The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches

More than a century ago, the Serbian engineer and astronomer 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. This paper shows 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. This result has been obtained in this paper 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 paper 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 paper after adopting a cartesian coordinate system; therefore, it works toward the safety of the arch.