1. Introduction
Many historic heritages in the world are dry-joint masonry structures, e.g., Hadrian’s Wall in the UK and Temple of Olympian Zeus in Greece. They were made of individual bricks or stones, which were bound together by mortar. However, it is widely found that the mortar degraded severely or lost completely over ages, and even more, it was not used at all. In civil engineering, foundation settlement is common and has detrimental effects, especially for such dry-joint historic structures due to a lack of bonding strength between individual bricks or stones. Thus, it is of theoretical and practical significance to study the behavior, especially the potential damage or failure mode, for such precious heritage masonry structures under settlement.
In Heyman [
1], leaning towers caused by uneven foundation settlement were investigated, and a rule was developed for the maximum inclination for the safety of masonry towers. Atamturktur et al. [
2] detected structural damages caused by the settlement of buttresses in Beverly Cathedral, whose arch crowns and walls separated and deformed seriously. Milani et al. [
3] performed case studies on three inclined masonry clock towers, and results showed that tilting significantly reduces the bearing capacity of masonry structures and increases seismic vulnerability. Drougkas et al. [
4] investigated the cracking of the nave wall of St. Jacob’s Church by foundation settlement. Some further research on the influence of foundation differential settlement on structural integrity can be referred to [
5,
6,
7,
8,
9,
10]. The respective authors have demonstrated that differential settlement increases the probability of structural failure, especially when there is no mortar or the strength of mortar is low, and thus these dry joint assemblies are suitable for considering a discrete modeling approach to masonry.
Experimentation is a direct way of studying the responses of masonry structures by differential settlement. Giardina et al. [
11] performed a 1/20 scaled model test on a stone masonry façade to evaluate its failure mechanism subjected to differential settlement. Portioli and Cascini [
12] investigated the collapse of rectangular masonry walls under foundation differential settlement experimentally and identified the locking failure mode. Romano and Ochsendorf [
13] studied and compared the mechanical behavior of various Gothic masonry arches under differential settlement through experiments, and the obtained results showed that pointed arches could withstand larger support displacement than circular arches.
Due to high costs in conducting physical tests, numerical simulation has become increasingly popular in recent decades. The finite element method (FEM) has been employed in evaluating the behavior of masonry structures caused by differential settlement. Alessandri et al. [
14] analyzed the cracking of a masonry façade under foundation differential settlement based on 2D homogenized nonlinear finite element model. Landolfo et al. [
15] employed similar homogenized FE model and predicted the failure modes of two-story masonry wall façades induced by differential settlement. Truong-Hong and Laefer [
16] devised FE models and investigated the influence of window shape and size, block orientation, and lintels on the failure of masonry walls caused by excavation subsidence. Malena et al. [
17] combined the piecewise rigid displacement (PRD) method and the FE approach and studied the failure of masonry arch bridge caused by pier displacement.
On the other hand, the discrete element method (DEM) is a discontinuous computational technique for analysing the responses of masonry structures. Bui et al. [
18] employed the DEM in their research and simulated the in-plane and out-of-plane behavior of dry-jointed masonry walls under support differential settlement. Baraldi et al. [
19] developed a DEM model and evaluated the nonlinear behavior of masonry panels with regular textures subjected to in-plane forces. Sarhosis et al. [
20] studied the structural behavior of a two-story colonnade under static and dynamic loads by using the commercial DEM code-UDEC and identified the major factors affecting the stability of colonnades. Foti et al. [
21] simulated the collapse of masonry cross vaults induced by support displacement by using a commercial DEM software-3DEC, and compared their results with those from tests.
Besides FEM and DEM, other numerical methods, such as limit analysis and the PRD method, have also been used to study the failure behavior of masonry structures under differential settlement. Gagliardo et al. [
22] investigated the failure mechanism of a masonry church façade under support differential settlement based on the rigid block limit analysis. Iannuzzo et al. [
23] addressed the stability of 2D masonry structures under large support displacement with the PRD method.
Lately, the combined finite-discrete element method (FDEM) is an advanced numerical approach developed by Munjiza in 1990s [
24]. In FDEM models, structures are fully discretized into number of elements, and FE formulation is incorporated within these elements, enabling accurate estimate on structural deformation and interaction forces. The details about the FDEM can be found in the literatures [
25,
26,
27]. Recently, the FDEM has been used to simulate the failure of brittle/quasi-brittle solids under static/quasi-static and dynamic loads [
28,
29,
30,
31,
32,
33]. Regarding the failure of masonry structures, Chen et al. [
34] employed the FDEM to investigate the collapse of dry-jointed masonry arches subjected to support movement, and investigated the effects of geometry and friction coefficient. Chen et al. [
35] simulated the behavior of masonry walls subjected to support differential settlement with the FDEM, considering floor load and block fracture. Pepe et al. [
36] investigated the effect of geometry, opening and region of differential settlement on the failure behavior of masonry structures with the FDEM. Smoljanović et al. [
37,
38,
39,
40] also used the FDEM to model and analyze masonry structures.
In this study, the behavior of historic masonry structures due to foundation differential settlement was simulated with the 2D FDEM program “Y” developed by Munjiza [
41]. It was designed to demonstrate some concepts explained in [
25]. In order to use the FDEM program “Y”, an input file describing the investigated structure needs to be prepared firstly. A flowchart is shown in
Figure 1. Relevant data is updated in the database after each time step, and the database is accessible for the computation of next time step.
The aim of this paper is to show the capability of FDEM on modeling historic masonry heritages subjected to differential settlement. This paper intends to provide a new and effective tool to predict the potential damage or failure mode of historic masonry heritages under various differential settlement scenarios, which is highly beneficial to protect these precious historic heritage structures against settlement risks and also to enrich the literature on FDEM masonry applications. The present study did not require Heyman’s hypothesis, which assumes zero tensile strength, infinite compressive strength, and no sliding of blocks [
42], and furthermore, the influence of block fracture was taken into account. The simulation results were compared and validated with the data from the literature, providing insights into the protection of these historic masonry structures. Layout for the rest of the paper is as follows.
Section 2 introduces the methodology, particularly the fundamentals of the FDEM.
Section 3 presents numerical examples on three types of historic masonry structures (i.e., Natività della Beata Vergine Maria church in Bondeno, Pompeii colonnade, and Spanish Deba arch bridge) subjected to foundation differential settlement. In
Section 4, the fracture behavior of masonry units, which was ignored in
Section 3, is included, and the simulation results are compared with the counterpart results in
Section 3. Finally, concluding remarks are summarized in
Section 5.