Appraisal of the overburden mass and boundary conditions on the rocking behaviour of the vertical spanning strip wall

Abstract

Unreinforced masonry structures are particularly vulnerable to seismic events. Specifically, local out-of-plane mechanisms form recurrently, posing a serious threat of collapse. Among them, a Vertical Spanning Strip Wall (VSSW) mechanism occurs when a portion of a wall is solely constrained at its top and base, experiencing out-of-plane bending across its height. During the formation of the mechanism and up to collapse, the VSSW presents a highly nonlinear dynamic behaviour governed by the presence of an overlying wall or diaphragm and a diversity of diaphragm-to-wall connections. The present work simulates the dynamics of a VSSW, accounting for the influence of the overburden mass and a variety of boundary conditions through a two-rigid-body single-degree-of-freedom model. In addition, this study examines the energy losses of the VSSW and provides closed-form expressions to estimate the angular coefficient of restitution either analytically for predictive purposes or experimentally for characterisation campaigns. Finally, a series of illustrative comparative free- and forced-rocking analyses highlight the importance of the overburden mass and boundary conditions on the dynamic stability of the VSSW.

Keywords: Boundary conditions; Coefficient of restitution; Out-of-plane collapse mechanism; Rocking dynamics; Unreinforced masonry structures; Vertically spanning strip wall.

Fig. 1

Scheme of the vertical spanning strip wall: a geometry and displaced configuration with boundary conditions for b positive (counter-clockwise) and c negative (clockwise) rotation of the lower body “1”

Fig. 2

Boundary conditions on top of the vertical spanning strip wall: a structural detail during the construction stage (reproduced from [33] with permission from Springer Nature), b schematic view of the diaphragm-wall connection (reproduced from [31] with permission from Elsevier), c schematic view of the displaced wall, and d–f view of illustrative boundary conditions that the present model replicates

Fig. 3

a Intermediate hinge height as a function of the overburden mass (or load) for various boundary (overload) conditions, and b instability angle as a function of the overburden mass (or load) for various boundary (overload) conditions, assuming a fixed hinge height $h_1/h=0.7$

Fig. 4

Impact instances scheme and associated impulses from a a counter-clockwise rotation to b impact and c clockwise rotation of the lower body

Fig. 5

Angular coefficient of restitution $e_{\theta }$ for different wall aspect ratios and overburden mass, assuming a fixed hinge height $h_1/h=0.7$ and alternative boundary conditions: a “clamped”, b “pinned”, and c asymmetric

Fig. 6

Comparison of free-rocking response between the present and the Sorrentino et al. [13] model. Vertical spanning strip wall details: $2h=6\text{[m]}$, $2b=0.5\text{[m]}$, $N=0.5W$, ${\lambda }_{\mathit{BC}}=2$, ${\lambda }_N=1$ and $m_T=0$

Fig. 7

Comparison of free-rocking response among different boundary conditions: a “clamped”, b “pinned”, c asymmetric, and different overburden scenarios: overload $N=0.5W$ (solid black line), and mass on top $m_T=0.5m_{\mathit{tot}}$ (dashed red line). Vertical spanning strip wall details: $2h=6\text{[m]}$ and $2b=0.5\text{[m]}$

Fig. 8

Extraction of the angular coefficient of restitution $e_{\theta }$ of the free-rocking response shown in Fig. 8 with the use of Eq. (17)

Fig. 9

Overturning spectra of a VSSW for different overburden mass: a–c $m_T=0.2m_{\mathit{tot}}$, d–f $m_T=0.5m_{\mathit{tot}}$ and (g–i) $m_T=1.0m_{\mathit{tot}}$, and boundary conditions: a, d, g “clamped”, b, e, c “pinned”, and c, f, i asymmetric. Vertical spanning strip wall details: $2h=6\text{[m]}$, $2b=0.5\text{[m]}$ and $N=0$

Fig. 10

Overturning spectra of a VSSW with a $2h=2\text{[m]}$, $2b=0.5\text{[m]}$, $m_T=0.5m_{\mathit{tot}}$, $N=0$, “clamped” boundary conditions under a positive-to-negative sine pulse, and b $2h=6\text{[m]}$, $2b=0.5\text{[m]}$, $m_T=0.5m_{\mathit{tot}}$, $N=0$, asymmetric boundary conditions under a negative-to-positive sine pulse

Fig. 11

Overturning fragility curves of vertical spanning strip walls with base width a, d, g 0.35, b, e, h 0.5, and c, f, i 0.7 [m], aspect ratio $h/b$ a–c 12, d-f 12, and g–i 15, for different overburden and boundary condition scenarios