=124. The Masonry Arch.=—The masonry arch is so old that its origin
is lost in antiquity, but its complete theory has been developed with
that of other bridge structures only within the latest period. It is
only possible here to give some of the main features of that theory and
a few of the fundamental ideas on which it is based. It is customary
among engineers to regard the masonry arch as an assemblage of blocks
finely cut to accurate dimensions, so that the assumption of either a
uniform or uniformly varying pressure in the surface of contact between
any two may be at least sufficiently near the truth for all practical
purposes. Although care is taken to make joints between ring-stones
or voussoirs completely cemented or filled with a rich cement mortar,
it is usually the implicit assumption that such joints do not resist
tension. As a matter of fact many arch joints are capable of resisting
considerable tension, but, in consequence of settlement or shrinkage,
cracks in them that may be almost or quite imperceptible frequently
prevent complete continuity. It is, therefore, considered judicious to
determine the stability of the ordinary masonry arch on the assumption
that the joints do not resist tension.
In these observations it is not intended to convey the impression that
no analysts treat the ordinary arch as a continuous elastic masonry
mass, like the composite arches of steel and concrete. Although much
may be said in favor of such treatment for all arches, it is believed
that prolonged experience with arch structures makes it advisable to
neglect any small capacity of resistance to tension which an ordinary
cut-stone masonry joint may possess, in the interests of reasonable
security.
The ring-stones or voussoirs of an arch are usually cut to form
circular or elliptic curves, or to lines which do not differ sensibly
from those curves. The arch-ring may make a complete semicircle, as
in the old Roman arches, or a segment of a semicircle; or the stones
may be arranged to make a pointed arch, like the Gothic; or, again,
a complete semiellipse may be formed, or possibly a segment of that
curve. When a complete semiellipse or complete semicircle is formed,
the arches are said to be full-centred, and in those cases they spring
from a horizontal joint at each end. On the other hand, segmental
arches spring from inclined joints at each end called skew-backs.
=125. Old and New Theories of the Arch.=—In the older theories of the
arch, considered as a series of blocks simply abutting against each
other, the resultant loading on each block was assumed to be vertical.
In the modern theories, on the other hand, the resultant loading on any
block is taken precisely as it is, either vertical or inclined, as the
case may be. Many arches are loaded with earth over their arch-rings.
This earth loading produces a horizontal pressure against each of the
stones, as well as a vertical loading due to its own weight. In such
cases it is necessary to recognize this horizontal or lateral pressure
of the earth, as it is called, as a part of the arch loading.
It is known from the theory of earth pressure that the amount of that
pressure per square foot or any other square unit may vary between
rather wide limits, the upper of which is called the abutting power
of earth, and the latter the conjugate pressure due to its own weight
only. If _w_ is the weight per cubic unit of earth and _x_ the depth
considered, and if φ be the angle of repose of the earth, the abutting
power per square unit will have the value:
1 + sin φ
_p_ = _wx_ ---------. (64)
1 - sin φ
while the horizontal or conjugate pressure due to the weight of earth
only will be:
1 - sin φ
_pʹ_ = _wx_ ---------. (65)
1 + sin φ
The use of these formulæ will be illustrated by actual arch
computations.
[Illustration: FIG. 36.]
[Illustration: FIG. 37.]
Fig. 36 is supposed to show a set of ring-stones for an arch of any
curvature whatever. The joints _LM_ and _ON_ represent the skew-backs
or springing joints, while _R_ and _R₁_ represent the supporting
forces or reactions with centres of action at _aʹ_ and _a₁_. The ring
is divided into blocks or pieces by the joints at _a_, _b_, _c_, _d_,
and _e_, the resultant loading or force on each block being given by
the lines with arrow-heads and numbered 1, 2, 3, 4, 5, 6, and 7. Fig.
37 represents a force polygon constructed in the ordinary manner by
laying off carefully to scale the two reactions _R_ and _R₁_, together
with the loads or forces numbered 1 to 7, inclusive. By constructing
the so-called polygonal frame in the ring-stones of Fig. 36 in the
usual manner with its lines or sides parallel to the radiating lines in
Fig. 37, as shown by the broken lines, the points _a_, _b_, _c_, etc.,
are found where the resultant forces cut each joint. The line drawn
through those points thus determined is called the line of resistance
of the arch. Obviously, if that line of resistance be determined, the
complete stability or instability of the arch, as the case may be, will
be established. Furthermore, the complete determination of the force
polygon in Fig. 37, and the corresponding polygonal frame drawn in the
arch-ring, constitute all the computations involved in the design of an
arch.
