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Seismic capacity and multi-mechanism analysis for dry-stack masonry arches subjected to hinge control

  • S.I. : 10th IMC conference
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Bulletin of Earthquake Engineering Aims and scope Submit manuscript

Abstract

Masonry arches are vulnerable to seismic actions. Over the last few years, extensive research has been carried out to develop strategies and methods for their seismic assessment and strengthening. The application of constant horizontal accelerations to masonry arches is a well-known quasi-static method, which approximates dynamic loading effects and quantifies their stability, while tilting plane testing is a cheap and effective strategy for experimentation of arches made of dry-stack masonry. Also, the common strengthening techniques for masonry arches are mainly focusing on achieving full strength of the system rather than stability. Through experimentation of a dry-stack masonry arch it has been shown that the capacity of an arch can be increased, and the failure controlled by defining hinge positions through reinforcement. This paper utilizes experimentally obtained results to introduce: (1) static friction and resulting mechanisms; and (2) the post-minimum mechanism reinforcement requirements into the two-dimensional limit analysis-based kinematic collapse load calculator (KCLC) software designed for the static seismic analysis of dry-stack masonry arches. Computational results are validated against a series of experimental observations based on tilt plane tests and good agreement is obtained. Discrete element models to represent the masonry arch with different hinge configurations are also developed to establish a validation trifecta. The limiting mechanism to activate collapse of arches subjected to hinge control is investigated and insights into the optimal reinforcement to be installed in the arch are derived. It is envisaged that the current modelling approach can be used by engineers to understand stability under horizontal loads and develop strengthening criteria for masonry arches of their care.

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Acknowledgements

This research was partially supported by the Global Challenge Research Fund provided by British Academy (CI170241). We also thank our colleagues from Newcastle University who provided insight and expertise in the area of experimental testing.

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Authors and Affiliations

  1. Department of Architecture, Built Environment and Construction Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Gabriel L. Stockdale & Gabriele Milani

  2. School of Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

    Vasilis Sarhosis

Authors
  1. Gabriel L. Stockdale
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  2. Vasilis Sarhosis
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  3. Gabriele Milani
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Correspondence to Gabriel L. Stockdale.

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Appendix: Equilibrium equations

Appendix: Equilibrium equations

1.1 Notation list

[BCj]

Balance matrix for mechanism Type j

f gi

Gravitational force of element i

h a

Horizontal reaction force for hinge point a

M a

Reaction moment for slip joint a

{qj}

Constants vector for mechanism Type j

{rj}

Reaction vector for mechanism Type j

v a

Vertical reaction force at hinge point a

α a

Angle relationship between the reaction vector, block boundary line and friction angle for slip joint a (see Sects. 2.3 and 2.4)

Δx a,b

Horizontal difference between hinge points a and b

Δx CMi,b

Horizontal distance between element i’s center of mass and hinge point a

Δy a,b

Vertical difference between hinge points b and a

Δy CMi,b

Vertical difference between element i’s center of mass and hinge point b

λ a

Collapse multiplier for constant horizontal acceleration

θ t

Tilting plane rotation angle

1.2 Type I mechanism: horizontal acceleration

$$\left[ {BC_{I} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g3} } \\ 0 \\ {f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{I} } \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {h_{4} } \\ {v_{4} } \\ {\lambda_{a} } \\ \end{array} } \\ \end{array} } \right]\quad \left\{ {q_{I} } \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} } \\ {f_{g2}\Delta x_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g3} } \\ { - f_{g3}\Delta x_{3,CM3} } \\ \end{array} } \\ \end{array} } \right]$$

1.3 Type I mechanism: horizontal acceleration and gravity decomposition

$$\left[ {BC_{It} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g3} } \\ 0 \\ {f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{II} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {h_{4} } \\ {v_{4} } \\ {\lambda_{a} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]\quad \left\{ {q_{II} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}l} { - f_{g1} \sin \left( {\theta_{t} } \right)} \hfill \\ {f_{g1} \cos \left( {\theta_{t} } \right)} \hfill \\ { - f_{g1} \cos \left( {\theta_{t} } \right)\Delta x_{1,CM1} + f_{g1} \sin \left( {\theta_{t} } \right)\Delta y_{CM1,1} } \hfill \\ { - f_{g2} \sin \left( {\theta_{t} } \right)} \hfill \\ {f_{g2} \cos \left( {\theta_{t} } \right)} \hfill \\ {f_{g2} \cos \left( {\theta_{t} } \right)\Delta x_{2,CM2} + f_{g2} \sin \left( {\theta_{t} } \right)\Delta y_{2,CM2} } \hfill \\ { - f_{g3} \sin \left( {\theta_{t} } \right)} \hfill \\ {f_{g3} \cos \left( {\theta_{t} } \right)} \hfill \\ { - f_{g3} \cos \left( {\theta_{t} } \right)\Delta x_{3,CM3} + f_{g3} \sin \left( {\theta_{t} } \right)\Delta y_{3,CM3} } \hfill \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.4 Type II mechanism: horizontal acceleration

