A Semi-Discrete Approach for the Numerical Simulation of Freestanding Blocks

This chapter addresses the numerical modeling of freestanding rigid blocks by means of a semi-discrete approach. The pure rocking motion of single rigid bodies can be easily studied with the differential equation of motion, which can be solved by numerical integration or by linearization. However, when we deal with sliding and jumping motion of rigid bodies, the mathematical formulation becomes quite complex. In order to overcome this complexity, a Semi-Discrete Model (SMD) is proposed for the study of rocking motion of rigid bodies, in which the rigid body is considered as a mass element supported by springs and dashpots, in the spirit of deformable contacts between rigid blocks. The SMD can detect separation and sliding of the body; however, initial base contacts do not change, keeping a relative continuity between the body and its base. Extensive numerical simulations have been carried out in order to validate the proposed approach.

FEM/DEM Approach for the Analysis of Masonry Arch Bridges

The problem of masonry arch bridges load carrying capacity is studied by means of a coupled FEM/DEM 2D approach. The numerical model relies into a triangular discretization of the domain with embedded crack elements that activate whenever the peak strength is reached. The proposed approach can be regarded as a combination between Finite Elements allowing for the reproduction of elastic strain into continuum and DEM, suitable to model frictional cohesive behavior exhibited by masonry structures even at very low levels of external loads. The aforementioned numerical approach is applied to masonry arch bridges interacting with infill. A preliminary validation of the procedure is addressed for the prediction of the masonry arches limit state behavior where the stones are supposed infinite resistant and plastic hinges can occur exclusively on mortar joints, modeled as cohesive frictional interfaces. The sensitivity of the infill role varying mechanical properties of the infill is extensively discussed.

The DDA Method

“DDA” stands for “Discontinuous Deformation Analysis”, suggesting that the displacement field of the analyzed domain shows abrupt changes on the element boundaries in the model. This chapter introduces the theoretical fundaments of DDA: mechanical characteristics of the elements together with the basic degrees of freedom, contact behavior, the equations of motion and their numerical integration with the help of Newmark's beta-method taking into account contact creation, loss and sliding with the help of an open-close iteration technique. Finally, a short overview on practical and scientific applications for masonry structures is given.

Numerical Study of Discrete Masonry Structures under Static and Dynamic Loading

Much of the world's architectural heritage consists of Unreinforced Masonry (URM) structures whose preservation is a topical subject. To prevent possible collapse of multi-block systems in hazardous conditions, a promising tool to investigate their structural response is represented by numerical modelling with the Discrete Element Method (DEM). Gothic buttresses of trapezoidal and stepped shapes are first analysed comparatively under static loading, defining the optimal configurations. Numerical results are verified against the analytical predictions of overturning and sliding resistances, based on a continuum approximation of masonry. The DEM is then successfully adopted to assess the first-order seismic behavior of arches and buttressed arches with different shapes as compared to predictions based on limit analysis. A systematic investigation on dynamic behavior failure domains and on modes of collapse of URM structures is finally performed for varying input parameters, as needed to gain more confidence on the numerical results.

Discrete Finite Element Method for Analysis of Masonry Structures

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.

Vulnerability Assessment of Damaged Classical Multidrum Columns

Multi-drum columns are articulated structures, made of several discrete bulgy stone blocks (drums) placed one on top of the other without mortar. The multi-drum column is a typical structural element of temples of the Classical, Hellenistic and earlier Roman period. Despite the lack of any lateral load resisting mechanism, these columns have survived several strong earthquakes over the centuries. The Chapter focuses on the effect of past drum dislocations on the vulnerability of classical columns and presents a performance-based framework for their seismic risk assessment. The vulnerability is numerically calculated through response estimations using detailed three-dimensional models based on the Discrete Element Method. Conditional limit-state probabilities are calculated and appropriate performance criteria are suggested. The proposed methodology is able to pinpoint cases where past damage affects the vulnerability of such structures and can serve as a valuable decision-making tool.

Numerical Modelling of Masonry Dams Using the Discrete Element Method

This work concerns the numerical modelling of masonry dams using the Discrete Element Method. It begins with a review of the history of masonry dams and their behaviour. A numerical tool based on the Discrete Element Method developed specifically for the structural assessment of masonry dams is then presented. The mechanical calculations performed by the tool are discussed in detail, together with the approach used for the modelling of passive anchors and the modules for seismic analysis and hydromechanical analysis. Structural and hydraulic analyses of a diverse set of existing masonry dams conducted using the tool are then presented. The Discrete Element Method is shown to be capable of reproducing the structural behaviour of masonry dams and identifying their likely failure mechanisms as required for structural safety evaluations.

Micro-Modeling Options for Masonry

In this chapter, a review of the available methods and their challenges to simulate the mechanical behavior of masonry structures are presented. Different micro-modeling computational options are considered and compared with regard to their ability to define the initial state of the structure, realism in simulation, computer efficiency and data availability for their application to model low bond strength masonry structures. It is highlighted that different computational approaches should lead to different results and these will depend on the adequacy of the approach used and the information available. From the results analysis it is also highlighted that a realistic analysis and assessment of existing masonry structures using numerical methods of analysis is not a straight forward task even under full knowledge of current conditions and materials.

Discrete Element Modeling of Masonry-Infilled Frames

In this chapter, the different modeling strategies for simulating the behavior of masonry infilled frames are investigated. Particular emphasis is given on the suitability of the Discrete Element Method (DEM) to accurately represent the mechanical behavior, strength and ductility of concrete and brickwork masonry infilled frames. Within DEM, masonry infill panels are represented by individual bricks and blocks separated by zero thickness interfaces representing mortar joints. The assumptions adopted, the numerical implementation and the advantages and disadvantages of modeling masonry infilled frames using the discrete element method are discussed. This ‘discontinuum' approach, an alternative to modeling masonry as a homogenized continuum, is particularly suited for studying the mechanical behavior and interaction between the individual masonry brick/blocks and their interaction with the framed structure.

On the Mechanical Behavior of Masonry

In this chapter, a review on the mechanical behaviour of masonry is presented. The aim is to establish a base of knowledge and understanding of masonry that will underpin its mechanical characteristics and will inform the decisions towards the selection of the computational tool used which are going to be described in the following chapters. Initially, a brief description of the factors that influence the mechanical response of masonry and the variation of the material properties are discussed. The review then considers the possible causes of cracking in masonry and the different failure modes that may occur during loading. Principal findings from the review are summarised at the end of the chapter.