Snapping of hinged arches under displacement control: Strength loss and nonreciprocity

We investigate experimentally and numerically the response of hinged shallow arches subjected to a transverse midpoint displacement. We find that this simple system supports a rich set of responses, which, to date, have received relatively little attention. We observe not only the snapping of the arches to their inverted equilibrium configuration, but also an earlier dynamic transition from a symmetric to an asymmetric shape that results in a sudden strength loss. Moreover, we find that the response of plastically deformed arches is nonreciprocal with respect to the loading direction. Finally, we discover that, while elastically deformed arches always snap to the inverted stable configuration, for plastically deformed ones there is a critical rise below which the structures are monostable.

Figure 1

(a)–(b) Snapshots of our (a) plastically deformed arch and (b) elastically deformed arch before and after mounting on the end supports. (c)–(d) Forces recorded during the tests for the (c) plastically and (d) elastically deformed shallow arches subjected to Loading 1 (continuous lines) and Loading 2 (dashed lines). (e)–(f) Snapshots of the (e) plastically and (f) elastically deformed shallow arches subjected to Loading 1.

Figure 2

(a)–(b) Forces recorded when indenting an elastically deformed arch according to Loading 1 (a) at $\alpha =1$ and 15 mm/s and (b) with $\delta =0$ and 12 mm. (c)–(d) Snapshots of our elastically deformed arch when tested at (c) $\alpha =1$ mm/s and with (d) $\delta =12\mathrm{mm}$. Note that for clarity in (a) we report the force as a function of midspan position, $w\left(L/2,t\right)$.

Figure 3

(a) Comparison between the experimentally measured (black-thick lines) and numerically predicted (green and red thin lines) reaction force $|{\overline{Q}}^{\left(1\right)}|$ for our plastically (left) and elastically (right) deformed arches subjected to Loading 1. (b) Computed mode amplitudes $a_i$. (c) Comparison between the experimentally measured (thick lines) and numerically predicted (thin lines) position $\overline{w}$ for Loading 1.

Figure 4

(a)–(c) Numerically predicted evolution of ${\overline{Q}}^{\left(j\right)}$ for different values of rise when indenting at $\alpha =15$ mm/s. Parametric study by varying ${\overline{e}}^{\left(j\right)}$: plastic shallow arch under (a) Loading 1 and (b) Loading 2, (c) elastic shallow arch; (d) sudden force drop amplitude $\mathrm{\Delta }{\overline{Q}}^{\left(j\right)}$ and (e) difference in the snapping time $\mathrm{\Delta }\overline{T}={\overline{t}}_{\mathrm{snap}}^{\left(1\right)}-{\overline{t}}_{\mathrm{snap}}^{\left(2\right)}$.

Figure 5

Numerically predicted evolution of $|Q^{\left(1\right)}|$ for different indentation rates $\alpha \in$ [0.01,15] mm/sec. In these analyses we consider an elastically deformed shallow arch with rise ${\overline{e}}^{\left(1\right)}\sim$ 125. The circular markers corresponds to the quasistatic solution.

Figure 6

Arches with low rises ${\overline{e}}^{\left(1\right)}\le 9$: computed quasistatic reaction forces $|{\overline{Q}}^{\left(1\right)}|$ of (a) plastic and (b) elastic shallow arches for Loading 1 with corresponding modes amplitudes $a_i$ (c), (d) and positions $\overline{w}$ (e), (f). The red star represents when the inverted equilibrium condition ${\overline{e}}_2$ is reached by the indenter.