$$\left[ {BC_{II} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ { - \tan \left( {\alpha_{1} } \right)} \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 1 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g3} } \\ 0 \\ {f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{II} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {h_{4} } \\ {v_{4} } \\ {\begin{array}{*{20}c} {\lambda_{a} } \\ {M_{1} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]\quad \left\{ {q_{II} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} } \\ {f_{g2}\Delta x_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g3} } \\ {\begin{array}{*{20}c} { - f_{g3}\Delta x_{3,CM3} } \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.5 Type III mechanism: horizontal acceleration

$$\left[ {BC_{III} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\tan \left( {\alpha_{1} } \right)} \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {\tan \left( {\alpha_{4} } \right)} \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g3} } \\ 0 \\ {f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{III} } \right\} = \begin{array}{*{20}l} {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {h_{4} } \\ {v_{4} } \\ {\begin{array}{*{20}c} {\lambda_{a} } \\ {\begin{array}{*{20}c} {M_{1} } \\ {M_{4} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]\quad } \hfill \\ \end{array} \left\{ {q_{III} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} } \\ {f_{g2}\Delta x_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g3} } \\ {\begin{array}{*{20}c} { - f_{g3}\Delta x_{3,CM3} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.6 Type IV mechanism: horizontal acceleration

$$\left[ {BC_{IV} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\tan \left( {\alpha_{1} } \right)} \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {\tan \left( {\alpha_{3} } \right)} \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g3} } \\ 0 \\ {f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{IV} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {h_{4} } \\ {v_{4} } \\ {\begin{array}{*{20}c} {\lambda_{a} } \\ {\begin{array}{*{20}c} {M_{1} } \\ {M_{3} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]\quad \left\{ {r_{IV} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} } \\ {f_{g2}\Delta x_{2,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g3} } \\ {\begin{array}{*{20}c} { - f_{g3}\Delta x_{3,CM3} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.7 Type V mechanism: horizontal acceleration

$$\left[ {BC_{V} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\tan \left( {\alpha_{1} } \right)} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 1 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,2} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\Delta x_{2,3} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{V} } \right\} = \begin{array}{*{20}l} {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{3} } \\ {v_{3} } \\ \end{array} } \\ {\lambda_{a} } \\ \end{array} } \right]} \hfill \\ \end{array} \quad \left\{ {q_{V} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} } \\ {f_{g2}\Delta x_{2,CM2} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.8 Type VI mechanism: horizontal acceleration

$$\left[ {BC_{VI} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\tan \left( {\alpha_{1} } \right)} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 1 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ {\Delta x_{1,2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{2,4} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{2,4} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {f_{g2} + f_{g3} } \\ 0 \\ {f_{g2}\Delta y_{2,CM2} + f_{g3}\Delta y_{2,CM3} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{VI} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{2} } \\ {h_{4} } \\ {v_{4} } \\ \end{array} } \\ {\lambda_{a} } \\ \end{array} } \hfill \\ \end{array} } \right]\quad \left\{ {q_{VI} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} } \\ { - f_{g1}\Delta x_{1,CM1} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g2} + f_{g3} } \\ { - f_{g2}\Delta x_{2,CM2} - f_{g3}\Delta x_{2,CM3} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$

1.9 Type VII mechanism: horizontal acceleration

$$\left[ {BC_{VII} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\tan \left( {\alpha_{1} } \right)} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ 1 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ { -\Delta y_{3,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ { -\Delta x_{1,3} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 1 \\ 0 \\ {\Delta y_{3,4} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 1 \\ { -\Delta x_{3,4} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {f_{g1} + f_{g2} } \\ 0 \\ { - f_{g1}\Delta y_{CM1,1} - f_{g2}\Delta y_{CM2,1} } \\ \end{array} } \\ {\begin{array}{*{20}c} { - f_{g3} } \\ 0 \\ { - f_{g3}\Delta y_{3,CM3} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$
$$\left\{ {r_{VII} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {h_{1} } \\ {v_{1} } \\ {h_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{3} } \\ {h_{4} } \\ {v_{4} } \\ \end{array} } \\ {\lambda_{a} } \\ \end{array} } \hfill \\ \end{array} } \right]\quad \left\{ {q_{VII} } \right\} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {f_{g1} + f_{g2} } \\ { - f_{g1}\Delta x_{1,CM1} - f_{g2}\Delta x_{1,CM2} } \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {f_{g3} } \\ { - f_{g3}\Delta x_{3,CM3} } \\ \end{array} } \\ 0 \\ \end{array} } \hfill \\ \end{array} } \right]$$

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Stockdale, G.L., Sarhosis, V. & Milani, G. Seismic capacity and multi-mechanism analysis for dry-stack masonry arches subjected to hinge control. Bull Earthquake Eng 18, 673–724 (2020). https://doi.org/10.1007/s10518-019-00583-7

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