Abstract

More than a century ago Milutin Milankovitch presented a remarkable formulation for the thrust-line of arches that do not sustain tension, and by taking radial cuts and a polar coordinate system he published for the first time the correct and complete solution for the theoretical minimum thickness, t, of a monolithic semicircular arch with radius R.

In this seminar we show that Milankovitch’s solution, t/R=0.1075, is not unique and that it depends on the stereotomy exercised. The adoption of vertical cuts which are associated with a cartesian coordinate system yields a neighboring thrust-line and a different, slightly higher value for the minimum thickness (t/R=0.1095) than the value computed by Milankovitch. We show that this result can be been obtained with a geometric and a variational formulation.

The Milankovitch minimum thrust-line derived with radial stereotomy and our minimum thrust-line derived with vertical stereotomy are two distinguishable, physically admissible thrust-lines which do not coincide with R. Hooke’s catenary that meets the extrados of the arch at the three extreme points. Furthermore, the seminar shows that the catenary (the “hanging chain”) is not a physically admissible minimum thrust-line of the semicircular arch although it is a neighboring line to the aforementioned physically admissible thrust-lines. The minimum thickness of a semicircular arch that is needed to accommodate the catenary curve is t/R=0.1117―a value that is even higher than the enhanced minimum thickness t/R=0.1095 computed in this work after adopting a vertical stereotomy; therefore, it works towards the safety of the arch.

As in the case of gravity loads, the value of the minimum horizontal acceleration that is needed to convert an arch into a four-hinge mechanism depends on the direction of rupture at the imminent hinge locations. Vertical ruptures yield the minimum rupture acceleration.