the three unknown quantities, whatever may be
the curvature of the rib.
A
⎲
⎳ _nMy_ = 0. (58)
B
If the reactions and horizontal thrust _H_ are found, stresses in every
member may readily be computed and the complete design made.
If the arch is three-hinged, as in Fig. 35, the condition that the
bending moment must be zero at the crown _C_ under all conditions of
loading gives a third statical equation independent of the elastic
properties of the structure which, in connection with equations (36)
and (37), give three equations of condition sufficient to determine the
two vertical reactions and the horizontal thrust _H_. In this case, as
has already been stated, no elastic equations of condition are required.
The determination of the end reactions, bending moments, and horizontal
thrust _H_, in these various cases, is all that is necessary in order
to compute with ease and immediately the stresses in every member of
the rib. These computations are obviously the final numerical work
required for the complete design of the structure. These procedures are
always followed, and in precisely the manner indicated, in the design
of arched ribs by civil engineers, whether the rib be articulated,
i.e., with open bracing, or with a solid plate web, like those of the
Washington Bridge across the Harlem River.