Question
Does RING give an upper or a lower-bound solution? Could an upper-bound approach provide an unsafe outcome?
Answer
RING uses an upper-bound (kinematic) analysis approach. In general, upper-bound and lower-bound (equilibrium) limit analysis methods are duals of each other, giving the same result under ideal conditions. The key distinction is that, in an upper-bound approach, equilibrium is always satisfied, but yield is not necessarily checked everywhere. This means parts of the system could, in theory, be overstressed - hence, giving an upper estimate of the true collapse load.
However, for rigid block masonry arch analysis, as used in RING, this distinction becomes less significant. In such models, failure is assumed to occur through yielding at the joints, not through the masonry blocks themselves. As a result, upper- and lower-bound approaches typically yield the same collapse load, provided the analysis includes a full search over the possible mechanisms. In practice, for simple arches, this search is computationally efficient, and RING is designed to find the true limit load, within its modelling assumptions.
A simple thrust line analysis, commonly used in hand calculations and other tools, is only able to detect failure due to hinging at joints (e.g., when the thrust line touches the edge of the masonry), and this failure mode is correctly identified by both upper- and lower-bound methods.
If crushing within joints is considered (by specifying a compressive yield stress), upper- and lower-bound analyses should still match, assuming consistent assumptions are applied. Likewise, shear failure along the joints must also be checked. Note that all limit analysis methods, including RING, are affected by non-associative shear behaviour, which can slightly overestimate the collapse load. More detail on this can be found in Section 17.1.4 of the RING User Manual.
It’s also important to recognise that, while the structural behaviour of the arch is rigorously modelled using limit analysis, backfill effects, such as load spreading and passive resistance, are included in a simplified, representative manner, and do not strictly follow limit analysis theory.
Finally, regarding the risk of unsafe outcomes: while a guessed upper-bound mechanism (as in hand calculations) could underestimate the true collapse load, RING uses numerical optimisation to find a precise critical mechanism. For rigid block arches, this approach is reliable, and will typically return an accurate, and often exact, estimate of the collapse load, again, within the modelling assumptions.