Masonry structures are comprised of a finite number of distinct interacting rock blocks that have a length scale relatively comparable to the structure. Therefore, they are ideal candidates for modeling as discrete systems. This chapter covers the Discrete Finite Element Method (DFEM) developed by the author to model discontinuous media consisting of blocks of arbitrary shapes. The DFEM is based on the finite element method incorporating contact elements. The DFEM considers blocks as sub-domains and represents them as solid elements. Contact elements are used to model block interactions such as sliding or separation. In this chapter, through some illustrative examples, the applicability of the DFEM to static and dynamic analysis of masonry structures, including arch bridges, walls, slopes, and underground openings, is discussed. The DFEM provides an efficient tool for researchers and practical engineers in designing, analyzing, and studying the behavior of masonry structures under static and dynamic loadings.
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Masonry structures, such as arch bridges, are the primary engineering structures designed. Most historical bridges, temples, walls, walls with openings, building facades, arches, and towers, which are found all over the world, are masonry structures. A sample masonry arch bridge is shown in Figure 1. Because of modern civilization and land problems, there is a need to demolish some old masonry structures and replace them with modern structures. On the other hand, it is necessary to preserve some of these structures, which are historically valuable. Thus, analysis of these structures and, if required, repairing and reinforcing them against failure, is of paramount importance. Therefore, a well-defined numerical analysis method for these structures is needed.
Figure 1. Masonry Arch Bridge, (Ghari Koprisi), Tabriz, Iran
The discrete element method (DEM), originally developed by Cundall (1971), is an innovative numerical method for solving a wide spectrum of problems involving the interaction of rock, soil, and structures and is a widely recognized technology for modelling geomaterials. The DEM is applied to the simulation of problems characterized by severe discontinuities. The DEM assumes that the analyzed structure can be modelled as an assembly of rigid particles interacting among themselves. The overall behavior of the system is determined by cohesive and frictional contact laws (Cundall, 1971; Munjiza, 2004).
The DEM has been used for years in different industries (e.g., mining, civil, and nuclear waste disposal) to solve problems involving deformation, damage, fracturing, and stability of the fractured rock masses and masonry structures (among others: Baggio & Trovalusci, 1998, 1993; Cundall, 2011; Itasca Consulting Group, Inc., 2014; Giamundo et al., 2014; Lermos, 2007; Sarhosis & Sheng, 2014; Toth et al., 2009). Over the last 25 years, a number of different modeling techniques have been developed to simulate coupled hydro-mechanical problems with the DEM. These methods, used in different applications and for different modes of hydro-mechanical behavior and coupling, have been reviewed by Furtney et al. (2013). For example, the Universal Distinct Element Code (UDEC), developed by Itasca Consulting Group, Inc. (2014), is two-dimensional numerical software that simulates the quasi-static or dynamic response to loading of media containing multiple, intersecting joint structures. UDEC utilizes an explicit solution scheme that can model complex, non-linear behaviors. Models may contain a mix of rigid or deformable blocks. Deformable blocks are defined by a continuum mesh of finite-difference zones, with each zone behaving according to a prescribed stress-strain law. The relative motion of the discontinuities is also governed by force-displacement relations for movement in both the normal and shear directions. Joint models and properties can be assigned separately to individual discontinuities or sets thereof. The analysis of rock mass stimulation by fluid injection requires analytical tools, such as numerical models based on DEM, which can represent discontinuities explicitly (Damjanac et al., 2015). A similar approach for simulation of fracturing and hydraulic fracturing of rocks is based on combined finite element method (FEM) and DEM. The formulation of the method and some example applications have been published by Rougier et al. (2011, 2012), Zhao et al. (2015), and Lisjak et al. (2015).