422. We shall next examine the structure of a type of bridge, shown in
Fig. 58.
[Illustration: FIG. 58.]
It consists of two beams, A B, 4' long, placed parallel to each other
at a distance of 3"·5, and supported at each end; they are firmly
clamped to the supports, and a roadway of short pieces is laid upon
them. At the points of trisection of the beams C, D, struts C F and D E
are clamped, their lower ends being supported by the framework: these
struts are 2' long, and there are two of them supporting each of the
beams. The tray G is attached by a chain to a stout piece of wood,
which rests upon the roadway at the centre of the bridge.
423. We shall first determine the strength of this bridge by actual
experiment, and then we shall endeavour to explain the results in
accordance with mechanical principles. We can observe the deflection of
the bridge by the cathetometer in the manner already described (Art.
362). By this means we shall ascertain whether the load has permanently
injured the elasticity of the structure (Art. 367). We begin by testing
the deflection when a load is distributed uniformly, as the weights
are disposed in the case of Fig. 62. A cross is marked upon one of the
beams, and is viewed in the cathetometer. We arrange 11 stone weights
along the bridge, and the cathetometer shows that the deflection is
only 0"·09: the elasticity of the bridge remains unaltered, for when
the weights are removed the cross on the beam returns to its original
position; hence the bridge is well able to bear this load.
424. We remove the row of weights from the bridge and suspend the tray
from the roadway. I take my place at the cathetometer to note the
deflection, while my assistant places weights H H on the tray. 1 cwt.
being the load, I see that the deflection amounts to 0"·2; with 2 cwt.
the deflection reaches 0·43"; and the bridge breaks with 238 lbs.
425. Let us endeavour to calculate the additional strength which the
struts have imparted to the bridge. By Table XXIV. we see that a rod
40" × 0"·5 × 0"·5 is broken by a load of 19 lbs.: hence the beams of
the bridge would have been broken by a load of 38 lbs. if their ends
had been free. As, however, the ends of the beams had been clamped
down, we learn from Art. 411 that a double load would be necessary. We
may, however, be confident that about 80 lbs. would have broken the
unsupported bridge. The strength is, therefore, increased threefold by
the struts, for a load of 238 lbs. was required to produce fracture.
426. We might have anticipated this result, because the points C and D
being supported by the struts may be considered as almost fixed points;
in fact, we see that C cannot descend, because the triangle A C F is
unalterable, and for a similar reason D remains fixed: the beam breaks
between C and D, and the force required must therefore be sufficient to
break a beam supported at the points C and D, whose ends are secured.
But C D is one-third of A B, and we have already seen that the strength
of a beam is inversely as its length (Art. 388); hence the force
required to break the beam when supported by the struts is three times
as large as would have been necessary to break the unsupported beam.
Thus the strength of the bridge is explained.
427. As a load of 238 lbs. applied near the centre is necessary to
break this bridge, it follows from the principle of Art. 408 that
a load of about double this amount must be placed uniformly on the
roadway before it succumbs; we can, therefore, understand how a load
of 11 stone was easily borne (Art 423) without permanent injury to the
elasticity of the structure. If we take the factor of safety as 3, we
see that a bridge of the form we have been considering may carry, as
its ordinary working load, a far greater weight than would have crushed
it if unsupported by the struts and with free ends.
428. The strength of the bridge in Fig. 58 is greater in some parts
than in others. At the points C and D a maximum load could be
borne; the weakest places on the bridge are in the middle points
of the segments A C, D C, and D B. The load applied by the tray was
principally borne at the middle of D C, but owing to the piece of wood
which sustained the chain being about 18" long, the load was to some
extent distributed.
The thrust upon the struts is not so easy to calculate accurately. That
down C F for example must be less than if the part C D were removed,
and half the load were suspended from C. The force in this case can be
determined by principles already explained (Art. 420).