O Pewnym Zastosowaniu Transformacji Laplace'a przy Wyznaczaniu Ugięcia Łuków

A differential equation of the second order, (21), is derived for the  deflection of an arch, symmetrical in relation to the vertical axis, supported and loaded arbitrarily. The considerations of the paper concern parabolic arches of small curvature and small rise to span ratio, ζ=f/l. Fig. 1. If the deflections are assumed to be small, the expression for the bending moment in any cross-section takes the form (21) and a fourth order differential equation (22) results. If, on the other hand, the deflections are assumed to be great, the expression for the bending moment takes the form (41) and finally Eq. (43) is obtained.
In both cases the Laplace transformation can be used to obtain the so-called general equations for arch deflection, (31) and (48) respectively. In these equations the load is assumed to be composed of continuously distributed forces (uniformly, for instance) and concentrated forces and moments. A numerical example is presented of a doubly hinged arch subjected to a concentrated force acting at the key. The deflection at the point of action of the force is equal to 0,46 cm if Eq. (35) derived for more rigid arches is used. The more accurate Eq. (51) leads to the maximum deflection which is equal to 0,96 cm (See [4]).