The thrust _T₀_ at the crown, shown both in Fig. 36 and Fig. 37, is
frequently horizontal, although not necessarily so; its value is shown
by Fig. 37. In the older arch theories a principle was enunciated
called the “principle of least resistance.” The thrust _T₀_ is a
fundamental and so-called passive force. That is, its magnitude depends
not only upon its position, but also largely upon the magnitude of
the active forces which represent the loading on the arch-ring. Under
the principle of least resistance it was laid down as a fundamental
proposition, in making arch computations, that this passive force
_T₀_ must be the least possible consistent with the stability of the
structure. While this provisional proposition answered its purpose
well enough, there are other clearer methods of procedure which are
thoroughly rational and involve the employment of no extraneous
considerations other than those attached to the determination of
statical equilibrium.
A scrutiny of the conditions existing in Fig. 36 will show that if the
external forces or loadings on the individual blocks of the ring are
given, four quantities are to be determined, viz., the two reactions
_R_ and _R₁_ and their lines of action. Inasmuch as no elastic
features of the structure are to be considered, there are available
for the determination of these four quantities the three equations of
equilibrium, equations (35), (36), and (37), which are not sufficient
for the purpose. If one line of action, such as that of _R_, be located
by assuming its point of application _aʹ_, the three equations just
named will be sufficient for the determination of the remaining three
equations; and that is precisely the method employed. It is tentative,
but perfectly practicable. If, instead of assuming one of the points
of application of the reactions, we assume both of those points and
construct a trial polygonal frame, it will be necessary to use but two
of the three equations of statical equilibrium. For that purpose there
are employed equations (35) and (36), but in a graphical manner, which
will presently be illustrated.
=126. Stress Conditions in the Arch-ring.=—Before proceeding to the
construction of an actual line of resistance, a little consideration
must be given to the stress conditions in the arch-ring. As the joints
are considered capable of resisting no tension, the dimensions of the
arch-ring must be finally so proportioned that pressure only will exist
in each and every joint. If each centre of pressure, as _a_, _b_, etc.,
in Fig. 36, is found in the middle third of the joint, it is known from
a very simple demonstration in mechanics that no tension will ever
exist in that joint, although the pressure may be zero at one extremity
and a maximum at the other. This is the condition usually imposed in
designing an arch-ring to carry given dead or live loads. It is usually
specified that “the line of resistance of the ring must lie in the
middle third.” It must be borne in mind, however, that the stability
of the ring is perfectly consistent with the location of the line of
resistance outside of the limits of the middle third, provided it is
not so far outside as to induce crushing of the ring-stones. Whenever
that crushing begins the arch is in serious danger and complete failure
is likely to result.
=127. Applications to an Actual Arch.=—These principles will be applied
to the arch-ring shown in Fig. 38, in which the clear span _TU_ is
90 feet. The radius _CO_ of the soffit (as the under surface of the
arch is called) is 50 feet, the ring being circular and segmental. The
uniform thickness of the ring shown at the various joints is assumed
at 4 feet as a trial value. The loading above the ring to the level of
the line _EʹO_ is assumed to be dry earth weighing, when well rammed
in place, 100 pounds per cubic foot. The depth of this earth filling
at the crown _n_ of the arch is taken at 4 feet. The ring-stones are
assumed to be of granite or best quality of limestone, weighing 160
pounds per cubic foot. The thickness or width of arch-ring of one foot
is assumed, as each foot in width is like every other foot, and the
loads are taken for that width of ring. The rectangle _EJJʹEʹ_ is
supposed to represent a moving load covering one half of the span and
averaging 500 pounds per linear foot; in other words, averaging 500
pounds per square foot of upper surface projected in the line _EʹO_.
The total length of the arch-ring, measured on the soffit, is about
113 feet, and it is divided into ten equal portions for the purpose of
convenient computation. The radial joints so located are as shown at
_de_, _fg_, _hk_. From the points where these joints cut the extrados
(as the upper surface of the arch-ring is called) vertical broken lines
are erected, as shown in Fig. 38.
[Illustration: FIG. 38.]
