Limit equilibrium analysis of masonry arches

Abstract

In this paper, we revisit the limit equilibrium analysis of masonry arches. Firstly, the major contributions during the last three centuries associated with geometric and energy formulations are discussed, and subsequently, the paper explains that the problem of determining the minimum thickness of a masonry arch capable to support its own weight has multiple solutions. The infinite many neighboring solutions for the minimum thickness of a masonry arch result from the infinite many possible directions of rupturing that an arch with finite thickness may develop when becoming a mechanism. Given this infinite number of physically admissible rupturing directions, the energy approach expressed with the principle of stationary potential energy emerges as the most powerful tool to analyze masonry arches at their limit equilibrium state. The paper concludes that vertical rupturing is the most critical rupturing direction since it results to the largest value of the minimum thickness that an elliptical arch needs to support its own weight. For the common case where there is an intrados layer of voussoirs with physical joints perpendicular to the intrados, the initial rupture has to first follow the physical joint; therefore, the broken rupture pattern reported by Lamé and Clapeyron in 1823 corresponds to the larger value of the minimum allowable thickness.

Analytical and numerical analysis on the collapse modes of least-thickness circular masonry arches at decreasing friction

“…Arches of a general half-angle of embrace 0 <  <  (including for undercomplete and over-complete, horseshoe circular masonry arches) have been systematically analysed in analytical terms. Different solutions have been explicitly derived, and numerically explored, which appeared fully consistent with updated outcomes from a re-discussion by Heyman [4], and prior developments by Ochsendorf [11][12], as well as with classical earlier work by Milankovitch [13] (see Foce [14]), and several most recent attempts that meanwhile have appeared [15][16][17][18][19][20][21][22][23][24]. An earlier account on these developments was provided in SAHC10 conference paper [5]; later, a comprehensive analytical treatment with unprecedented closed-form explicit representations was provided in [6], while in [8], consistent comparisons were developed by a Discrete Element Method implementation, in the form of a Discontinuous Deformation Analysis (DDA) tool.…”

Basket-Handle Arch and Its Optimum Symmetry Generation as a Structural Element and Keeping the Aesthetic Point of View

“…It should be noted that of the infinite possible directions of the arch's fracture, the vertical is the most crucial as it results in the greatest minimum thickness value of an elliptical arch needed to support its own weight. In addition, in the case of mortars joining the elements that make up the arch perpendicular to the line of intrados, the fracture pattern is the greater value of the minimum thickness allowed [21]. According to this, for a semi arch of clear span or width 2d, height h and thickness t, exposed to its own weight, the minimum thickness obtained to maintain the equilibrium corresponds to the lowest ratio h/2d.…”

Structural Knowledge in the Homage Tower of the Alhambra Palace

“…As mentioned in the paragraph above, the horizontal force is needed and has to be greater than zero. There are several ways to discover the shape of "the true arch," or the line of thrust, inside of a designed arch: from analytical formulation Gil-Martín et al [5] to numerical approaches Alexakis H & Makris N [6,7]. Additionally, chain models may also give enough accuracy of "the true arch" shape,…”