elasticity, a deflection may be found by a simple calculation which is
an average of that which may be expected. Here again, simple theory is
better than complex, because of the ease with which it may be applied,
and because it gives results which are just as reliable.

The thirteenth point concerns the elastic theory as applied to a
reinforced concrete arch. This theory treats a reinforced concrete arch
as a spring. In order to justify its use, the arch or spring is
considered as having fixed ends. The results obtained by the intricate
methods of the elastic theory and the simple method of the equilibrium
polygon, are too nearly identical to justify the former when the arch is
taken as hinged at the ends.

The assumption of fixed ends in an arch is a most extravagant one,
because it means that the abutments must be rigid, that is, capable of
taking bending moments. Rigidity in an abutment is only effected by a
large increase in bulk, whereas strength in an arch ring is greatly
augmented by the addition of a few inches to its thickness. By the
elastic theory, the arch ring does not appear to need as much strength
as by the other method, but additional stability is needed in the
abutments in order to take the bending moments. This latter feature is
not dwelt on by the elastic theorists.

In the ordinary arch, the criterion by which the size of abutment is
gauged, is the location of the line of pressure. It is difficult and
expensive to obtain depth enough in the base of the abutment to keep
this line within the middle third, when only the thrust of the arch is
considered. If, in addition to the thrust, there is a bending moment
which, for many conditions of loading, further displaces the line of
pressure toward the critical edge, the difficulty and expense are