The horizontal line drawn to the left from _f_ gives the vertical
projection of that part of the extrados between _d_ and _f_, and the
horizontal earth pressure on _df_ will be precisely the same in amount
as that on the vertical projection of _df_, as just found. In the same
manner the horizontal earth pressure on that part of the extrados
between any two adjacent joints may be found. The mid-depths of these
vertical projections below the line _E′O_ are to be carefully measured
by scale and then used for the values of _x_ in equations (64) and
(65), which now become equations (66) and (67), as the angle of repose
φ is taken to correspond to a slope of earth surface of 1 vertical on
1½ horizontal.
_p = 3.51wx._ (66)
_pʹ = 0.285wx._ (67)
The horizontal earth pressures thus found are as follows:
_h₁_ = { 101,500 pounds; _h₃_ = { 30,625 pounds;
{ 8,700 ” { 2,625 ”
_h₂_ = { 59,500 ” _h₄_ = { 9,800 ”
{ 5,100 ” { 840 ”
These quantities _h₁_, etc., are found by multiplying the two
intensities _p_ and _p′_ by the vertical projections of the surface
on which they act. The larger values are found by equation (66) and
represent the abutting power of the earth, while the smaller values
are found by equation (67), and represent the horizontal or conjugate
pressure of the earth due to its own weight only. The actual horizontal
earth pressure against the arch-ring may lie anywhere between these
limits.
The weights of the moving load, earth, and ring-stones between each
pair of vertical lines and radial joints shown in Fig. 38 are next to
be determined, and they are as follows:
_W₁_ = 27,300 pounds; _W₆_ = 12,300 pounds;
_W₂_ = 27,900 ” _W₇_ = 15,550 ”
_W₃_ = 24,500 ” _W₈_ = 19,500 ”
_W₄_ = 21,300 ” _W₉_ = 19,400 ”
_W₅_ = 18,300 ” _W₁₀_ = 24,300 ”
The centres of gravity of these various vertical forces are shown in
Fig. 38 at the points _W₁_, _W₂_, etc. The triangles of forces shown
in that figure and composed, each one, of a vertical and horizontal
force as described, are laid down in actual position on the arch-ring,
as shown. All data are thus secured for completing the force polygon
and polygonal frame or line of resistance. It will be assumed that
the reactions _R_ and _R′_ cut the springing joints at _c_ and _a_,
respectively, one third of the width of the joint from the soffit,
and it will further be assumed that _b_, the mid-point of the joint
at the crown, is also in the line of resistance. The assumption of
the location of these three points is made for the reason, as is well
known, that with a given system of forces a polygonal frame may be
found which will pass through any three points in the ring.
[Illustration: FIG. 39.]
The force polygon _B_, 1, 2, 3, ..., 10, _A_, Fig. 39, is then drawn
with the loadings on each ring segment found as already explained.
The horizontal forces are taken as represented by the smaller values
of _h₁_, _h₂_, _h₃_, _h₄_. Other force polygons with larger values of
these horizontal forces were tried and not found satisfactory. Having
constructed the force polygon and assumed the trial pole _Pʹ_, the
radial lines are drawn from it as shown in Fig. 39. The polygonal
frame shown in broken lines in Fig. 38 results from this trial pole.
The frame practically passes through _b_ and _c_, but leaves the ring,
passing outside of it, above the joint _VU_. The point _q_ in this
frame is vertically above _a_. The “three-point” method of finding the
frame that will pass through _a_, _b_, and _c_ was then employed. The
line _A6_, Fig. 39, was drawn; then _P′D_ was drawn parallel to _qb_,
Fig. 38 (not shown); after which _PD_ was drawn parallel to _ab_,
until it intercepted the horizontal line _PQ_, the line _PʹQ_ having
previously been drawn parallel to _qc_ (not shown). The final pole
_P_ was thus found. The polygonal frame shown in full lines in the
arch-ring was then drawn with sides parallel to the lines radiating
from _P_, all in accordance with the usual methods for such graphic
analysis. That polygonal frame lies within the middle third of the
arch-ring, although at three points it touches the limit of the middle
third. The arch, therefore, is stable.
This construction shows that, with the actual loading of the ring, a
line of resistance can be found lying within the middle third; its
stability under the conditions assumed is, therefore, demonstrated.
It does not follow that the line of resistance as determined must
necessarily exist, since there may be others located still more
favorably for stability. This indetermination results from the fact
already observed that the equations of statical equilibrium are not
sufficient in number to determine the four unknown quantities (the
two horizontal and the two vertical reactions); but the process of
demonstrating the stability of the arch-ring is simple and sufficient
for all ordinary purposes. The line of resistance found, if not the
true one, is so near to it that no sensible waste of material is
involved in employing it. This indetermination has prompted some
engineers and other analysts to consider all arch-rings as elastic,
thus obtaining other equations of condition. While such a procedure may
be permissible, it is scarcely necessary, and perhaps not advisable, in
view of the fact that many joints of cut-stone arches become slightly
open by very small cracks, resulting possibly from unequal settlement,
quite harmless in themselves, having practically no effect upon the
stability of the structure.
=128. Intensities of Pressure in the Arch-ring.=—It still remains to
ascertain whether the actual pressures of masonry in the arch-ring are
too high or not. The greatest single force shown in the force polygon
in Fig. 39 is the reaction _R_, having a value by scale of 122,000
pounds, under the left end of the arch, and it is supposed to act at
the limit of the middle third of the joint. Hence the average pressure
on that joint will be
122,000 × 2
----------- = 61,000 pounds per square foot.
4
This value may be taken as satisfactory for granite or the best quality
of limestone.
Again, it is necessary in bridges, as in some other structures, to
determine whether there is any liability of stones to slip on each
other. In order that motion shall take place the resultant forces
acting on the surface of a stone joint must have an inclination to
that surface less than a value which is not well determined and which
depends upon the condition of the surface of the stone; it certainly
must be less than 70°. The inclination of every resultant force in Fig.
38 to the surface on which it acts is considerably greater than that
value and, hence, the stability of friction is certainly secured.
=129. Permissible Working Pressures.=—The working values of pressures
permissible on cut-stone and brick or other masonry must be inferred
from the results of the actual tests of such classes of masonry in
connection with the results of experience with structures in which the
actual pressures existing are known. It is safe to state that with
such classes of material as are used in the best grade of engineering
structures these pressures will generally be found not to exceed the
following limits:
Concrete, 20,000 to 40,000 pounds per square foot.
Cement rubble, same values.
Hard-burned brick, cement mortar joints, 30,000 to 50,000 pounds per
square foot.
Limestone ashlar, 40,000 to 60,000 pounds per square foot.
Granite ashlar, 50,000 to 70,000 pounds per square foot.
The masonry arch is at the same time the most graceful and the most
substantial and durable of all bridge structures, and it is deservedly
coming to be more and more used in modern bridge practice. One of the
greatest railroad corporations in the United States has, for a number
of years, been substituting, wherever practicable, masonry arches for
the iron and steel structures replaced. The high degree of excellence
already developed in this country in the manufacture of the best grades
of hydraulic cement at reasonable prices, and the abundance of cut
stone, has brought this type of structure within the limits of a sound
economy where cost but a few years ago would have excluded it. It is
obviously limited in use to spans that are not very great but yet
considerably longer than any hitherto constructed.
[Illustration]
[Illustration: FIG. 40.—Elevation of Luxemburg Bridge and Sections of
Main Span.]
=130. Largest Arch Spans.=—The longest arch span yet built has been but
recently completed in Germany at the city of Luxemburg. This bridge has
a span of 275.5 feet and a rise of 101.8 feet. It is rather peculiarly
built in two parallel parts separated 19.5 feet in the clear, the space
between being spanned by slabs or beams of combined concrete and steel.
The arch-ring is 4.75 feet thick at the crown and 7.18 feet thick at a
point 53.14 feet vertically below the crown where it joins the spandrel
masonry. The roadway is about 52.5 feet wide and 144.5 feet above the
water in the Petrusse River, which it spans.
[Illustration: Cabin John Bridge, near Washington, D. C.]
The longest arch in this country is known as the Cabin John Bridge of
220 feet span and 57.5 feet rise. It is a segmental arch and is located
a short distance from the city of Washington, carrying the aqueduct for
the water-supply of that city. These lengths of span may be exceeded in
good ordinary masonry construction, but the high degree of strength and
comparative lightness which characterize the combination of steel and
concrete will enable bridges to be built in considerably greater spans
than any yet contemplated in cut-stone masonry.